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University
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dissertation entitled
FRICTION-INDUCED VIBRATION IN LINEAR ELASTIC MEDIA
WITH DISTRIBUTED CONTACTS
presented by
Choong—Min Jung
has been accepted towards fulï¬llment
of the requirements for _
Ph . D . degree in Mechanical Engineering
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FRICTION-INDUCED VIBRATION IN LINEAR
ELASTIC MEDIA WITH DISTRIBUTED CONTACTS
By
Choong—Min Jung
A DISSERTATION
Submitted to
Michigan State University
in partial fulï¬llment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Mechanical Engineering
1999
ABSTRACT
FRICTION-INDUCED VIBRATION IN LINEAR
ELASTIC MEDIA WITH DISTRIBUTED CONTACTS
By
Choong-Min Jung
When there is friction between two parts in contact relative motions may generate
vibrations and noise which can cause serious problems in applications. In this study
friction-induced vibrations in elastic media subjected to distributed contacts are in-
vestigated in order to understand mechanisms responsible for generations of noise
and vibrations. We investigated system stability and stick-slip oscillations to explain
friction-induced vibration in linear elastic media with distributed contacts.
A one-dimensional elastic media with ï¬xed-end boundary conditions are investi-
gated. The system is marginally stable when the coefï¬cient of friction is a constant.
Under ï¬xed-end boundary conditions distributed friction leads to a non-self-adjoint
system. A non-self-adjoint eigenvalue problem and an eigenvalue problem based on a
proper inner product are reviewed as alternative methods in handling non-self-adjoint
systems. A contradictory result between the exact and an assumed mode projection
based on the non-self-adjoint formulation is presented as a cautionary example.
Under periodic boundary conditions the one-dimensional system is destabilized
with a constant coefficient of friction. The destabilizing phenomena occur in the
form of unstable traveling waves propagating in the direction of the slider velocity.
External and internal damping play stabilizing roles in system stability. By construct-
ing a discretized lumped-parameter model, the non-symmetric eigenvalue problem is
studied. A negative-slope in friction-velocity curve destabilizes the system.
Stick-slip oscillations are analyzed with the lumped-parameter discretized model.
An algorithm for handling nonlinear stick-slip oscillations is presented. Series of
detachments over whole domains and localized small-grouped stick-slip oscillations
are observed. Effects of system parameters on stick-slip oscillations are considered
as well. Under high normal loads, the frequency of the series of detachments is
lowered and frequency of small-grouped motions is increased. Sustained stick-slip
oscillations are observed when the friction-velocity curve is discontinuous (Ms > me)
and the system is linearly unstable. With the help of ï¬nite element analysis dynamic
behaviors of one- and two-dimensional linear elastic systems are investigated.
Copyright © by
Choong—Min Jung
1999
To my God and family
ACKNOWLEDGMENTS
This research was possible with the help of many individuals, most notably my
advisor, Dr. Brian Feeny. I would like to express my sincere gratitude to him for his
patience and encouragement. Thanks also goes to the members of my dissertation
committee: Drs. Steven Shaw, Alan Haddow, and Charles MacCluer for their many
valuable suggestions to my research work.
I would also acknowledge all professors who taught me the knowledge in engi-
neering and mathematics. Professors Brian Feeny, Steven Shaw, Ronald Rosenberg,
Clark Radcliffe, Hassan Khalil, Byron Drachman, Richard Phillips, Jerry Schuur of
Michigan State University, Professors Dae-Gab Gweon of KAIST, and Chang-seop
Song of Hanyang University in Korea deserve thanks for their enthusiastic lectures.
I also appreciate the help of Dr. Farhang Pourboghrat for conversations about my
research, and Dr. David Wiggert and Mr. Nevin Leder for corrections in my dis-
sertation. A special appreciation goes to Dr. Jen Her of the Ford Motor Co., who
provided partial support and motivation to the topic of this research.
Chris Hause, Polarit Apiwattanalunggarn, Reimund Keiser, Mark Berry, Y i Wu,
Ming Liao, and Brian Olson were a great team in the Dynamics and Vibrations Lab-
oratory. I would also like to thank Pastor Jung Kee Lee, who inspired me spiritually.
vi
Special thanks to my mother, who always prays for me, and to all family mem-
bers who always encouraged me. My wife, Yoonkoum Ji, who dedicated herself in
supporting my study and taking care of my son, Jin Hong, deserves thanks for this
accomplishment.
Thank you, God.
vii
TABLE OF CONTENTS
LIST OF TABLES
LIST OF FIGURES
CHAPTER
1 INTRODUCTION
1.1
1.2
1.3
1.4
1.5
Motivation .................................
Literature Review .............................
1.2.1
1.2.2
1.2.3
1.2.4
Dynamic Instability due to Friction
Stick-Slip Oscillations induced by Friction ...........
Destabilized Waves due to Friction
ooooooooooooooo
System Properties related to Friction ..............
Proposed Research ............................
Contributions ...............................
Thesis Organization ............................
2 FRICTIONAL SLIDING IN A ONE-DIMENSIONAL MEDIUM
2.1
2.2
2.3
2.4
2.5
2.6
2.7
Introduction ................................
Equation of Motion ............................
Exact Solution ...............
The Non-Self-Adjoint Eigenvalue Problem ............. '. .
Eigenvalue Problem based on a Proper Inner Product .........
000000000000000
Nonconvergence of Galerkin’s Method ..................
Conclusion .................................
3 FRICTIONAL SLIP WAVES IN AN ELASTIC MEDIUM
3.1
3.2
Introduction ................................
One-Dimensional Elastic System .....................
3.2.1
Stability Analysis of Elastic Waves
000000000000000
3.2.2 Addition of External Damping ..................
viii
xi
xii
010in
10
13
15
17
18
19
21
21
22
25
28
35
40
48
50
50
53
53
58
3.2.3 Addition of Internal Damping .................. 61
3.3 Two-Dimensional Elastic System .................... 65
3.4 Conclusion ................................. 70
STABILITY ANALYSIS IN A LUMPED PARAMETER MODEL 72
4.1 Introduction ................................ 72
4.2 Stability Criteria ............................. 74
4.3 A Lumped-Parameter Model under Fixed Boundary Conditions . . . 76
4.3.1 A Stability Analysis of an Undamped System ......... 76
4.3.2 Addition of Damping ....................... 84
4.4 A Lumped-Parameter Model under a Periodic Boundary Condition . 90
4.4.1 A Stability Analysis of an Undamped System ......... 90
4.4.2 Addition of Damping ....................... 94
4.5 Conclusion ................................. 96
STICK—SLIP OSCILLATIONS 97
5.1 Introduction ................................ 97
5.2 Numerical Aspects of Stick-Slip Phenomena .............. 100
5.3 Stick-Slip Oscillations with Fixed Boundary Conditions ........ 104
5.3.1 Conditions of Numerical Simulations .............. 105
5.3.2 Investigations of Stick-Slip Oscillations ............. 106
5.4 Parameter Effects on Stick-Slip Oscillations .............. 113
5.4.1 Effects of Normal Loads ..................... 114
5.4.2 Effects of Driving Speed ..................... 128
5.4.3 Effects of the Poisson’s ratio ................... 135
5.4.4 Effects of Friction Characteristics ................ 142
5.4.5 Effects of Other Parameters ................... 146
5.5 Stick-Slip Oscillations: Modal Projection Method ........... 150
5.6 Stick-Slip Oscillations under Periodic Boundary Conditions ...... 158
5.7 Conclusion ................................. 161
FINITE ELEMENT ANALYSIS 163
6.1 Introduction ................................ 163
6.2 One-Dimensional System ......................... 164
6.2.1 Formulation and Algorithm for Nonlinear Finite Element Analysisl64
6.2.2 Eigenvalue Comparison ...................... 167
6.2.3 Numerical Results ........................ 171
6.3 Two-Dimensional System ......................... 181
ix
6.4 Conclusion ................................. 188
7 CONCLUSIONS AND FUTURE WORKS 189
BIBLIOGRAPHY 196
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6.1
6.2
LIST OF TABLES
The typical system parameters selected for numerical simulations in
Chapter 5 ..................................
The approximate peak-to-peak amplitudes in displacement responses
by changing the normal load. ......................
The approximate peak-to—peak amplitudes in velocity responses by
changing the normal load. ........................
The maximum peak-to-peak amplitudes in displacement response by
changing the driving speed. .......................
The maximum peak-to—peak amplitudes in velocity responses by chang-
ing the driving speed ............................
The maximum peak-to—peak amplitude in displacement responses by
changing the Poisson’s ratio ........................
The maximum peak-to—peak amplitudes in velocity responses by chang-
ing the Poisson’s ratio. ..........................
Equations of motion, boundary conditions, and projected motions on
modal coordinates for the stick-slip oscillations with fixed boundary
conditions. Here 07 = ’21, t" = tA/ZTE, 453â€"(x) = 6% sin(j11(:—;:fl),
wf‘â€($) = e‘% sin(3'11(15_:":—"l) .........................
The equation of motion, boundary condition, and projection to modal
coordinates for pure sliding oscillations under the ï¬xed boundary con-
dition. ...................................
The approximate modal frequencies by applying ï¬nite element analysis.
The numerical results including 10, 20, and 40 elements are presented
with the exact frequencies .........................
The approximate modal frequencies from the lumped-parameter
model. The numerical results including 10, 20, and 40 masses are
presented with the exact frequencies. ..................
xi
105
116
129
136
136
152
154
169
169
Figure
1.1
1.2
2.1
2.2
2.3
2.4
2.5
2.6
LIST OF FIGURES
A schematic diagram for the bushing squeaking noise problem in an
automotive suspension system. Squeaking noise is generated on the
contact surface between the rotating shaft and the rubber bushing
material ...................................
A typical spring-mass model which has been used for explanations of
friction-induced vibrations .........................
A schematic diagram for a one-dimensional elastic medium subjected
to distributed friction. Friction between a moving belt (a moving rigid
body) and an elastic medium induces vibrations and noise. A medium
is under a ï¬xed boundary condition. A frictionless linear bearing is
installed on top of a medium so as to allow axial motions of an elastic
medium ...................................
The exact static solution u3(a:) by changing 6 in the one-dimensional
system. Here ,6 is in the range of 0.1 to 1.0 with increments of 0.1. In
this example a = 4.0 ............................
The exact static solution u,(a:) by changing a in the one-dimensional
system. Here a is in the range of 1.0 to 10.0 with increments of 1.0.
In this example 6 = 1.0. .........................
The exact static strain solution %31 by changing ,6 in the one-
dimensional system. Here [3 is in the range of 0.1 to 1.0 with increments
of 0.1. In this example a = 4.0. .....................
The exact static strain solution %9 by changing a in the one-
dimensional system. Here a is in the range of 1.0 to 10.0 with in-
crements of 1.0. In this example 6 = 1.0 .................
The ï¬rst three eigenfunctions in the dynamic solution. (6%3’ sin(j7r:1:),
where j = 1, 2, 3 with a = 4.0.) .....................
xii
23
26
27
28
29
2.7
2.8
2.9
2.10
2.11
3.1
3.2
3.3
3.4
3.5
A contradictory result: Eigenvalue trajectories versus a in the one-
dimensional friction system by applying the assumed mode method
with two modes included. (a) Imaginary and (b) real parts of the
eigenvalues versus a are shown. The selected assumed modes are
(Ji,-(2:) = \/2sin(j1r:c) forj = 1,2 ......................
A contradictory result: Imaginary parts of the eigenvalues versus a
by applying the assumed mode method in the one-dimensional friction
system. (a) 3 modes, (b) 4 modes, and (c) 5 modes are included. The
selected assumed modes are (5,-(115) = flsinUms). ...........
A contradictory result: Real parts of the eigenvalues versus a by apply-
ing the assumed mode method in the one-dimensional friction system.
(a) 3 modes, (b) 4 modes, and (c) 5 modes are included. The selected
assumed modes are (133(27) = ï¬sinUvrx). ................
Comparison between the exact and approximate eigenvalues: The
square roots of the exact eigenvalues, \//\—,- = (/ (j7r)2 + “72, are shown
with the solid lines. The mode projected approximate eigenvalues ob-
tained from a self-adjoint system are shown with dotted line. The
selected assumed modes are @(m) = x/23in(j7r:1:) for j = 1, 2, 3, 4, 5.
Eigenvalues obtained from the non-self-adjoint system by projecting
the exact eigenfunctions ¢j(:r) = V2605“c sin(j7r:r:). ...........
The unstable characteristic solutions for the undamped, periodic
boundary conditioned model. (a) Imaginary and (b) real parts of the
characteristic solution versus a are shown. ...............
The imaginary parts of the characteristic solutions including the exter-
nal damping coefï¬cient d. The maximum value of the imaginary parts
is presented in the parameter domains a and d. In this example k = 1.
The real parts of the characteristic solutions including the external
damping coefficient d. The real parts of the characteristic solution
corresponding to the maximum imaginary value is presented in the
parameter domains a and d. In this example k = 1. ..........
The imaginary parts of the characteristic solutions including the ex-
ternal damping coefï¬cient d. In this example a = 1.0 ..........
The imaginary parts of the characteristic solution including the internal
damping coefï¬cient 7. The maximum value of the imaginary parts is
presented in the parameter domains a and 'y. In this example k = 1.
xiii
42
43
44
46
47
56
59
60
61
63
3.6
3.7
3.8
3.9
3.10
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
The real parts of the characteristic solution including the internal
damping coefficient 7. The real parts of the characteristic solution
corresponding to the maximum imaginary value is presented in the
parameter domains 0 and ’y. In this example I: = 1 ...........
The imaginary parts of the characteristic solutions including the inter-
nal damping coefficient 7. In this example a = 1.0. ..........
A schematic diagram for a semi-inï¬nite, two-dimensional elastic
medium in contact with a moving rigid body. .............
The imaginary parts of the characteristic solution L in the parameter
plane of friction coefficient p and Poisson’s ratio 11. ..........
The real parts of the characteristic solution L in the parameter plane
of friction coefï¬cient )1 and Poisson’s ratio 12 ...............
A schematic diagram for the undamped, lumped-parameter model sub-
64
64
69
jected to distributed friction. Fixed end boundary conditions are applied. 77
Trajectories of the eigenvalues versus friction coefï¬cient )1 in the un-
damped, lumped-parameter model. ...................
Static equilibria by increasing the friction coefï¬cient it. Here it is in-
creased from 0.0 to 0.7 by 0.07. .....................
The non-symmetric eigenvectors corresponding to the three lowest
eigenvalues. ................................
The discontinuous coefficient of friction [1 versus relative velocity IV —
:5,- |. The coefficients of friction are represented by ,u = sign (V—dci) {c1+
age-c3lv'iil}, where c1 = 0.1, 03 = 1.0, and Cg = 0.1 for the dashed line,
C2 = 0.2 for the dotted line, 02 = 0.3 for the dash dot line, and c2 = 0.4
for the solid line. .............................
The locus of eigenvalues with varying normal loads for the damped
model under a ï¬xed boundary condition. The normal load is increased
by 5.0 N. Here 7 = 0.01,d = 0,c1 = 0.1,c2 = 0.2, and C3 = 5.0 are
selected. ..................................
The detailed presentations of the eigenvalues by increasing the normal
loads .....................................
A schematic diagram for the lumped-parameter model with a periodic
boundary condition. ...........................
The locus of eigenvalues with varying u for the undamped lumped—
parameter model under a periodic boundary condition. Here n is in-
creased by 0.05. ..............................
xiv
82
83
84
88
92
4.10 The detailed presentation of the trajectories of the eigenvalues by in-
4.11
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
creasing the friction coefï¬cient. .....................
Trajectories of the eigenvalues for the damped lumped-parameter
model under a periodic boundary condition. Here '7 = 0.05,d =
0.05, CI = 0.1, 02 = 0.2, and c3 = 5.0 are selected .............
Velocity responses of stick-slip oscillations for the lumped-parameter
model. (The selected parameters are in Table 5.1.) ..........
A contour plot of the stick-slip response in velocity. (The selected
parameters are in Table 5.1.) .......................
A power spectral density of the velocity responses. (The selected pa-
rameters are in Table 5.1.) ........................
Displacement responses of stick-slip oscillations for the lumped-
parameter model. (The selected parameters are in Table 5.1.)
Strain presentations from the displacement responses. (The selected
parameters are in Table 5.1.) .......................
Sticking events versus time. The mark ‘*’ indicates “the stick stateâ€
and the others (the blanks) indicate “the slip state†for each mass.
(The selected parameters are in Table 5.1.) ...............
State-space (displacement versus velocity) presentations for several po-
sitioned masses. (The 5th, 9th, 13th, and 17th positioned masses are
shown.) The selected parameters are in Table 5.1 ............
The stick-slip displacement responses for normal loads of (a) N0 = —1
N, (b) N0 = —5 N, (c) N0 = —10 N. The other parameters are in Table
5.1 ......................................
The stick-slip velocity responses for normal loads of (a) N0 = —1 N,
108
109
110
111
112
113
118
(b) N0 = —5 N, (c) N0 =2 -—10 N. The other parameters are in Table 5.1.119
Projections of state-space trajectories (displacement versus velocity)
for several positioned masses (the 5th, 9th, 13th, and 17th positioned
masses) for normal loads of (a) N0 = —1 N, (b) N0 = —5 N, (c)
N0 = —10 N. The other parameters are in Table 5.1. .........
Power spectral density of the stick-slip velocity responses for normal
loads of (a) N0 = —1 N, (b) N0 = —5 N, (c) N0 = —10 N. The other
parameters are in Table 5.1 ........................
Stick-slip strain responses for normal load of (a) N0 = —1 N, (b)
N0 = —5 N, (c) N0 = -10 N. The other parameters are in Table
5.1 ......................................
XV
124
125
5.13
5.14
5.15
5.16
5.17
5.18
5.19
5.20
5.21
5.22
5.23
5.24
5.25
Stick-slip responses in contour plot of velocity for normal load of (a)
N0 = —-1 N, (b) N0 = —5 N, (c) N0 = —10 N. The other parameters
are in Table 5.1 ...............................
Sticking events versus times for normal loads of (a) N0 = —1 N, (b)
N0 = —5 N, (c) N0 = —10 N. The other parameters are in Table 5.1.
The stick-slip responses for driving speeds of (a) V = 0.1m/s, (b) V =
0.5m/s, (c) V = 1.0m/s. The other parameters are in Table 5.1. . . .
Sticking events for driving speeds of (a) V = 0.1m/s, (b) V = 0.5m /s,
(c) V = 1.0m/s. The other parameters are in Table 5.1 .........
Power spectral density of the stick-slip velocity responses for driving
speeds of (a) V = 0.1m/s, (b) V = 0.5m/s, (c) V = 1.0m/s. The other
parameters are in Table 5.1 ........................
Projections of state-space trajectories of the stick-slip response by
changing the driving speeds of (a) V = 0.1m/s, (b) V = 0.5m/s, (c) V
= 1.0m /s. The other parameters are in Table 5.1 ............
Sticking events versus time for Poisson’s ratios of (a) V = 0.0, (b)
V = 0.1, (c) V = 0.4. The other parameters are shown in Table 5.1. . .
The stick-slip response in strains for Poisson’s ratios of (a) V = 0.0, (b)
V = 0.1, (c) V = 0.4. The other parameters are shown in Table 5.1. . .
The stick-slip responses in contour plot of velocity for Poisson’s ratios
of (a) V = 0.0, (b) V = 0.1, (c) V = 0.4. The other parameters are
shown in Table 5.1 .............................
Power spectral density presentations of the stick-slip responses of ve-
locity for Poisson’s ratios of (a) V = 0.0, (b) V = 0.1, (c) V = 0.4. The
other parameters are shown in Table 5.1 .................
Projections of state-space trajectories for Poisson’s ratios of (a) V =
0.0, (b) V = 0.1, (c) V = 0.4. The other parameters are shown in Table
5.1 ......................................
Sticking events versus time for the following friction-speed relations:
(a) a discontinuous function, u, 2 pk = 0.1, (b) a discontinuous func-
tion, [1, = 0.3, m, = 0.1, (c) p = 01 + age-“3'3“â€, where cl = 0.3,
c2 = 0.2, and c3 = 0.1. The other conditions are in Table 5.1 ......
Projected state-space trajectories for the following friction-speed rela-
tions: (a) a discontinuous function, u, = M = 0.1, (b) a discontinuous
function, u, = 0.3, m, = 0.1, (c) u = 01 + CQC-C3li’-Vl, where c1 = 0.3,
(:2 = 0.2, and c3 = 0.1. The other conditions are in Table 5.1 ......
xvi
126
127
131
132
133
134
137
138
139
140
141
144
145
5.26
5.27
5.28
5.29
5.30
5.31
5.32
5.33
5.34
5.35
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
Stick events versus time for stiffnesses of (a) k = 1, (b) k = 2, (c)
k = 3. The other conditions are in Table 5.1 ...............
Projected state-space trajectories for stiffness values of (a) k = 1, (b)
k = 2, (c) k = 3. The other conditions are in Table 5.1 .........
A possible velocity proï¬le showing the stick-slip motions in a continu-
ous one-dimensional system: High dimensional model ..........
A possible velocity proï¬le showing the stick-slip motion in a continuous
one-dimensional system with bounded boundary condition model: Low
dimensional model. ............................
Schematic diagrams showing stick-slip oscillations. (a) a pure sliding
stage, (b) a growing sticking stage, (c) an enlarged sticking stage, and
(d) a shrinking sticking stage. ......................
Stick-slip responses in displacement by applying the modal projection
method. Here p, = 0.3, and m, = 0.1. (Displacement variations with
respect to static equilibria are shown.) .................
Sticking region versus time by applying the modal projection method.
A contour presentation of the velocity response for the model with
periodic boundary conditions. Here d = 0.01, y = 0.01,/1, = 0.3, and
m, = 0.1. The other parameters are in Table 5.1 .............
Stick events versus time for the periodic boundary condition model. .
A state space diagram for several positioned masses under a periodic
boundary condition model .........................
A schematic diagram of the system used in ï¬nite element analysis. The
model is composed of twenty elements. The top rigid body is stationary
without friction. The lower rigid body moves at 1 inch/sec to the
positive 1: direction. There is friction between the elastic material and
the lower moving body. Selected nodes are shown. ..........
The friction coefï¬cient versus relative velocity. u = sign (V —
u)pk% arctan(J-’%l), where C = 0.1 for dotted line and C = 0.01 for
solid line. .................................
The distributions of friction forces versus time at nodes on contact.
Stress distributions (011) versus time at nodes on contact. ......
Velocity response at node 19. ......................
Friction force versus velocity at nodes 11, 19, 27, and 35. .......
Friction force versus displacement at nodes 11, 19, 27, and 35.
Displacement versus velocity at node 19. ................
xvii
148
149
150
151
155
156
157
158
159
160
168
171
175
176
178
179
180
6.9 A schematic diagram for a two-dimensional elastic medium under dis-
tributed contact. The lower rigid body moves at 1 inch/sec to the
positive :2: direction. Selected nodes are shown .............. 181
6.10 Stress distributions (011) versus time for nodes at the contact. . . . . 184
6.11 Friction forces versus time at nodes on contact. ............ 185
6.12 Displacement versus velocity at node 102. ............... 186
6.13 Friction force versus displacement at nodes 146 and 190. ....... 187
xviii
CHAPTER 1
INTRODUCTION
1 .1 Motivation
It is well known that troublesome noise and oscillations occur in systems subjected
to frictional contact. Friction-induced vibrations and accompanying noise are serious
problems in many industrial applications, for example brake systems in automobiles,
wheel and rail systems in trains, water-lubricated bearing systems in ships, robot
joint systems, and machine-tool/work-piece systems in manufacturing. These various
forms of vibrations are undesirable not only because of their detrimental effects on the
performance of the mechanical systems, but also as sources of discomfort in operating
environments.
Consider the example of frictional slip and stick-slip vibrations in the stabilizer
bar of automobile suspension systems. Figure 1.1 shows a simpliï¬ed schematic model
of the elements in the squeak system. The clamped rubber bushing is represented as
the annulus. The outer surface of the annulus is ï¬xed, and there is frictional con-
/ Rutiwmmng
//%
Figure 1.1. A schematic diagram for the bushing squeaking noise problem in an
automotive suspension system. Squeaking noise is generated on the contact surface
between the rotating shaft and the rubber bushing material.
tact between the rotating shaft and the rubber bushing at the interface. Under some
circumstances, the rubber-on-steel contact between the bushing and the shaft gen-
erates an annoying, high-frequency, frictional squeaking noise. Such noise problems
motivate us to study vibrations and dynamics of the bushing system.
Previous studies which related how the friction generates such unwanted noise and
vibrations have shown that causes of friction-induced vibrations and noise depend on
numerous factors including:
0 Friction force characteristics with respect to relative sliding speed
0 Clamping torque producing normal contact pressure
0 Humidity and heat generation on the contact surface
Figure 1.2. A typical spring-mass model which has been used for explanations of
friction-induced vibrations.
0 Random roughness on the contact surface
0 Material nonlinearities
o Modal coupling effects in frictional steady sliding
In order to understand the dynamic system behavior of the elastic medium, which
results from self-excited vibrations and stick-slip oscillations, an appropriate math-
ematical model which explains the distributed friction effect is required. However,
only a limited number of studies have considered the influence of distributed friction
contact on system behavior. Most of the mathematical models in previous studies
have been based on simpliï¬ed, discretized, low-degree-of-freedom models. For a basic
example, a spring-mass on a frictional moving belt model (Figure 1.2) has been used
to explain friction-induced vibrations in previous studies. Such a model has limi-
tations, and cannot describe the dynamics of a continuum subjected to distributed
frictional contacts.
One of the particular phenomena occurring in a frictionally excited elastic media
is the vibration in the form of waves. According to research on deformable elas-
tic materials, oscillations in the form of waves are generated by distributed friction
(Schallamach [26], Martins et al. [25], and Adams [23, 24]) and some of waves can
destabilize the overall system. Therefore, an investigation on frictional waves in an
elastic medium is required to understand the generating mechanisms of the noise and
vibrations caused by distributed friction.
Furthermore, when stick-slip oscillations occur over distributed contact surfaces,
stick-slip motions can be observed at the interface. This means that within the
system domain, micro-scale stick and slip regions are observed on the contact surface
of the continuum. This produces difficulties in deï¬ning the system conï¬gurations and
leads to complicated responses in the continuous elastic medium. The experimental
investigations by De Togni et al. [99] and Vallett and Gollub [101], who dealt with
the distributed friction contacts, revealed mechanisms responsible for friction-induced
vibrations including the stick-slip responses. However, there have been few analytic
investigations regarding stick-slip oscillations in a continuum and detailed system
behaviors that lead to vibrations and noise in a continuum have not been investigated.
Possible mechanisms for generating vibration and noise are hypothesized to be lin-
ear instability and nonlinear limit cycles. For linear instability we look for eigenvalues
of a system under a steady sliding condition and identify criteria for instability by
varying system parameters. Such instability means growth in vibration and is usually
expected to lead to stick-slip limit cycle behaviors. It is possible that a linearly sta-
ble system can have a stable stick-slip limit cycle. Thus by setting initial conditions
representing the bushing system we seek the possibility of sustained stick-slip limit
cycle behaviors.
The primary concerns of this study are determining the mechanisms
that can generate noise and vibrations in distributed friction systems, and
understanding the dynamic behaviors of the system.
Emphasizing the structural stability, several issues regarding system properties
are considered in this study. System properties which are introduced by distributed
friction contact are investigated. Effects of damping and boundary conditions on
system stability are considered in this investigation.
1 .2 Literature Review
1.2.1 Dynamic Instability due to Friction
Experimental and analytical investigations for noise and vibrations induced by friction
have shown that numerous system parameters have influenced on dynamic system
instability, which may result in chattering, squeaking, squealing noise. Although
it is not easy to distinguish system parameters as independent factors influencing
system stability, categorizations based on their functions will show several primary
parameters responsible for friction-induced vibrations and noise.
Crucial parameters which dominate system stability are friction force characteris-
tics with respect to relative speed, dependency of normal loads, coordinate couplings
by friction contact, random roughness of contact surface, temperature and humidity
around the contact surface, transient or time-dependent state variables, geometric
nonlinearities, frictionally destabilized waves on elastic materials, boundary condi-
tions and so on. In this section, previous studies related to dynamic stability are
organized and presented in order to understand principal mechanisms of friction-
induced vibrations and noise.
Characteristics of friction-speed relation
One of the main parameters which affects system stability is the slope of the friction-
speed relation. Brockley et al. [65] investigated fundamental mechanisms of friction-
induced vibrations of a system composed with a spring-damper-mass on a frictional
moving belt. The results suggested a critical sliding belt speed must be exceeded in
order to attenuate oscillations induced by friction. The operating under the critical
sliding speed, which depends on damping, normal loads, system stiffness, and friction
force characteristics, limited an incidence of vibrations and reduced the amplitude of
oscillations. They emphasized that the friction-speed curve plays an important and
crucial role in an occurrence of self-excited vibrations. Experimental veriï¬cations
were also conducted by Brockley and Ko [66].
Moreover, variations of the friction characteristics also effect stability. Cocker-
ham [67] presented analyses of stick-slip and sliding stability by using a discontinu-
ous friction model which consist of different coefficients of friction in acceleration and
deceleration. Additionally, nonlinear variations in coefï¬cients of friction during oscil-
lation cycles were analyzed by Antoniou et al. [47]. Some researchers included ideas of
the discontinuous properties of static and kinetic frictional coefficients associated with
time dependency in modeling processes (Brockley [65], Gao et al. [45, 46], Tworzydlo
et al. [52]). For example, experimental studies by Gao et al. [45, 46] showed that the
rate of increase in static friction coefficient on sticking time is a crucial parameter in
addition to friction-speed effects on stick-slip and steady sliding motions.
Such friction-speed relations, sometimes with time-dependent forms, have influ-
enced the effective system damping and destabilized systems in many applications,
e.g., brake systems (Friesen [69], Abdelhamid [70], Black [71]), hearing systems
(Bhusha [73] Simpson and Ibrahim [74], Krauter [75]), and manufacturing systems
(Ulsoy [59], Palmov [31], Paslay [34], Dareing [33], Dawson [35], Belyaev [32]).
In the study by Krauter [75], unstable high-frequency vibrations, which result in
as squealing noise, were originated from the growth of unstable vibration modes. The
quantity most affected the onset of instability was the slope of friction-speed relation
and effective modal structural damping. In the research on water-lubricated compli-
ant rubber bearings (Bhusha [73] and Simpson and Ibrahim [74]), experimental and
analytical approaches by modeling of the system ascertained that instability mainly
depends on the negative slopes of friction properties. Stability analysis for machining
systems in manufacturing industries have conï¬rmed the importance of friction-speed
relations on system stability as well (band saw system (Ulsoy [59]), drilling process
(Palmov [31], Paslay [34], Dareing [33], Dawson [35], Belyaev [32]), musical instrument
analysis (Schelleng [92]), audio system (Majewski [68]), and turbine blades systems
(Pfeiffer [91]).
Detailed explanations about dynamic stability by the effects of friction-speed re-
lations have been summarized in works by Nakai and Yokoi [72] and Ibrahim [93, 94].
Coupling instability associated with normal loads
Observations of experimental phenomena for friction-induced vibrations of multi-
degree—of-freedom systems have provided another signiï¬cant mechanism responsible
for system instability: a coupling instability associated with normal loads.
Tolstoi [48] investigated experimental kinetic friction systems in the presence of
vibrations and informed that negative friction-speed slopes and frictional self-excited
vibrations are closely associated with the freedom of normal displacement of the
slider. Later, several researchers have conï¬rmed that the self-excited oscillations
were accompanied by normal displacement of sliding elements (Aronov et al. [61],
Sakamoto [49, 50], T worzydlo et al. [51, 52] and Dweib and D’Souza [55, 56]).
By using experimental works, Aronov et al. [61] showed when the normal load
reaches a critical value, which depends on the system rigidity, high frequency self-
excited vibrations are generated. These oscillations exhibited coupling between a
lateral and a normal degree of freedom. In their series of works (Aronov et al. [62,
63, 64]), stiffness couplings have signiï¬cant effects on the normal load at which a
transition takes place from mild to severe friction and wear.
Further investigations related to coupled self-excited vibrations were performed
by Dweib and D’Souza [55, 56]. They determined four different friction regions,
such as linear, nonlinear, transient, and self-excited vibration region, as the normal
load increases. The self-excited vibrations occurred under high normal loads and a
small equivalent kinetic coefï¬cient of friction. By using the linear stability theory
the conditions which caused the steady state sliding motions to become unstable
oscillations were presented.
A numerical study by Tworzydlo et al. [52] has conï¬rmed that coupling between
the rotational and normal modes was the primary mechanism responsible for self-
excited oscillations. Oscillations with high-frequency stick-slip motions produced sig-
niï¬cant reductions of the apparent kinetic coefï¬cients of friction (Sakamoto [49, 50],
Gao et al. [45, 46]).
Coordinate coupling instabilities have been found in many applications as well.
Nakai and Yokoi [72, 57] investigated the squeal mechanisms of band brakes in order
to develop effective treatments of reduction or elimination of squealing noise. They
showed that squealing noise caused by frictional forces were originated from the cou-
pling between two modes of the brake band. Experimental studies on the disc brake
squeal (Ichiba and Nagasawa [54]) and multiple modes coupling effects have been
found in several applications as well (Hulten [58], Abdelhamid [70], Sherif [60]).
Surface roughness and other environmental effects
The dependence of surface treatment and environmental conditions on system sta-
bility have been investigated by several researchers who were interested in dynamic
behaviors with tribological effects. Surface roughness allowed the interlocking be-
tween two contacted bodies and also promoted the normal vibrations of the slider in
real situations. Soom et al. [42, 43] investigated the oscillations caused by the in-
teractions of normal and frictional forces when surface roughness is considered. The
normal oscillating forces were generated due to the surface irregularities being swept
through the contact region during sliding, and destabilized the system (Soom and
10
Chen [41], Hess and Soom [40]).
Environmental conditions, such as temperature and humidity of the contact also
contributed to the system stability. Bhushan [73] investigated basic phenomena for
frictional sliding and stick—slip oscillations of the water-lubricated rubber bearing in
ships. The mechanism for noise generation was stick-slip motions of rubber at the
interface. Generated noise was closely related contact conditions, such as roughness,
temperature, and humidity. In humid conditions some dry spots deve10ped during
sliding, which would make nonuniform friction forces over the surface. This caused the
bearing to undergo stick-slip oscillations, resulting in chattering and squealing noise.
From an experimental study by Nakai and Yokoi [57] conditions for generating the
screaming noise were changed by the surface treatment and temperature of interfacial
surfaces. At high temperatures, slopes in the friction-speed curves became more
negative than at the ordinary operating temperature condition and resulted in severe
noise and vibrations. Stick-slip amplitudes by the effect of humidity showed that in
high humidity condition the stick-slip oscillations are apt to occur (Gao et al. [45, 46]).
1.2.2 Stick-Slip Oscillations induced by Friction
When elastic systems are driven by friction forces, the motions of the elastic body
may not continuous, but may be intermittent and proceed by processes of stick-slip
oscillations. The occurrences of stick-slip motions are unpredictable and system be-
haviors including chattering, squealing, squeaking noise and even chaotic phenomena
are expected.
11
The analytical approaches for stick-slip behaviors have been performed by many
researchers. The periodically forced, single—degree-of-freedom system was considered
by Den Hartog [102]. He has made the exact solutions for the systematic steady
state responses. Later, Hundal [104] studied the analytical solutions in closed form
of continuous sliding and stick-slip motions. Dynamic responses and stability of a
system having discontinuous static and kinetic coefficients of friction were investigated
by Shaw [105].
Some simple deterministic systems are chaotic when they subjected to friction
forces. Feeny [106] and Feeny and Moon [107] provided chaotic motions in a harmon-
ically forced spring-mass—damper system. They used different friction law models and
showed the system dynamics in terms of maps for non-smooth systems. The extended
analyses associated with phase space reconstructions (Feeny and Liang [108]) and
a wavelet analysis in low dimensional characteristics (Liang and Feeny [109]) were
conducted as well. Other investigations related to two-degree-of—freedom frictional
systems were found in several works (Yeh [103], Pratt and Williams [110]).
Meanwhile, for investigations of non-periodic forcing systems, a mass on a fric-
tional moving belt has been used as a typical model for explaining stick-slip motions.
(Refer to Figure 1.2 for system conï¬guration.) The analytical solutions and exper-
imental data were given by Banerjee [83] and Bo and Pavelescu [82] for influences
of kinetic friction on stick-slip motions. Additionally, the influence of friction—speed
relation in the stick-slip motions were investigated by You and Hsia [80] and Capone
et al. [81] with graphical techniques.
Nonlinear phenomena introduced by stick-slip motions were investigated by sev-
12
eral researchers. Stelter and Sextro [85] and Popp and Stelter [78] investigated the
frictional system characterizations with one— and two-degree-of-freedom systems and
provided the bifurcation behaviors of deterministic systems. Period doubling and
Hopf bifurcations were observed in parameter variations and a jump phenomenon
in amplitude of responses of frictional systems was founded. Later, Galvalnetto et
al. [86, 87] investigated the stick-slip vibration with a two-degree-of-freedom mechan-
ical system and the global dynamics was characterized by using a Poincaré map.
Dynamics in two-degree-of-freedom stick-slip oscillations were studied by Awre-
jcewicz and Delfs [89, 90]. They showed the qualitative changes in equilibria by
changing system parameters along with integration techniques for stick-slip motions
in numerical aspects. In addition, Pfeiffer [91] studied dynamics of turbine blades
as a multi-dimensional stick-slip system. Studies for self-excited and stick-slip mo-
tions have been found in several works (Popp and Stelter [78], Popp [79], Hinrichs et
al. [84]).
Most of the previous research has dealt with low-degree-of-freedom systems, which
did not include distributed friction effects. However, real systems always have areas of
contact and sometimes that could have major influences on dynamic characteristics.
The model consist of blocks of masses have been used to describe the dynamics for
multi-dimensional systems and also used for earthquake fault analysis (Carlson and
Langer [97, 98], Carlson et al. [96], Takayasu [95]). Carlson and Langer [97, 98] have
investigated global stick-slip behaviors of a multi-degree—of-freedom system. They
also provided the system slipping instability and analyzed earthquake events.
Vallette and Gollub [101] studied the stick-slip motions with spatiotemporal dy-
13
namic systems and explained the experimental behaviors of stick-slip motions in
terms of propagating waves. The instability occurred as consequences of Schallamach
waves [26] of detachment. Studies on the elastomeric friction system were found in
the works by De Togni et al. [99] and Rorrer [100].
Analysis related to the stick-slip oscillations in elastic systems, especially for elastic
continua, is difï¬cult since stick-slip motions are unpredictable and generate variable-
degree—of-freedom systems. In this study, a discretized lumped-parameter model is
established and its stability is analyzed (Chapter 4). Dynamic system behaviors
including the stick-slip motions are numerically performed with various system pa-
rameters (Chapter 5).
1.2.3 Destabilized Waves due to Friction
When an elastic continuum is subjected to speciï¬c boundary conditions, materials
have been known to generate troublesome noise and vibrations. Such noise and
vibrations are originated from unstable motions around a contact surface. In steady
frictional sliding, unstable friction-induced waves have been reported.
For semi-inï¬nite, homogeneous, isotropic, materials having free surface, waves
propagating around surface, known as Rayleigh waves, were observed in elastic ma-
terials. The Rayleigh wave, which has an exponentially decaying amplitude with the
distance from the free surface, propagates along the free surface of the elastic body
(Fung [37]). On the other hand, when two different materials bonded together, there
are waves between the bonded interfaces, called Stoneley waves [30], which are similar
14
in nature to Rayleigh waves. Barnett et al. [29] investigated a variant of the Stone-
ley wave, namely a slip wave between two anisotropic elastic half-spaces in sliding
contact. Surface waves involving interface separation and unbonded interface were
investigated by Comninou and Dundurs [28].
When elastic media were subjected to distributed friction it was reported that
entire systems were destabilized due to unstable waves. In experimental works with a
continuous system, such as rubber on a moving rigid body, the unstable wave, called
a Schallamach wave, has been observed by Schallamach [26] and Best et al. [27]. The
unstable motions caused by static instability in the vicinity of the front part of contact
prOpagates as a wave of detachment sequence by the effect of static buckling.
Recently, Martins et al. [25] and Adams [23] investigated the wave propagation
in distributed friction contact in two-dimensional systems by analytic methods. A
mathematical model for a inï¬nite compressed elastic medium was established and
unstable waves were found under a condition of a constant coefï¬cient of friction.
Adams [24] studied a tensioned beam subjected to friction and found unstable solu-
tions in sliding contact as a one-dimensional system. These works showed that under
a constant coefï¬cient of friction two sliding materials caused unstable oscillations,
which travel from front to rear in presence of friction. They found that the solutions
have properties of non-symmetry by the effect of friction and instabilities are caused
by the coupling of various degrees of freedom in the form of waves.
Togni et al. [99], Rorrer et al. [100], Vallette et al. [101] experimented on continu-
ous materials in contact. Rorrer et al. [100] experimented with elastomer on a sliding
body and revealed four different frequency regimes of sliding, such as steady state
15
sliding, low frequency self-excited motions, high frequency motions and stick-slip mo-
tions. He showed that the stick-slip motion did not require a negative slope in the
friction-velocity curve. Vallette et al. [101] also investigated the unstable phenomena
using the stretched latex membranes in contact with a translating rod. He claimed
that the instability can occur even without a negative slope in friction-speed relation
as a consequence of wave of detachment, known as Schallamach waves [26].
In this thesis, friction-induced waves in an elastic medium are investigated in
Chapter 3. Instability mechanisms in the presence of friction are presented in terms
of the traveling wave mechanics.
1.2.4 System Properties related to Friction
Elastic systems subjected to nonconservative forces, such as friction forces or follower-
type traction forces, become unstable either statically or dynamically (Ziegler [17],
Beda [14]). Those instabilities also can be found in the area of aero-elasticity systems
(Dowell [8], Higuchi [9, 10]), friction involved systems [25], and some speciï¬c bound-
ary conditioned systems (Meirovitch and Hagerdorn [2], Meirovitch and Kwak[1]).
Unlike conservative systems, nonconservative systems can have dynamic instability
called flutter instability. The flutter destabilizing phenomena were investigated by
Herrmann and Bungay [11] and Herrmann and Jong [12]. Plaut [15] and Seyranian
and Pedersen [13] showed theoretical investigations about nonconservative instabil-
ities including system properties. Plaut [15] and Beta [14] formulated the material
stability conditions and classiï¬ed the generic loss of stability scenarios in dynamic
16
systems as fundamental researches.
By dealing such system stability which typically described by partial differential
equations (PDEs), evaluations of such system stability have primarily depends on
system eigenvalues, which can be evaluated after the model reduction. Due to inï¬-
nite dimensionality of partial differential equations, continuous systems are generally
difï¬cult to analyze. Moreover the system subjected to speciï¬c boundary conditions
sometimes does not admit closed form solutions. These difliculties can be avoided
when the system is approximated by eliminating the spatial dependence through
discretization in space. There are two major classes of approximated discretization
procedures—one based on expansion of the solution in ï¬nite series of given func-
tions, and the other is consisting of simply lumping the system properties. Galerkin’s
method is the most appealing and reduces a continuous system to n—degree-of-freedom
system by assuming the solution with series of assumed functions.
However in applying the approximate method (Galerkin’s method), careful at-
tentions should be needed in handling eigenvalue problems. Based on the previ-
ous studies (Prasad and Herrmann [4], Meirovitch [2]), the Galerkin’s approximate
method does not provide an estimated magnitude of the error involved, nor does it,
in general, guarantee convergence for non-self-adjoint systems. That statement about
non-convergence has been proved by Bolotin’s works [16]. Bolotin [16] investigated
the membrane exposed to a flow in research of aero-elasticity and showed that the
non-convergence of Galerkin’s method to that particular system. He showed that the
example of “flutter paradox†in the membrane flow and gave the range of application
for its method in order to reduce its dimension.
17
Diprima and Sani [6] studied for the convergence of the Galerkin’s method for the
beam subjected to the non-conservative forces and Prasad and Herrman [4] and Peder-
sen and Seyranian [5] investigated the general non-self-adjoint problem. The proof of
convergence for the non-self-adjoint system can be found in few simple problems (Kan-
torovich [7], Diprima [6]). Several recent studies have investigated the convergence
of non-self-adjoint systems by using modiï¬ed candidate functions (lV'leirovitch [1, 2],
Hagedorn [3]).
1.3 Proposed Research
The goal of this study is to investigate the dynamics of friction-induced vi-
brations in a continuous elastic medium subjected to distributed frictional
contact.
In order to understand dynamic stability and system behaviors by effects of fric-
tion, a mathematical model for a continuous elastic medium subjected to distributed
friction is established. The non-self adjointedness, which is the intrinsic property
introduced by friction, is shown in this study and the eigenvalue problem associated
with the non-symmetric property is investigated. In order to show the feasibility for
applying approximate discretization methods the exact and approximate eigenvalues
are compared.
Wave dynamics involving friction effect in one- and two-dimensional continuous
elastic medium are shown by imposing periodic boundary conditions. Stability anal-
yses including external, internal, and frictional damping, are performed with the
18
lumped-parameter model.
Stick-slip oscillations dependent on spatial and temporal motions are analyzed
numerically. Visual presentations of stick-slip oscillations of the elastic system are
provided and mechanisms related to generating noise are explained with various sys-
tem parameters. The numerical results of stick-slip oscillations are veriï¬ed by ï¬nite
element analysis.
1.4 Contributions
The chief contribution of this study is the dynamic analysis of an elastic medium
which are subjected to distributed frictional contact. Summaries of contributions are
as follows.
0 The construction of a mathematical model of continuous system with driving
friction can extend the scope of research from discretized systems to continuous
systems. Most of the previous work has focused on discrete, low-degree—of-
freedom systems. With the aid of modeling work, a mathematical description
of continuous elastic system subjected to friction can be established.
0 Using the established model, parameter effects on system stability can be ana-
lyzed in order to show the mechanisms how friction generates noise and vibra-
tion.
0 The validity of a discretization method—Galerkin’s method—is examined. A
non-convergence of Galerkin’s projection in calculating eigenvalues in this study
19
provides a cautionary example on the blind application of projection method.
0 It has been known that the system stability closely depends on its boundary
conditions. The wave dynamics for one- and two-dimensional periodic boundary
condition models are provided to show the possible causes of unstable waves in
presence of friction.
o Stick-slip motions of the space and time dependent system explain how the
distributed friction generates noise and vibration in an elastic medium. Visual-
izations of stick-slip motions in high-dimension are shown.
0 Veriï¬cation by ï¬nite element analysis assures the validity of this study.
1 .5 Thesis Organization
The remainder of this dissertation is organized as follows. Chapter 2 covers funda-
mental topics basic to the thesis. A mathematical model of a one-dimensional elastic
medium subjected to distributed frictional contact is derived. The exact solution
of the partial differential equation is obtained. The eigenvalue problem is non-self-
adjoint, and the self-adjoint transformation method is given as an alternative. An ap-
proximate discretization method is applied to show the validity of Galerkin’s method
to this problem.
In Chapter 3, mathematical models of frictional slip waves in one- and two-
dimensional systems are provided with periodic boundary Conditions. System sta-
bility, including the effects of general damping, is determined to explain the existence
20
of unstable traveling slip waves in elastic systems.
In Chapter 4, a lumped-parameter model is established and its pure-sliding fric~
tional stability is obtained. By including general damping effects and nonlinear fric-
tion characteristics the system instability which initiates self-excited motion is eval-
uated.
In Chapter 5, using the model developed in the previous chapters, stick-slip vibra-
tion is simulated and then interpreted in terms of mechanisms of noise and vibration.
The numerical algorithms which deal with state-dependent boundary conditions are
explained and the visual presentations of stick-slip vibrations are shown. The trends
of behavior due to changing parameters are considered.
In Chapter 6, the stick-slip vibrations are simulated by using ï¬nite element anal-
ysis. The numerical algorithms used in the ï¬nite element analysis are given and the
comparisons between the approximate and exact solution are made. Numerical re-
sults for one- and two-dimensional elastic systems with parameter effects are provided
as well.
In Chapter 7, the conclusion and summaries of the research conducted, lessons
learned and directions for the future works are presented.
CHAPTER 2
FRICTIONAL SLIDING IN A
ONE-DIMENSIONAL MEDIUM
2. 1 Introduction
In most of the previous research related to friction-induced vibrations, low-degree-
of-freedom, discretized models have been used in order to explain dynamic stability
of frictional sliding and stick-slip vibrations. For example, a simple model composed
a spring-damper-mass on a moving rigid body has usually been used. Despite its
simplicity in modeling and analysis, such a system may have limitations in showing
characteristic features of an elastic medium subject to distributed friction. Especially,
in order to investigate a continuum in contact with a large area, such as in the
suspension bushing that motivates this study, a proper continuous model which can
capture dynamic features is required. In order to understand dynamic behaviors
of a continuous system a one-dimensional continuous system under ï¬xed boundary
21
22
conditions is investigated in this chapter.
This chapter is organized as follows. A mathematical model for a one-dimensional
elastic material subjected to distributed friction contact is established. Because of
the friction the boundary value problem is non-self-adjoint. The system properties
related to non-self-adjointness are presented, and the general eigenvalue problem,
which covers the non-self-adjoint eigenvalue problem, is explained. Using a proper
inner product, the transformation from a non-self-adjoint problem to a self-adjoint
problem is shown in this chapter. A cautionary example in applying an approximate
discretization method for ï¬nding the system eigenvalue is presented as well. The
effect of distributed friction on the system stability is explained based on the system
eigenvalues.
2.2 Equation of Motion
Consider a system shown in Figure 2.1. A linear elastic medium, placed between
a moving belt (a moving rigid body) and a frictionless linear bearing, represents a
one-dimensional, undamped, continuous system in distributed sliding contact. The
friction coeflicient is considered as a constant. Although a non-linear coeflicient of
friction has been known to be one of the crucial factors for system stability, the fric-
tion coefï¬cient is assumed to be a constant with respect to relative speed. (Non-linear
friction coefï¬cient effects on dynamic stability are mainly discussed in Chapter 4.)
In addition, any parameters having random properties, such as roughness of con-
tact surface, are not included in this development in order to emphasize on dynamic
23
Bearing
Elastic media 777 1 .
in: as“) â€W 1,1 /
_.—>
Figure 2.1. A schematic diagram for a one-dimensional elastic medium subjected to
distributed friction. Friction between a moving belt (a moving rigid body) and an
elastic medium induces vibrations and noise. A medium is under a ï¬xed boundary
condition. A frictionless linear bearing is installed on top of a medium so as to allow
axial motions of an elastic medium.
stability by effects of uniform properties of materials. Moreover, any non-uniform
motions, such as stick-slip motion or loss of contacts are not included in this devel-
opment. (Demonstrations of stick-slip motions by using a lumped-parameter model
are presented in Chapter 5.)
A system composed of a linear elastic medium undergoes a axial sliding. An
equation of axial motion for undamped elastic medium is
603(x, t) 6211
— = — 2.1
A(x) 62: + f(x, t) 12%,, ( )
where A(:r) is a cross sectional area of elastic medium, p is a mass density of elastic
material, o,(1:, t) is a stress over the cross section, u(a:, t) is an axial displacement, and
f (2:, t) is a friction force per unit length. Applying linear stress-strain relation, stress
24
is expressed as 03(x, t) = E 63(1), t), where E is Young’s modulus of the material.
The friction force including Poisson’s ratio effect per unit length is given by
f(.’13,t) : —/10'y(.’13,t) : _/-‘{00 + VOICE, 15)}, (2'2)
where p is a friction coefï¬cient, oy(a:,t) is a contact normal stress, and 00 is a pre-
loaded normal stress per unit length, which should be always less than zero (com-
pression) to generate friction force and maintain contact to the sliding rigid body.
By considering the linear strain-displacement relation, 61(x,t) = 9%?2, a non-
dimensional equation of motion is obtained by
0221 Bu 6211.
(9.70"2 813* g at“2 ( )
The dimensionless parameters used in equation (2.3) are a = â€7:1, fl = —%°é, cc“ =
%, and t* = t , where l denotes contact length and 17* and t* are the dimensionless
£13
.48
coordinate and time, respectively. For the sake of simplicity, the notation * will be
neglected in the following development.
The boundary conditions are
u(0, t) = u(1, t) = 0. (2.4)
For a typical system subjected to distributed friction contact a ï¬xed boundary condi-
tion is selected. Stability analyses with a different boundary condition, e. g., a periodic
boundary condition, are analyzed in Chapter 3.
25
2.3 Exact Solution
The exact solution for the equation (2.3) satisfying the boundary condition (2.4) is
obtained by using the separation of variables method. Consequently, the exact solu-
tion, u(:r, t), which is composed with the static solution u,(:1:) and dynamic solution
ud(:1:, t), is
u(:1:, t) = u,(a:) + ud(:r, t), (2.5)
in which
u,(:z:) — —(—é—5,—fl—:T)(ealf — 1) + 513:, (2.6)
and
ud(x, t) = : V267†sin(j7r:c) {aj COS(Cth) + bj sin(wjt)}, (2.7)
where natural frequencies of wj = (jvr)2 + “72, and aj, bj are constants determined
by initial conditions.
The exact static solution in equation (2.6) by changing fl and a are shown in
Figure 2.2 and Figure 2.3, respectively. Figure 2.2 depicts the variation in the static
solution u,(a:) for 6 in the range of 0.1 to 1.0 with increments of 0.1 under a condition
of a = 4.0. As 6 increases, i.e., as normal loads and contact length increase, or
Young’s modulus decreases, the non-symmetric static solution along the .1: axis gets
26
displacement us(x)
.0 .0 P
N 00 -h
.0
A
O 0.2 0.4 0.6 0.8 1
position x
Figure 2.2. The exact static solution 218(3) by changing B in the one-dimensional
system. Here 6 is in the range of 0.1 to 1.0 with increments of 0.1. In this example
a = 4.0.
larger. Figure 2.3 provides the trends of static solutions under variations in a from
1.0 to 10.0 with increments 1.0 with ,6 = 1.0. oz influences the asymmetry of u,(:c).
Static strain distributions, deï¬ned by diff), under changes 6 and a are shown in
Figure 2.4 and Figure 2.5, respectively. The static strains are increased by increasing
6 and a. High tensile regions are observed at the front, while high compressive regions
are located at the rear on the :1: axis. Moreover, a sensitivity on the variation of the
parameters is clearly shown. In the compressive region, i.e., where %3 < 0, stress
variations for varying 6 are larger than those of the tensile region because of the
existence of Poisson’s ratio.
Importances of the strain and stress in elastic material have been found by Schal-
lamach [26] and Krauter [75] through their experimental and theoretical studies. The
27
0.7 .
a = 10.0
0.6 ~
?0.5 " .4
‘1» increasing a
3
E 0.4
o
E
8 .
g 0.3
.3 a=
‘3 0.2 - ]
a = 1.0
0.1 L
O r 1 1 n
0 0.2 0.4 0.6 0.8 1
position x
Figure 2.3. The exact static solution u,(z) by changing a in the one-dimensional
system. Here a is in the range of 1.0 to 10.0 with increments of 1.0. In this example
6 = 1.0.
expectation of buckling by the effects of tangential stress has been found to be the
source of wave propagations in elastic materials. Detailed works related to stick-slip
motion are presented in Chapter 5.
The ï¬rst three exact modes shapes, which depend on parameter a in equa-
tion (2.7), are shown in Figure 2.6. Increasing oz influences the shapes of the free-
vibration unsymmetric eigenfunctions. However, it does not destabilize system. In
other words, a determines the modes shapes, which are non-symmetric along the :z:
axis, and a affects the natural frequencies in equation (2.7). This is a conservative
system when or is a constant. It should be noted variations of a do not destabilize
the dynamic system under ï¬xed boundary conditions with a constant coefï¬cient of
friction.
28
1
0.5
p = 0.1
.5 -0 5 \
s†-1 -
3
,5 -1 5 B = 1 O
S
175 __2 ..
increasing [3
-2.5 ~
-3 -
‘3'50 0.2 0.4 0.6 0.8 1
position x
Figure 2.4. The exact static strain solution 513$?) by changing ,6 in the one-dimensional
system. Here 6 is in the range of 0.1 to 1.0 with increments of 0.1. In this example
oz = 4.0.
2.4 The Non-Self-Adjoint Eigenvalue Problem
Numerous systems encountered in structural dynamics are belonging to distinct
eigenvalues and self—adjoint. This means that such systems have symmetric prop-
erties. When a system is self-adjoint eigenvalues and eigenfunctions are real quan-
tities. Moreover, the eigenfunctions are orthogonal to each other. However, struc-
tural systems which endure aerodynamic forces, friction forces, and follower forces
have been reported to lose their symmetries and have non-self-adjoint properties
(Meirovitch [18], Martins et al. [25], Dowell [8], Higuchi [9, 10]). The orthogonal
relations and the expansion theorem which have been deve10ped on the bases of self-
adjoint properties are no longer applicable to the non-self-adjoint systems.
29
2" a=1QO
a = 1.0 \‘\\
fl —2 ~ \.
3
.30)
.s '4 "
t! . .
1;; mcreasmg a
-6 -
_8 -
'1 0o 0.2 0.4 0.6 0.8 1
position x
Figure 2.5. The exact static strain solution d—ï¬Ã©ï¬‚ by changing a in the one-
dimensional system. Here a is in the range of 1.0 to 10.0 with increments of 1.0.
In this example 6 = 1.0.
Although non-self-adjoint systems can be transformed to self-adjoint systems
by deï¬ning a proper inner product, the problem of non-self-adjointedness can also
be handled through similar procedures of the self-adjoint cases (MacCluer [20],
Hochstadt [21]). (The techniques associated with transformations of system prop-
erties are presented in the next section.)
Let us review the general eigenvalue problem, which includes non-self adjoint
problems. Suppose that a solution u(a:, t) is represented as equation (2.5). Then the
term 06 in equation (2.3) is eliminated by static solution u,(:1:) in equation (2.6), and
a dynamic equation of motion in terms of ud(:1:, t) is obtained as
8221,, Bud _ 0271,;
3.32 “.97- 8t2' (28)
30
the first mode
$
N
l
N
the third mode
dimensionless displacement
o
_4 _
the second mode
'60 0.2 0.4 0.6 0.8 1
position x
Figure 2.6. The ï¬rst three eigenfunctions in the dynamic solution. (8%“? sin(j7r:z:),
where j = 1, 2, 3 with a = 4.0.)
Let the dynamic solution of equation (2.8) can be represented in the form,
ud(x, t) = <I>(:c)Q(t). Then the eigenvalue problem is given by
82(1) 8(1)
__ _ = q) .
8:132 ads: A , (2 9)
with the boundary conditions of
<I>(0) = <I>(1) = 0. (2.10)
The eigenvalue problem represented with system operator L is
L<I> = A<I>, (2.11)
31
where the linear operator in equation (2.11) is deï¬ned by
def (12 d
= —— —. 2.12
L (12:2 + ads: ( )
We introduce the classical deï¬nition of an inner product of
< f,g >qg/f(:r)g(:r) dz. (2.13)
Then, the operator L has always an adjoint operator L" deï¬ned by
< \II,L<I> > = < L*\II,<I> >. (2.14)
And the original system and its adjoint system can be written as
Mr = )‘id’ia (2-15)
ij = A3215, (2.16)
where A, and A; are real or complex eigenvalues corresponding to L and L", respec-
tively. The operator L" is called as the adjoint operator of L and the set of eigenfunc-
tions 1b]- (j = 1, 2, . . .) is said to be adjoint to the set of eigenfunction o,- (z’ = 1, 2, . . .)
over the deï¬ned classical inner product (2.13).
A large class of structural dynamic systems with conservative forces are self-
adjoz'nt, which means that the two operators L and L“ are identical, L = L*, and
the two sets of eigenfunctions are the same for the corresponding eigenvalues. In such
32
case orthogonality is expressed as
<¢,,¢,>=/D¢,¢,dx=o, we)" i,j=1,2,...oo. (2.17)
By using the orthogonality, coefï¬cients of any function w(:r, t) = $11 ¢j(:r)qj(t) can
be written as
oo
(Ii =< (131310 >=< 451, Z <15ij >- (2-18)
i=1
This is called as the expansion theorem for self-adjoint system.
However, if the linear operator L is not the same to the adjoint operator L",
L 79 L“, the system is non-self-adjoint, and the orthogonality in equation (2.17) does
not hold. For the case in which L 79 Lâ€, multiplying equation (2.15) by 1,0,, and
equation (2.16) by (15,-, and then integrating over the interval D yields
<¢,,L¢,> = [Dz/)chp,dx=A, f0 quï¬idm, (2.19)
< ¢i,L*7,bj > = / ¢iL*'¢‘jd$ ‘2 A3/ (25,2/2jdx.
D D
Subtracting equations (2.19) leads to
(A, — A;) [D (bitbjda: = 0. (2.20)
33
Hence, if A,- 7é A;
<¢,,w,->=/D¢,w,dx=o, iaéj i,j=1,2,...oo. (2.21)
This is the bz’orthogonalz'ty of eigenfunctions ¢,- and 1b], which means an eigenfunction
of L corresponding to an eigenvalue A,- is orthogonal to an eigenfunction of L* cor-
responding to A}, where the A,- is distinct from A}. The non-self-adjoint operator L
has the same eigenvalues as the operator L". The general expansion theorem related
to non-self-adjoint systems, called the dual-expansion theorem, is presented in the
works by Meirovitch [18] and MacCluer [20].
Let us return to the problem of interest. In order to seek the adjoint operator L"
of this study, we examine the adjoint operator L"' deï¬ned in the equation (2.14):
l\IIL<I>d 11: d2 d (M
[0 SE — A (—‘(E§+ag;) (I) (2.22)
dd) , 1 d<I> d\II
_ d<I> , dip,
_ \1:( + <I>)|0 (Dds (0 /a<I>—dz—/ cï¬-dx
ll
9
“.1
e
9..
“H
where the boundary conditions of equation (2.4) have been accounted for. Thus the
adjoint operator of this study is
“at :11_ _d_
L — (13:2 adx (2.23)
34
with zero boundary conditions. Note that the adjoint operator L* in equation (2.23)
is not identical to the operator L in equation (2.12). Assuming that dynamic solutions
of this study can be represented by
umn=icmtm (us
then, by multiplying adjoint eigenfunction 1b,, and using biorthogonality in equation
(2.21), coefï¬cients qj(t) are obtained as
qJ-(t) =< ((15,116: >=< 1,1213% ¢j($) qj(t) > . (2.25)
1:1
Thus the biorthonormal relations of the eigenfunctions are
1
/0 $101?) ¢j(33)d$ = (513‘, (2.26)
where
1,0,(113) = V262†sin(z'7r:1:), (2.27)
(VJ-(:13) = \/26%xsin(j7r:r), i,j=1,2,...,oo.
By multiplying the normalized adjoint eigenfunction, win), with equation (2.9)
and integrating from 0 to 1, an inï¬nite set of decoupled ordinary differential equations
is obtained by
35
Z mï¬ijj + Z 161'ij = 0, Z = I, 2, . . . , 00, (2.28)
i=1 1:1
where
1
mij = < 143% >=/0 ¢i¢jd$ = 513', (2-29)
1
[9,-j = < ¢i,L¢j >2] 1911102161113 = (V1265): = (j77)2 + 02/4, i,j = 1,2,. . . , OO.
0
Consequently, the projection by using the adjoint eigenfunctions in the non-self-
adjoint system yields the set of decoupled ordinary differential equations. In addition,
it is veriï¬ed that eigenvalues derived from general eigenvalue problems are the same
as the exact solutions derived in the previous section.
2.5 Eigenvalue Problem based on a PrOper Inner
Product
The eigenfunctions derived in the previous section are not mutually orthogonal since
the system has a non-self-adjoint operator. However, it is folklore that such a non-self-
adjoint problem can be cast as self-adjoint by using a proper inner product (MacCluer
[20]). In this section, the method for choosing a proper inner product which enables
the system to be self-adjoint is reviewed. Then this method is applied to the problem
of interest in order to suggest an alternative way in solving the general eigenvalue
36
problem.
The general second order partial differential equation in the form of
Po($)—x“ + meg—Z + mm + mm = o, (2.30)
with the auxiliary homogeneous boundary conditions
Cit/($0)
dx
619050)
(1:1:
(ii/(331)
dz
C131(331)
d2:
a0y(:r0) + a1 + 023/(151) + a3 = 0, (2.31)
boy($0) + bi + (bl/(371) + b3 = 0
is deï¬ned on the interval (170,231). This is the Sturm-Liouville problem subject to
homogeneous boundary conditions (Hochstadt [21], Powers [22]). Suppose that the
coefï¬cients p0(:z:) and p3($) are positive and the p0(:z:),p1(:r), and 123(1) are twice
differentiable. Let
p<a= >=efroii‘5d“ q<x>=p———2(â€)p(†————) . (2.32)
and multiply equation (2.30) by weighting function £55. Then
die (23—) :x}+{q< )+Ag( my» =0, (2.33)
which is a more convenient self-adjoint form. Thus by multiplying equation (2.30) by
the weight function £3), the system is shown to be self-adjoint.
Consider the problem of interest in equation (2.9) again. According to the self-
37
adjoint transform in equation (2.33), the equation of motion (2.9) can be transformed
to self-adjoint system by using the weight function e‘“.
Thus the eigenvalue problem in self-adjoint form is given by
d —aqu) _ —023
—%{e d$}—A{e }<1>. (2.34)
The eigenvalue problem represented with self-adjoint operator L and a weight function
w(z) = eԠis
L<I> = Aw(z)<I>, (2.35)
where the self-adjoint linear operator L is deï¬ned by
~ def d e_a$ d
— ‘E E}' (2.36)
The self-adjointness of operator L is veriï¬ed by taking the classical inner product
(2.13) and integrating by parts, such that
~ 1 d d<I>
\II,L<I> = _ f _ —aa=_ .
< > 0 ‘Ild${e dz )dz (2 37)
_ _a,.d<I> , 1 _a$d<D dill
‘ 6 dz†'0 / dz d
— [1 e-azd—Qd—Wdz
— 0 dz dz
38
In addition, the positive deï¬niteness also can be shown from the fact that
- 1 d d<1>
= _ _ -ax_ 2.
<<I>,L<I>> [(de{e dx}dz ( 38)
â€(1(1) —az
= “ ‘d—q’lo‘l’ +/Ole {£3— :leCC
=/: rag—‘0? —:-)x}2d >0
is always nonnegative. It is equal to zero only if <I>(z) is a constant throughout the
domain. Because of the boundary condition (2.4), however, this constant must be
zero, which would imply a trivial solution. It follows that the operator L in equa-
tion (2.36) is positive deï¬nite. Therefore, the non-self-adjoint operator L described
in equation (2.12) is transformed to the self-adjoint positive deï¬nite operator L in
equation (2.36) by taking the weight function 6““.
Identical results are also obtained by taking the weighted inner product which
deï¬ned as
< M >wdél / f<x>g<x>w<x> dz. (2.39)
where w(z) is weight function. By choosing a weight function w(z) : 6“â€, we can
verify the self-adjointedness with respect to the weighted inner product as
< <I>,L\II >w=< \II, L<I> >,,,, (2.40)
where the operator L is deï¬ned in (2.12).
39
The equation of motion in equation (2.34) is identical to the equation of axial free
motion for an elastic rod having varying stiffness 6‘0“†and varying mass distribution
e‘a“ without friction.
The discretized equation of motion can be presented by taking the Lagrange for-
mula. Suppose that the solution ud(z, t) can be written as a series:
= f: (It-(2:) not). (241)
where ([5]-(z) can be any admissible function without loss of generosity. The kinetic
and potential energies of a continuous system have integral expressions. The kinetic
energy can be written in the familiar form of
6 t
T(t)- 2 —/0 e-a${———— “‘4‘†———)-}2dz. (2.42)
In the similar expression, the potential energy can be written as
_2[)1 e—az{_a—aud(: t) }2d£L‘ (2.43)
The natural boundary conditions are of no concern here because they are automati-
cally accounted for in the kinetic and potential energies. Consider Lagrange’s equa-
tions for conservative systems, namely,
dBT 8T6V
d—t-(a—-Tj)— 67j+6—_jr=0’ ]=1,2,...,OO. (2.44)
40
The equation of motion in discretized form is obtained by
Z Trig—(122m + Z kijTjU) = 0, (2.45)
I—l dt2 ':I
J— .7
where
1
m..- = [0 e-“:¢.(:c)¢j(x)dx. (2.46)
_ 1 —azd¢i(x) d¢3($) - . __
k2] — [06 dz dz dz, z,]—1,2,...,oo.
By selecting the set of (VJ-(z) as normalized eigenfunction in equation (2.46),
i.e., (Ji,-(z) 2 \/2e%"’ sin(j7rz) from the results of the previous section, the dis-
cretized equations of motion are obtained. The eigenvalues for this system, which
are Aj = (j7r)2 + 0‘72, are identical to the exact solution (2.7). Thus, it is veriï¬ed that
the system having a form of non-orthogonality in its eigenfunctions is a minor matter,
and it is correctable by projecting under the proper inner product.
2.6 Nonconvergence of Galerkin’s Method
The exact solution from the previous section shows that this system’s dynamic stabil-
ity is not dependent on the system parameters. The system is neutrally stable, behav-
ing like an undamped vibration system with natural frequencies of wj = (j7r)2 + 943.
The natural frequencies depend on the parameter, oz, but stability does not depend
on it. The effect of ,6 changes the system’s static solution and has no influence on
the linear stability. With the addition of modal damping, the eigenvalues will have
41
negative real parts and steady sliding is expected to be asymptotically stable.
In this section the assumed mode projection—Galerkin’s projection—is applied in
the evaluation of system stability in order to verify a feasibility for applying an approx-
imate method. Even though the exact eigenvalue solutions have been obtained already
in the previous sections, an application of an approximate discretization method may
provides a cautionary example for its use.
Apply the assumed mode method to recast equation (2.3) to ordinary differential
equations. It should be noted that the equation (2.3) is cast as non-self-adjoint in
equation (2.9). Assuming that ud(z,t) can be represented with assumed modes sat-
isfying the geometric boundary conditions and p derivative in the partial differential
equation of order 2 p, where p = 1, such an approximate mode can be accepted as
one of the candidates. Thus the solutions are expressed by possible assumed modes.
Then
ud(z,t) = :dj(z)aj(t), (2.47)
where cij(z) = ï¬sinUvrz) is chosen as an approximate mode. After projecting with
these assumed modes, an approximate ordinary different equation of motion is
00 (120' 00
Ema—2?- + Zkijaj = f,, i = 1,2,. ..00, (2.48)
j=l dt j=l
42
N
Real parts ol eigenvalues
O
_2»-
Imaginary parts of eigenvalues
Figure 2.7. A contradictory result: Eigenvalue trajectories versus a in the one-
dimensional friction system by applying the assumed mode method with two modes
included. (a) Imaginary and (b) real parts of the eigenvalues versus a are shown.
The selected assumed modes are $j(z) = x/2sin(j7rz) for j = 1, 2.
where
mi]- : 611', kij =k3~+k3 (2.49)
(95‘ = (jfll254j
i—‘L li—J'I =odd,
[CA - 1.2-J?
ij ‘—
0 otherwise,
W 2' = odd,
f. _ I77
1 _
0 otherwise.
where k5 and [CA are symmetric and anti-symmetric stiffness matrix, respectively.
Focusing on the low-dimensional dynamics, the system can be approximated with
43
Imaginary parts of eigenvalues
Figure 2.8. A contradictory result: Imaginary parts of the eigenvalues versus a by
applying the assumed mode method in the one-dimensional friction system. (a) 3
modes, (b) 4 modes, and (c) 5 modes are included. The selected assumed modes are
(DJ-(z) = \/2sin(j7rz).
n—coupled ordinary differential equations. The real parts of the eigenvalues of this
system indicate predicted stability characteristics. The dependency of eigenvalues on
parameters by including two modes are shown in Figure 2.7. Instability apparently
occurs when the real part of an eigenvalue is positive at the critical condition a =
5.7 by a collision between two frequencies. This instability mechanism resembles
flutter, and has been seen as one of possible instability mechanisms, e.g, flow induced
vibrations (Bolotin [16]) and friction induced vibrations (Nakai [57]).
However, these results contradict the exact solution since it has no instability
mechanism in the exact solution by parameter (1 based on the results in the previous
44
1o 10 1°
8* ‘ 8* - 8* d
6- - 6~ - 6~ ~
a:
3:3, 4- -’ 4- w 4~ ‘
g
51:) 2f 1 2r / 2* /
s
B o a 0 \ 0 :
g —2— ] —2~ - -2~ -
3
SB -4' —4r ~ —4r u
a:
—6- - —6- - —6- -
—8- - —8- - —8~ —
_ . . — - ' —1o
100 5 1O 15 100 5 1O 15 O 5 1O 15
a or (I.
Figure 2.9. A contradictory result: Real parts of the eigenvalues versus a by applying
the assumed mode method in the one-dimensional friction system. (a) 3 modes, (b)
4 modes, and (c) 5 modes are included. The selected assumed modes are (VJ-(z) =
\/281H(j7l’$).
section. Bolotin [16] had investigated this “paradox†for flow across a membrane. The
work showed non-convergent characteristics in the assumed mode projections, and
gave a theoretical criterion for convergence based on the linear operator. According
to those results, conservative systems with second order operators are not guaranteed
to converge in assumed mode approximations.
Nonconvergence of this eigenvalue problem can be demonstrated by increasing the
number of assumed modes. Figure 2.8 and Figure 2.9 show the imaginary and real part
eigenvalues for 3 to 5 modes, respectively. Considering Figure 2.7 also, the two lowest-
frequency modes interact at a = 5.7, and a = 9.0 for two- and four-mode including
45
approximations, but do not interact for three- and ï¬ve-mode approximations. This
shows that the approximated solution by using assumed mode methods for ï¬nding
the smallest interaction value a diverges as increasing the modal coordinates. This
hints at faulty results when applying the assumed mode method to this problem.
The proof of nonconvergence has been shown by checking the matrix determinant
by Bolotin [16]. Consider the convergence of determinant in equation (2.48). The
equation (2.48) can be written as
d2a- °°
dt; +Qfai+772bijaj=0, i=1,2,...,oo. (2,50)
1:1
And the characteristic determinant becomes
I (Q? — â€511' + 77sz l = 0- (2-51)
Dividing the 1'â€, row by Q,- and the jth column by (23-, determinant A can be expressed
in the form of
A =[ 6ij + Cij [ . (2.52)
According to the works of Bolotin [16] and Kantorovich and Krylov [7], the inï¬nite
determinant converges if the double series
0000
Z Z l ca l (2.53)
i=1j=1
46
converges. The determinant is described as normal when it satisï¬es this condi-
tion (2.53). By checking the determinant of the equation (2.48), it diverges as taking
inï¬nite modes. Thus this series diverge and is not a normal determinant at all.
25
O
""""
uuuu
.u
N
O
"i
E
-.
eigenvalues
—A
0|
7
A
A
01 o
n. n. '
\
Figure 2.10. Comparison between the exact and approximate eigenvalues: The square
roots of the exact eigenvalues, \/A: = (/(j7r)2 + “72, are shown with the solid lines.
The mode projected approximate eigenvalues obtained from a self-adjoint system are
shown with dotted line. The selected assumed modes are (VJ-(z) = ï¬sinUnz) for
j = 1,2,3,4,5.
For veriï¬cation, we apply the same assumed mode projection in (2.47) to the
equation of motion (2.46), which has the self-adjoint form derived in the previous
section by taking the proper inner product. Applying the assumed mode, \/2 sin(z'7rz),
to the equation of motion (2.46) and using the proper inner product, the approximate
eigenvalues have been numerically calculated by parameter a.
47
25
20 r
J = 5 _...WMWWW
3 15 ' .... mmmmmmm r
‘4 .=4 M-..
> no..-
5 J = 3 â€wan-“WM...“
.91 0 MM
0 â€ï¬‚u-0"â€N a.
l = 2 ‘//ï¬ï¬‚u.uw "fl...
“MM...â€
5 " I = 1 WWW“. .4
1Mâ€.M
00 5 10 1 5
0t
Figure 2.11. Eigenvalues obtained from the non-self-adjoint system by projecting the
exact eigenfunctions ¢j(z) = x/2e%f sin(j7rz).
In this example, ï¬ve modes are selected for the discretized system. Figure 2.10
presents the exact and approximate eigenvalues based on the proper inner product
versus a. The low frequency approximation has good accuracy in eigenvalues cal-
culation. Though there are still slight deviations from the exact solution in high
frequency eigenvalue approximations, a more accurate approximation is expected by
including more modes. Consequently, a contradictory result has been avoided in eval-
uating the eigenvalue for self-adjoint system. (When we use the exact eigenfunctions,
¢j (z) = 6%3 sin(j7rz), on the non-self-adjoint system (2.9) and project with the exact
eigenfunctions, (15,-(z) = 6%$Sin(i7l'13), we have eigenvalues which are identical to the
exact eigenvalues, shown in Figure 2.11.)
There are investigations into the approximation of non-self-adjoint systems.
48
Meirovitch and Hagedorn [2] investigated the modeling of distributed non-self—adjoint
systems, such as damped boundary condition models. In using the method of weighted
residuals to produce the approximate solution to the eigenvalue problem, the displace-
ment of a non-self-adjoint system is ordinarily represented by a linear combination
of comparison functions, i.e., the functions that satisfy all the boundary conditions.
Because of difï¬culties in ï¬nding comparison functions the more feasible approach
consists of the construction of an approximate solution by using combinations of ad-
missible functions, called quasi-comparison functions, capable of satisfying all the
boundary conditions of the problem [2]. The similar approaches for solving the ap-
proximate solutions have been found in some literatures by Meirovich and Kwak [1]
and Hagedorn [3]. The proof of Galerkin’s method for non-self-adjoint boundary
value problems has been given by Diprima and Sani [6] and a sensitivity analysis in
the non-conservative problem by using adjoint variational method are presented by
Prasad and Herrmann [4] and Pedersen and Seyranian [5].
2.7 Conclusion
A one-dimensional continuous system with distributed sliding contact was investi-
gated in order to study the dynamic instability caused by friction. A partial dif-
ferential equation of motion was established and its exact solution was presented.
An eigenvalue problem in this non-self-adjoint system was shown and its solution was
provided with a different approach: using the proper inner product and a transforma-
tion to a self-adjoint system. A technique for choosing a proper inner product which
49
switches the system properties from non-self-adjoint to self-adjoint was reviewed. The
system can overcome the difï¬culties in evaluating the approximate eigenvalues with
the help of the prOper inner product.
A contradictory result between the exact solution and the assumed modes ap-
proximation in evaluating the eigenvalues was shown as a cautionary example. In
this case, non-convergence of the assumed modes method can be easily detected.
The exact solution shows the undamped system is neutrally stable for all parame-
ter values. The constant coefï¬cient of friction does not cause an instability. Boundary
conditions and non-linear friction force contributions to the system stability are in-
vestigated in Chapter 3 and Chapter 4, respectively.
CHAPTER 3
FRICTIONAL SLIP WAVES IN
AN ELASTIC MEDIUM
3. 1 Introduction
In the previous chapter, we saw that a one-dimensional elastic material with ï¬xed
end points under distributed friction did not undergo an instability when the friction
coefï¬cient is a constant. However, the ï¬xed end points may not be representative of
our motivational annular system. A ï¬rst correction might be to implement periodic
boundary conditions, which may enable traveling waves to exist.
The structural stability of waves generated in an elastic medium has been an inter-
esting t0pic for scientists and engineers. When there is contact between two materials,
waves, which are generated around the contact area, contain properties of dynamic
stability. Such dynamic stabilities have been used to explain the friction-induced
vibrations in an elastic medium, which are associated with chattering, squeaking,
50
51
squealing noise, and stick-slip oscillations.
In classical interpretations of causes for the noise and vibrations, analyses
dealing with discretized mathematical models have prevailed (Brockley et al.[65],
Sakamoto [49, 50]). Moreover, most of the causes cited for steady sliding instabili-
ties have been based on friction-speed relations: a decreasing coefï¬cient of friction in
sliding speed has played a primary role in instability of the system based on linear ,
stability criteria (Simpson [74], Nakai and Yokoi [72], Krauter [7 5])
Experimental and analytical studies associated with elastic continua have shown
that systems have various elastic waves (Stoneley [30], Barnett et al. [29], Dun-
durs [28]) and some of the waves destabilize the systems when elastic materi—
als were subjected to distributed friction forces (Schallamach [26], Martins[25],
Adams [23, 24]).
Observations of destabilized waves in elastic continua were performed by Schal-
lamach [26]. Experimentally, he observed that the relative motions between two
frictional members are due to waves of detachment crossing the contact area at high
speed, and that waves appear as moving folds or wrinkles on the surface of rubber.
When the tangential compressive stress reached a buckling state, the buckling of the
front edge induced detachment waves, known as Schallamach waves, which travel
from the front to rear.
Extended studies of occurrence conditions of such waves were investigated by Best
et al. [27], Martins et al. [25] and Adams [23, 24]. According to experimental studies
by Martins et al. [25], De Togni et al. [99], Rorrer et al. [100], and Vallette and
Gollub [101], dynamic instabilities can occur even in a condition with no decreasing
52
characteristics in friction-speed relations. For example, Vallette and Gollub [101]
measured spatiotemporal internal displacements of an elastic continuum subjected to
friction contact and proved that unstable traveling waves can occur even without a
decreasing friction coefï¬cient in sliding speed.
Using analytical approaches, the existence of destabilized waves in the presence
of friction was conï¬rmed by Martins [25] and Adams [23, 24]. Martins et al. [25]
showed that the intrinsic non-symmetry of Coulomb’s friction contributions to equa-
tions of motion and couplings of various degree-of-freedom play an important role in
generations of dynamic instabilities. Under a condition of large couplings in spatial
coordinates caused by friction stresses, steady sliding motions are dynamically un-
stable even in a constant coefï¬cient of friction. It is also claimed that a decrease of
coefï¬cient of friction with sliding speed is not a necessary condition for the occur-
rence of unstable elastic waves. The deve10ped studies were found in the works by
Adams [23, 24] for one- and two-dimensional elastic medium in contact. He deter-
mined the existence of unstable waves by using a one-dimensional model composed
of a beam-on-elastic foundation. A beam placed on a series of springs was used as
a qualitative model for two bodies in sliding contact. This analysis indicated that
steady state solutions are dynamically unstable for any ï¬nite sliding speed even with
a constant coefï¬cient of friction, due to interactions of complex modes of vibrations.
In this chapter investigations on the stability of waves of an elastic medium on
the condition under frictional steady sliding are presented in order to understand the
mechanisms which cause vibrations and noise. For a one-dimensional elastic system,
the presence of unstable waves in a continuum is investigated via the mathematical
53
model developed in the previous chapter. The fact that the system’s stability depen-
dent on the boundary condition is emphasized in this study. Through evaluations
of characteristic solutions of waves, explanations for instabilities under a condition
of a constant coefï¬cient of friction are provided. In addition, the effects of external
and internal damping on overall system stability are analyzed. For a two-dimensional
elastic system, a mathematical model of semi-inï¬nite, isotropic, linear material with
a periodic boundary condition is presented and its characteristic solution is investi-
gated. Effects of system parameters, such as Poisson’s ratio and a friction coefï¬cient,
on dynamic stability are shown.
3.2 One-Dimensional Elastic System
3.2.1 Stability Analysis of Elastic Waves
Unstable waves in elastic materials yield non-uniform motions, such as micro scale
stick-slip oscillations, or cause to loss of contact at the contact surface. Prior to
investigations of the nature of the non-uniform motions (discussed in Chapter 5 in
this study), primary causes for dynamic instability are investigated from a wave
dynamics point of view.
Consider the one-dimensional, undamped, elastic system developed in the previous
chapter. The dynamic equation of motion in the self-adjoint form is
6
—aa:a_u}_ —azfl 31
8:1: 6.7: _e 8t2' (')
54
However, these time periodic boundary conditions
u(0, t) = u(1,t), (3.2)
du(0, t) _ du(1, t)
dz _ dz ’
are considered. Note that the system parameter a (2 nu) in equation (3.1) is a
constant value, which represents a ï¬xed coefï¬cient of friction and Poisson’s ratio.
Considering periodic boundary conditions (3.2), solutions are assumed to have the
form
u(:c, t) = Real{ei2"k($_“)}, (3.3)
where k is a positive real number representing the angular frequency of solutions
along the x axis, as the term % shows the wave periods along the :2: axis. (See Figure
2.1 for the system conï¬guration. We will use the notation in equation (3.3) in this
chapter since references from wave dynamics in continua have used such notation in
their studies.)
Generally, c can be a complex value and plays an important role in dynamic system
stability. In the case of a real value of c, pure waves of constant shape are expected.
This implies that conservative non-dispersing waves exist in the elastic medium and
the system is in a neutrally stable state without damping. On the other hand, a
complex value of 0 contains information about the characteristics of the waves. This
can be easily expressed by c = R + I i, i.e., c composed with a real component R and
55
a imaginary component I . Thus the equation (3.3) can be rewritten as
u(:r, t) = Real{el2â€k(x‘mle%“t}. (3.4)
A positive R indicates that there is a wave propagating toward the positive direction
and a positive I indicates that there isan unstable wave which increases its motion
exponentially in time. On the other hand, a negative R indicates that there is a wave
propagating toward the negative direction and a negative I indicates that there is a
stable wave which decreases its motion exponentially in time. Thereby, the imaginary
component of the characteristic solutions represents the stability of the wave.
A characteristic equation of c obtained by substituting the equation (3.3) into
equation (3.1) is
2 _ i
c (1+ 27mi): O. (3.5)
The imaginary and real parts of the characteristic solution are
R = \J 1+ \/1+( 2(2()z/)27rk , (3.6)
(1 C!
47FkR 1+‘/1+(a/21rk)2
47rk 2
It is clear that without friction—the condition in which 0 equals zero—the char-
acteristic solution has pure real solutions for c and the traveling waves which keep
their wave shapes in time are pure sinusoidal functions.
56
When friction is considered, however, the characteristic equation yields general,
complex conjugate solutions for c. From the result in equation (3.6), the waves
propagating toward the positive 3: axis, which are represented by a positive value R,
are unstable waves. In other words, the amplitude of the propagating waves increases
in time. On the other hand, when the waves propagate toward the negative :1: axis
they are stable since they have negative imaginary components in the characteristic
solutions.
0.7 1.2
1.18-
0.6- 4
1.16" ‘
00.5" k-‘ ‘ 1.14. .4
'8 o
w 251.12 ~
EDA] g k=‘
E Q 1.1[
_EO.3~ “2 _ _ .
g 51:31.08
-0.2~ k-3 1.06"
1.04- k=2 «
0.1 -
1.02[ “'3
4 -10
O ‘ 1
O 5 1O 0 5 10
a a
(a) (b)
Figure 3.1. The unstable characteristic solutions for the undamped, periodic bound-
ary conditioned model. (a) Imaginary and (b) real parts of the characteristic solution
versus a are shown.
Figure 3.1 shows the imaginary and real parts of the characteristic solution cor-
responding to an unstable wave by increasing the parameter oz. Those solutions are
57
presented with various undetermined frequency factors Is. As (1 increases, i.e., as the
coefï¬cient of friction or Poisson’s ratio increases, waves traveling toward the positive
a: direction (the direction of the moving rigid body) are increasingly unstable in any
ï¬nite sliding velocity. Clearly, waves traveling toward the negative z direction (not
shown in Figure 3.1) are stable waves.
Similar trends associated with such unstable traveling waves were found in previ-
ous studies. Regarding the traveling direction of unstable waves, the direction of the
moving rigid body indicates the direction of the unstable waves (Martins et al. [25],
Adams [23, 24]). In addition the traveling unstable waves make whole systems un-
stable even if the coefï¬cient of friction is constant. By considering a beam subjected
to distributed friction, which was modeled mathematically as a fourth order partial
differential equation, Adams [24] proved that one-dimensional traveling waves make
whole systems unstable. He included random properties representing the roughness
of the contact surface in his modeling.
With the aid of this study, it has been analytically shown that elastic systems
subjected to distributed friction can also be unstable in the presence of a constant
coefï¬cient of friction, without including any random properties. As described earlier,
it is expected that a destabilizing wave phenomenon is one of the possible causes
for unstable motions. In real situations, such unstable waves are expected to yield
non-uniform motions, such as stick-slip oscillations or loss of contact in materials.
58
3.2.2 Addition of External Damping
An undamped elastic continuum subject to distributed friction is considered in the
previous section. In this section, effects of external damping on system stability
are considered. (External damping is deï¬ned as a relative dissipation between an
elastic material and a ground.) An equation of motion including an external damping
coefï¬cient d is
8 Bu u an
e _
—aa: _ —a:c 62
'5;{ 8:17} —6 {37+dg}. (3.7)
Applying the periodic boundary condition (3.2) again, the characteristic equation is
c2 + —c — (1+ 7) = 0. (3.8)
A search for analytical solutions of quadratic equation (3.8) with complex coefï¬cients
is not an easy job, so we apply numerical method in searching for solutions.
Since unstable waves affected by external damping are primary concerns of this
study, consider only the maximum value of imaginary parts in the characteristic equa-
tion (3.8). The maximum value of imaginary components determines the whole sys-
tem stability. Remember that a positive imaginary component indicates an unstable
traveling wave.
Figure 3.2 and Figure 3.3 provide the imaginary and real parts of equation (3.8) on
the parameter domains (1 and d, respectively. In Figure 3.2, the maximum imaginary
part is decreased by decreasing a or increasing d. In other words, a reduction of
59
p
N
J
p
d
l
I;
' . II
, II II
'0.1‘ " , III/I ’
IIIIIIIII/IllfIIIIII
IIIIIIIIIIIIIII;I’;I,I†"-
maginary parts of c
5’
Figure 3.2. The imaginary parts of the characteristic solutions including the external
damping coefï¬cient d. The maximum value of the imaginary parts is presented in the
parameter domains (1 and d. In this example k = 1.
friction, or an increase in external damping is required to stabilize the system. The
traveling speeds of waves, which are represented as the real parts of the characteristic
solution, are influenced by oz and d as shown in Figure 3.3. The speeds of waves
corresponding to the unstable ones are decreased by decreasing a or increasing al.
Figure 3.4 depicts trajectories of imaginary parts of c under variations in d for
several frequencies 1:. In this example, the overall system, which was unstable by
having complex conjugate pairs in imaginary characteristic solutions when d = 0.0,
becomes a stable system with sufï¬ciently large external damping d. The solutions
of c no longer have complex conjugate imaginary pairs when d is not equal to zero.
Beyond the point (1 = 1.0, all characteristic solutions can have negative imaginary
parts, which implies the system is fully stable. Low frequency terms, such as k = 1,
60
Real parts of c
Figure 3.3. The real parts of the characteristic solutions including the external damp-
ing coefï¬cient d. The real parts of the characteristic solution corresponding to the
maximum imaginary value is presented in the parameter domains a and d. In this
example k = 1.
are easily stabilized by increasing d, as indicated by the steep downward slopes with
increasing (1 in Figure 3.4.
Based on this interpretation, the system can undergo under an unstable condi-
tion; a condition that the system has some stable and unstable eigenvalues in its
parameters. (For example, the range 0 < d < 1.0 in Figure 3.4). Under such con-
dition, responses corresponding to the stable eigenvalues are damped out in time,
but responses corresponding to unstable eigenvalues can dominate the whole system
responses, generating squeaking or squealing noise and vibrations in experiments.
0.2
0.15*
61
Imapinary parts of c
‘ v u - v _ _ .
v .. ~ 7
v »
. -. .....
Figure 3.4. The imaginary parts of the characteristic solutions including the external
—— k=1
—4o— k=2 .,
—~— k=3
—u——— :4 z
——e— k=5
‘ ‘ii‘ '- is‘a‘-‘-‘—‘---.- .
“u..- -
b
damping coefï¬cient (1. In this example a = 1.0.
3.2.3 Addition of Internal Damping
Elastic materials such as rubber contain considerable internal damping. (Internal
damping is deï¬ned as a relative dissipation of strain energy in the materials.) Usu-
ally internal damping is stabilizing, but under some conditions, especially when there
are non-conservative forces, such internal damping is able to destabilize systems
(Bolotin [16], Hendricks [38], Shaw and Shaw [39], and Higuchi and Dowell [10]).
Effects of internal damping on the system being studied are not clear. In this section,
the effects of internal damping, referred to as structural damping, on system stability
are considered.
62
A stress-strain relation including internal damping is given by
6'
0I(x,t) = E€+Vé=E§B+V u
0:1: 873’ (3.9)
where E is the modulus of elasticity and V is the modulus of viscosity of the material.
Applying equation (3.9) to equations (2.1) and (2.2), an equation of motion including
internal damping 7 is given by
131
6:1:
_a_
6:1: (9th
8211 _m, 6211
Kama—U} +
82: 7
where 7 is an internal damping coefï¬cient deï¬ned as V/AE. A characteristic equation
obtained by substituting the periodic boundary condition is
2 2k'— —1—C£=. .
c+7(7rz (1)0 (+27rk) 0 (311)
Figures 3.5 and 3.6 show the imaginary and real parts of the characteristic solution
corresponding to the maximum imaginary value in the parameter domains a and
7, respectively. The imaginary parts attain negative values when increasing 7 or
decreasing a, as shown in Figure 3.5. Internal damping in elastic materials stabilize
the system as the external damping does. The speed of waves versus 7 and a is
shown in Figure 3.6. From these results, it is concluded that the system is stabilized
by increasing internal damping.
Figure 3.7 shows trajectories of imaginary parts of c under variations in 7 for
several frequencies k. Like external damping does, the system is stabilized beyond
63
Figure 3.5. The imaginary parts of the characteristic solution including the internal
damping coefï¬cient 7. The maximum value of the imaginary parts is presented in the
parameter domains a and 7. In this example k = 1.
the point 7 = 0.025. Note that the high frequency terms, such as k = 5 in Figure
3.7, are easily influenced and stabilized as increasing internal damping 7.
Instabilities induced by internal damping have been reported for some speciï¬c
systems such as rotation systems (Shaw and Shaw [39], Hendricks [38], Bolotin [16],
Iwan and Stahl [76] and Iwan and Moeller [77]) and systems with follower forces
(Higuchi and Dowell [9, 10]). In such systems, small internal damping can destabilize
the system. For the instability by modal interactions, i.e., the instability accompanied
by colliding of frequencies with changing parameters, it is reported that small internal
damping can destabilize whole systems.
64
1.0377)
c
—L
o
N
1
Real parts of
Figure 3.6. The real parts of the characteristic solution including the internal damp-
ing coefï¬cient 7. The real parts of the characteristic solution corresponding to the
maximum imaginary value is presented in the parameter domains a and 7. In this
example k = 1.
a = 1.0
0.3 F I I I
— =1
02’ —~— k=2 ‘
—+— k=3
0 1 ' _._ k: .
.2 , —e— k=5
g 0] .
8
b—O 1 - «
(U
.5
3"0-2 - -
.5.
-0.3 ~ -
k
-0.4]- <
-0.5 ‘ i ’
0 0.01 0.02 0.03 0.04 0.05
Figure 3.7. The imaginary parts of the characteristic solutions including the internal
damping coefficient 7. In this example a = 1.0.
65
3.3 Two-Dimensional Elastic System
A semi-inï¬nite, linear elastic medium in contact with a moving semi-inï¬nite rigid body
is considered as a two-dimensional elastic system subjected to distributed friction
(Martins et al. [25]).
/
\\
/
.A/
a\
Figure 3.8. A schematic diagram for a semi-inï¬nite, two-dimensional elastic medium
in contact with a moving rigid body.
In this section, dynamic stability as affected by a constant coefï¬cient of friction
is investigated (Figure 3.8). The dynamic equation of the linear elastic material with
respect to a static equilibrium state is represented by
8
2 — —
GV u+(z\+G)azA
l
:3.
(3.12)
6
0V2 + A+G—A
v ( )6],
||
:3.
66
2 .
where V2: 66:7 + :72,A = % + -g—Z,G— - 2713—1â€, and A- — W. The variables u
and v are displacements in the x and y directions with respect to the static equilibrium
state, respectively. By rewriting equation (3.12) in terms of the speed of free vibration
waves, the equations of motions are
8221 8221 62a 62v 6211
2 2 2
— + — c — — + —- = —,
62v 621; 6% 62v 62v
2 _ _ 2 _ 2 = _
Q4611? + ay2)+(cL OT)(axay + 83/2) atga
(3.13)
where 01, = ,/ A—‘L—pm and cT =\/§ represent the longitudinal and transverse speed of
waves, respectively (Fung [37], Saada [36]).
Boundary conditions on the contact surface at y = 0 are
v(a:, 0, t) = 0, (3.14)
ayx(a:, 0, t) = payy(x, 0, t),
where p is a coefï¬cient of friction, assumed to be a constant. The stress-strain
relations are am, = C(g—Z + g—Z), and am, = Ag: + (A + 2G)g—:. The boundary condition
in equation (3.14) implies that there is no loss of contact between the medium and
the moving rigid body.
The dynamic solutions subjected to periodic boundary conditions are assumed to
have the form
u(:r, y, t) = Ae"byeik($—Ct), (3.15)
67
v(:z:,y,t) = Be‘byeiklx"“),
where k is a positive real number, which represents an angular frequency along the :1:
axis, and b is a complex number, which contains a positive real value that allows ex-
ponentially decreasing oscillations in the y direction. Such boundary conditions have
been adopted in a development of the Rayleigh waves in elastic materials (Fung [37]).
Here a complex value 0 determines the stabilities of traveling waves induced by fric-
tion. Note that a imaginary value of 0 implies a neutrally stable wave.
Applying equation (3.15) to equation (3.13), nontrivial solutions are
â€(2 — 2\/1— L\/1 — T — T) =z'T\/1— L, (3.16)
where L and T are deï¬ned as (c/cL)2 and (c/cT)2, respectively. For a case of a
compressible linear elastic material, i.e., the range 0 S u < 1 / 2, the equation (3.16)
yields the following sixth order equation with respect to L (See Martins et al. [25] for
details.)
L2(L4 + (13L3 + a2L2 + a1L + a0) = 0. (3.17)
The coefï¬cients a0, a1, a2, and (13 are given by
a0 = 16n2r2(1+u272), (3.18)
a1 = —8u27(2+T(3+u2(4T—1))),
with T = cT/cL.
Figure 3.9. The imaginary parts of the characteristic solution L in the parameter
(12
as
68
= 41120 + 27(2 + r) + (u2(4r —1)+1)2,
= -2(u2(4T+ 1)),
,‘
a
.“c‘
.0
o1
.\
\\ \°\‘
\ s‘ ~ ‘\ ‘s‘
s ~
\ ‘s
s:.:
‘ o
,.
o“
s
go
o
o“.
\\
Q‘
g»
‘Q
‘ o
‘
‘.
\
axe
net“
‘
‘5‘
Imaginary parts of L
\. §
1 \§
S$§
~\
‘\
Q.‘
010
plane of friction coefï¬cient )1 and Poisson’s ratio 11.
A numerical analysis of equation (3.17) is performed in the domains )1 and V.
Figure 3.9 and Figure 3.10 show the imaginary and real parts of L in equation (3.17).
The zeros in the solutions over the parameter domains, which are located in regions
of small 11 and small 11, indicate that there are no nontrivial solutions which satisfy
the boundary conditions in (3.14). Based on the relation of c = :lzx/L CL, the positive
imaginary and real components of L, which are shown by the non-zero values in
69
Real parts of L
.0
Figure 3.10. The real parts of the characteristic solution L in the parameter plane of
friction coefï¬cient )1 and Poisson’s ratio V.
Figure 3.9 and Figure 3.10, correspond to the ï¬rst and third quadrants of the complex
plane c. Remember that the positive imaginary value of c implies an unstable wave
and the sign of the real part of 0 determines the direction of the wave. Thus the
solutions located in the ï¬rst quadrant in the complex plane 0 (Real(c) > 0 and
Imaginary(c) > 0) indicate unstable waves traveling toward the positive x axis. On
the other hand, the solutions located in the third quadrant in the complex plane
c (Real(c) < 0 and Imaginary(c) < 0) indicate stable waves traveling toward the
negative :5 axis.
Therefore, according to these numerical solutions, the two—dimensional elastic
medium has unstable traveling waves, even with a constant coefï¬cient of friction,
if Poisson’s ratio and a friction coefï¬cient are large enough. The numerical results
70
from the two-dimensional elastic systems are consistent with the one-dimensional sys-
tem developed in the previous sections. Both systems have unstable traveling waves,
which propagate toward the direction of the moving rigid body, in any ï¬nite speed
of the rigid body. (A two-dimensional elastic medium under distributed contact with
ï¬xed ends boundaries is numerically investigated by applying ï¬nite element analysis
in Chapter 6.)
3.4 Conclusion
In this chapter, the dynamic stability of frictional steady sliding in one- and two-
dimensional systems was investigated. Under the periodic boundary condition un-
stable traveling waves in a one-dimensional elastic system were found to be depen-
dent upon a constant coefï¬cient of friction and Poisson’s ratio. It was demonstrated
that high coupling in the coordinates due to Poisson’s ratio destabilizes the two-
dimensional elastic continuum.
It was concluded that a decreasing coefficient of friction is not a necessary condi-
tion for the occurrence of dynamic instability. In addition, the characteristic analysis
showed that dynamic instability occurs in the form of self-excited, unstable, traveling
waves. The stabilizing effects by adding external and internal damping were studied.
The system imposed by the ï¬xed boundary conditions, presented in Chapter 2,
has no instabilities under the condition of a constant friction coefï¬cient. Thus the
neutrally stable condition exists for the undamped one-dimensional system. However,
according to the results of this chapter, the same system under the periodic boundary
71
condition becomes unstable because of the unstable traveling waves.
It should be noted that these analyses were based on the steady state frictional slid-
ing stability. Thus any noise and vibrations originated from the non-uniform motions
should be analyzed by different approaches. Chapter 5 illustrates these phenomena.
CHAPTER 4
STABILITY ANALYSIS IN A
LUMPED PARAMETER MODEL
4.1 Introduction
In order to investigate friction-induced vibrations and noise generated from large
frictional contact surface, for example regarding bushing squeaking noise in a vehi-
cle suspension system, noise generation from a band/drum brake system, and jerky
motions of a clutch engagement, a construction of a mathematical model which can
explain the dynamic behaviors including stick-slip oscillations is essential process in
system analysis.
However, it could be a painstaking job in handling the continuous model to explain
dynamic phenomena induced by friction. For a example, when the system undergoes
stick-slip motions, the prediction of stick-slip motions are difï¬cult through analytical
approaches since they are dependent on the system states and occur anywhere over
72
73
the domain of contact. In order to overcome such difï¬culties, it is necessary to build
a descretized model based on space coordinates.
In Chapter 2, we saw that improper discretization of the PDE led to misleading
results. If we are going to apply lumped parameter models to nonlinear stick-slip
studies later, it is important to investigate the quality such a discretization. One way
to gage the discretization is through a linear stability study, and its comparison with
previous analytical results.
Most of the works in the previous chapters were investigated based on a assump-
tion that the friction coefï¬cient is a constant with respect to the relative speed. How-
ever, it has been reported that the frictional damping plays a crucial role in system
stability and should be included in stability analysis. Theoretical and experimental
studies by Brockley [66, 65], Tolstoi [48] have shown that a single degree-of-freedom
model with negative slope in friction-speed relation is unstable and leads to self-
excited vibrations.
In applied systems, for example automotive brake systems (Friesen [69],
Abdelhamid[70]), aircraft brake systems (Black [71]) and water lubricated bearing sys-
tems (Simpson [74], Krauter [75], Bhushan [73]), the friction-speed relation strongly
influences the overall system stability, and the cause of noise and vibrations. Such
noise generation mechanism was broadly investigated by Nakai and Yokoi [72] and an
importance of resultant structural damping has been recognized by Krauter [75] with
linear analysis for generation of squealing noise caused by dry friction. By using two
degree-of-freedom nonlinear models, Simson [74] and Krauter[75] have revealed that
chattering and squealing noise are generated due to the resultant structural damping.
74
In this chapter, a lumped-parameter, multi degree-of-freedom model is constructed
as a discretization of the previous continuous system. (This model is adopted for
investigations of stick-slip oscillations in Chapter 5.) Non-symmetric properties in the
eigenvalue problems, which are usually introduced by friction, are properly handled
and analyzed. Stability analyses including damping, such as external, internal, and
frictional damping, are evaluated.
4.2 Stability Criteria
The condition of stability, i.e., the boundary between stable and unstable domains,
naturally depends on the parameters of the system such as boundary conditions,
distribution of loads, system damping, and nonlinearities of materials. For most
of the classical investigations related to the theory of elastic stability, the external
forces are expressed through the potential energies and the problems usually have
self-adjoint properties. Because the external forces have potentials, loss of stability
can take place only in the form of static instability—divergence—which has zero
frequency.
On the other hand, when a system contains nonconservative forces, for exam-
ple panels or shells in air flow, follower forces in elastic materials, and systems in-
cluding dry friction forces, the instability may occur either dynamically—ï¬utter—or
statically—divergence (Bolotin [16], Ziegler[17]).
For investigations of a stability of multi degree-of-freedom system the deï¬nition
of linear stability with respect to eigenvalues is explained as follows. When the linear
75
system has the form of a homogeneous matrix equation, then the eigenvalue problem
is represented as
Lezo, (as
where the system matrix L depends on the real load parameter and the complex
eigenvalue, A = a + iw. We write the matrix L as a linear function of real matrices,
specifying the dependence on A explicitly by
L = A2M + AC + K, (4.2)
where the stiffness matrix K, damping matrix C, and the mass matrix M can be
non-symmetric matrices in general.
The dynamic stability of the system is determined quantitatively by the eigen-
values, A,. = a, + in), for r = 1,2,. ..,n. The eigenvalue with the maximum real
part is the important one and for this am“. since the stability of the whole system
is dominated by this value. Let wan,“ be the imaginary eigenvalue corresponding to
the amâ€. Then motions and stability are categorized by
Stable motion if am“. < 0
Critical motion (marginally stable) if am†2 0
Flutter instability if am†> 0 and won,†75 0
Divergence instability if amax > 0 and won,†= 0
76
Base on these stability criteria, the possible unstable motions are evaluated by eigen-
values of the system.
4.3 A Lumped-Parameter Model under Fixed
Boundary Conditions
4.3.1 A Stability Analysis of an Undamped System
Investigations of the dynamic behaviors and stabilities of systems having consider-
ably large rubbing surfaces may encounter some difï¬culties in the evaluation of the
system eigenvalues since the classical approximate method which relies on the modal
coordinates may no longer valid and their convergence of eigenvalues are not guar-
anteed as discussed in Chapter 2. There is no reason to expect other discretizations
to converge, either. But we investigate the performance of other discretizations in
hope that those difï¬culties are overcome, so that the discretization can be applied to
nonlinear studies with some conï¬dence.
In this section by using the lumped-parameter discretization method the continu-
ous system is simpliï¬ed to a multi degree-of-freedom model and the system stability
is analyzed. (Analyses using the ï¬nite element method are shown in Chapter 6.)
Consider a system shown in Figure 4.1, which shows the lumped-parameter model
from the continuous system in Figure 2.1. The mass blocks connected to linear
springs are placed on the moving belt. There are frictional forces between the mass
blocks and the moving belt. In this model, the each mass block plays a role not
77
77 77777777 7+
Figure 4.1. A schematic diagram for the undamped, lumped-parameter model sub-
jected to distributed fn'ction. Fixed end boundary conditions are applied.
only as a lumped-mass, but also as a discrete elastic mass which can contract and
elongate based on the Poisson effect due to the forces exerted around the mass. Since
the normal displacement is restricted as shown in Figure 4.1, the contraction and
elongation influence the normal load, which causes the variation of the friction forces.
The equation of motion for undamped it}. mass is written as
mm,- (t) + k{—.’L‘,'_1(t) + 21'5“) — $i+1(t)} + fi(t) = 0, (4.3)
where m is a mass of each mass block, I: is a spring stiffness, x,(t),:i:.-(t), and :i':.-(t)
represent the displacement, velocity and acceleration of it), mass, respectively and
f,-(t) is the friction forces on the in. mass. Here the mass and stiffness are lumped
from the evenly distributed system. Let us include the Poisson’s ratio effect. Then
the friction force is
M0 = I‘M-(t) = #lNo + I’M-Mt) - xs—1(t)}], (4-4)
78
where u is a friction coeflicient and N0 is the normal load on each block, which is a
negative constant value (No < 0), and N,(t) is the resultant normal load including
the Poisson’s ratio effects (N,(t) < 0). Thus the undamped equation of motion for
the it), mass block is
431(7) — (1+ HV)$1—1(T)+(2 + V#)$i(7') " 33:41“) + #NO/k = 0, (4-5)
1
61
where T = wpt, w; = k/ m and the time derivative () denotes . This is a difference
equation of motion of the continuous system in equation (2.3).
It has been known that the system stability is closely dependent on its boundary
conditions. Firstly, consider a ï¬xed boundary condition of
=0 mm
The equation of motion for the undamped system is expressed by
MX+KX=Fm up
where
79
2+u,u —1 0 0 0
-(1+1/,u) 2+V/1 —1 0 0
K: 0 —(1+1/11) 2+Vp 0 0 ’
O 0 O —(1+1/;1) 2+V,u
l. -
and
F0 = —pN0/k [1,1, . . ., 1]T.
The matrix I denotes a unit matrix. The stiffness matrix K is non-symmetric due to
the effect of friction and the Poisson’s ratio. The eigenvalues for the dynamic systems
are evaluated with respect to the static equilibria. Thus, the eigenvalue problem is
represented by
AM<P = K<I>. (4.9)
Since K 7:9 KT, the orthogonal relations obtained from the symmetric properties are
no longer valid. Furthermore, the expansion theorem derived from the symmetric
relations can not be applied to decompose any arbitrary vectors in terms of a set of
eigenvectors.
Let us briefly discuss the general eigenvalue problem, which covers the non-
symmetric prOperties in equation (4.9), and then return to the problem of interest.
80
Consider the transposed eigenvalue problem associated with equation (4.9) and write
it in the form
AM\II = KTw. (4.10)
The eigenvalues of equation (4.10) are the same as those of equation (4.9). On the
other hand, the eigenvectors of equation (4.10) are different from those of equation
(4.9). Consider two distinct solutions of equation of (4.9) and (4.10). These solutions
satisfy the equations
AiM¢i = K65†Z = 1,2, . . .,n, (4.11)
and
AjMw, = Kij, j : 1,2,...,n. (4.12)
The equation (4.12) can also be written in the left eigenvector form by
Ajwa = ï¬x, j = 1,2,. . .,n. (4.13)
Multiplying equation (4.11) on the left by d)? and equation (4.13) on the right by ()5,-
and subtracting one results from the other, then
(A1 — /\j) wJTQbi = O, (4.14)
81
so that for distinct eigenvalues
333,70, A,¢/\,-, 2',j=1,2,...,n. (4.15)
This means that the left eigenvectors and right eigenvectors of the system correspond-
ing to distinct eigenvalues are orthogonal. It should be stressed that the eigenvectors
are not mutually orthogonal in the same ordinary sense as those associated with the
Hermitian matrix. Indeed, the two sets of eigenvectors <1),- and (I), are biorthogonal.
The fact that the eigenvectors d),- and 1,1), are biorthogonal permits to formulate an
expansion theorem for general case. Assuming that any vector can be represented by
inï¬nite sum of eigenvectors there is a choice of expanding any arbitrary truncated n-
vector x in terms of the eigenvector (b,- or 1b,. Assuming that the truncated expansion
in terms of d),- closely approximates x, then
x = <I>q, (4.16)
where q = [q1, q2, . . . , qn]T is the vector of associated coefï¬cients. Thus the coefï¬cients
are obtained by
q = \IJTx. (4.17)
Similarly, an expansion in terms of the eigenvector w,- has the form
x = ‘Ilr, r = <I>Tx, (4.18)
82
where r 2 [T1, r2, . . . , Tn]T is the vector of associated coefï¬cients associated with 1,0,.
This procedure, which treats the non-symmetric eigenvalue problem in the lumped—
parameter system, corresponds to the non-self adjoint eigenvalue problem in the con-
tinuous system discussed in Chapter 2.
20
‘3 ....................................................................................................
$14: ........... q
2
g oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
:12“- ................................ _1
O
o:
’510’ .......................................... 4
5‘ .........................................................
.E 8~ .
O!
m onn.....ouo..oou.noonnoon.accounoono.o.....n.l
g 6~ ........................................
4'— ooooooooooooooooooooooooooooooooooooooooooooooooooooooooo u
2~ .
0 L
O 2 4 10 12 14
6 8
u (x 20)
Figure 4.2. Trajectories of the eigenvalues versus friction coefï¬cient 11 in the un-
damped, lumped-parameter model.
Let us return to the problem of interest. Figure 4.2 shows the numerical results
of the eigenvalues by changing the friction coefï¬cient )1, which is assumed to be
constant with respect to the relative speed. The calculated eigenvalues can also
be compared to the exact eigenvalues of the continuous system, shown in Figure
2.10. As )1 increases the frequencies simply increase and no destabilizations are found
in Figure 4.2. This result shows a close approximation to the exact eigenvalues
83
0.35 r ‘
.0
to
I
l
_O
N
01
T
Static equilibria
o
m
9
_L
01
0.1
: : 3 2 Z 2 Z I 10 Z I 2 2 1.5 3 2 2 2 2
Position
Figure 4.3. Static equilibria by increasing the friction coefï¬cient 11. Here 11 is increased
from 0.0 to 0.7 by 0.07.
of the continuous system. Comparing with the exact eigenvalues the eigenvalues
obtained from the lumped-parameter model are usually underestimated, which are
explained in Chapter 6. In Chapter 2 the contradictory example in evaluation of
eigenvalues was presented. In this section, fortunately, there are no contradictory
results in evaluating the approximate eigenvalues in the lumped-parameter model
since the numerical method used in this study (MATLAB) utilizes an adjoint property
in evaluating the eigenvalues. We can expect that the numerical analysis using such
an algorithm generate reliable results.
Figure 4.3 shows the dependence of the static equilibria on the coefï¬cient of fric-
tion )1. Comparing this result to Figure 2.3, which shows the static equilibria obtained
from the continuous model, indicates similar trends. Figure 4.4 shows eigenvectors
84
0.15 7 I . a
-e— the first eigenvector
+ the second eigenvector
O 1 —+— the third eigenvector
0.05—
dimensionless displacement
s':
O
01
I
p
A
Y
2 4 6 8 1O 12 14 16 18 20
position
Figure 4.4. The non-symmetric eigenvectors corresponding to the three lowest eigen-
values.
corresponding to the three lowest frequencies. These results show a close approx-
imation to the exact eigenfunctions obtained from the continuous model in Figure
2.6. It is veriï¬ed that the non-symmetric system matrix produces the non-symmetric
eigenvectors. It is concluded that the system does not have any unstable motions by
the effects of parameter )1 assuming that the coefï¬cient of friction is constant with
respect to relative speed.
4.3.2 Addition of Damping
In this section by including damping we evaluate the dynamic stabilities and show
the effect of damping on the system stability. Three typical damping are included:
external, internal, and general friction damping. In the previous chapters we inves-
85
tigated the system stability by the effects of damping. Because it is difficult to ï¬nd
analytical solutions by including a negative-slope friction damping we did not adopt
the negative-slope friction damping in the previous model. In this section we evaluate
the system stability by including a negative-SIOpe friction damping.
The equation of motion including the damping is
miilt) + diilt) + â€ff—553171“) + 23.31“) ‘ ii+1(t)} (4-19)
+ k{—$i_1(t) + 21:10:) — $14.10)} '1' f1“) = 0,
where the parameter d, 7, and f,(t) represent the external, internal and frictional
damping, respectively. The frictional force including the eflect of Poisson’s ratio is
f1“) = HM“) (4.20)
= /L[N0 + V{k($i — 171.1) + 7(551— ii_1)}].
Let us assume the friction characteristics in friction-speed relations is
u = 14561) = sign(V - iii){us +013 - #k)e"ci"vâ€â€™"}, (4-21)
:: sign(V — 1L3){Cl + Cg e—C3IV—x',|},
where the u, and 11;, represent static and dynamic coefï¬cient of friction, respectively
(113 > pk) and the c3 has a positive value. The typical friction force f,(t), which is
dependent on the relative slipping speed between the 2'â€, mass and driving speed V,
86
-4 -5 -é —1 0 1 2 3 4
relative speed (V - dx/dt)
Figure 4.5. The discontinuous coeflicient of friction 11 versus relative velocity [V — :i:,-|.
The coefï¬cients of friction are represented by p = sign (V — in) {CI + 028—63lv_iil},
where c1 = 0.1, c3 = 1.0, and c2 = 0.1 for the dashed line, c2 = 0.2 for the dotted line,
c2 = 0.3 for the dash dot line, and c2 = 0.4 for the solid line.
is shown in Figure 4.5.
Substituting equation (4.20) to equation (4.19), the non-linear coupled equation
of motion is obtained as
5151 + ’Y{_(1 + I‘V)i'i-1+(d/'Y + 2 + â€ID-â€bi — 43:41} (4-22)
+ {—(1 + uu):r,-_1+(2 + [Ll/)IE,‘ — 1314.1} + [IND/k = 0,
where the time derivative ( ) implies 5"; with 7' = wpt, a): = k/ m = 1.
Examining the system’s dynamic stability can be facilitated by change coordi-
nates with respect to the static equilibrium state. The total displacements 23,-(t) are
represented by it,- + yi(t), where the i,- denotes the static equilibria satisfying the
equilibria status, i.e., :13,- = 515,- = 0, and the y,(t) indicates small displacement around
87
A
j
2~-~'
Imaginary part of eigenvaleus
oL
0.02
Normal loads '60 '0'02 Real parts of eigenvalues
Figure 4.6. The locus of eigenvalues with varying normal loads for the damped model
under a ï¬xed boundary condition. The normal load is increased by 5.0 N. Here
7 = 0.01, d = O, 01 = 0.1, c2 = 0.2, and c3 = 5.0 are selected.
the static equilibria. By taking the Taylor series expansion for the friction forces the
linearized equations of motion with respect to equilibria are obtained. The static
equilibria satisï¬es the following equation.
xx = F0, (4.23)
where the X is composed of [521,532, . . .,:En]T and F0 2 ——u(0)No/k[1, 1,. . ., 1]T. And
88
3 . . . 3 . . .
§ Normal load = -s N § Normal load = —1 5 N
'6 2 "I B 2 . “II
E "*l g â€H
3 1 - “a .9 1 *I
e ‘\ e ‘I‘
B '6
g 0 ‘ g 0
a 1 ï¬ 1 8. 1 i
B ' ’ n a- ' a ‘
2 ft". 2 i...
2’ -2 d" 79 -2. 0"“
E E
_3 A A a -3 A A A
—0.02 -0.01 0 0.01 0.02 -0.02 -0.01 0 0.01 0.02
Real parts ol eigenvalues Real parts ol eigenvalues
3 . . . 3 . . .
§ Normal load a -25 N g Normal load = .35 N
E 2 ’ ‘ E 2 ’
E N... g “‘1‘.
3 1 » "n < 3} 1 . "I!
0 O
a "\ l a "\
g 0r 3 0
8 1 l a a 1 l ,
E 9*". 5 fl".
'9 -2 1 1 I? -2 ’ fl 1
§ §
-3 4 A A -3 . a a
-0.02 -0.01 0 0.01 0.02 —0.02 -0.01 0 0.01 0.02
Real pans ol eigenvalues Real parts of eigenvalues
Figure 4.7. The detailed presentations of the eigenvalues by increasing the normal
loads.
the linearized equation of motion with respect to the equilibria is
Mi? + CY + KY = o, (4.24)
where M is the identity matrix, K is the same stiffness matrix to the undamped
model in (4.8), and the Y represents [311,312, . . . ,yn]T. The resultant damping matrix
89
C is
" W
cm —*y 0 0 0
-7(1 + 141(0)) C22 —) 0 0
C = 0 —’y(1 + 142(0)) C33 ... 0 0 , (4-25)
L 0 0 0 ... —*y(1 + 141(0)) cm,
where
Cï¬ = d + ’)’{2 + V(Cl + Cze—cav)} + (02638—63v)[l/{11_3i — 53;:4} ‘l' No/k]
The elements in C are evaluated with respect to the static equilibria and they are
affected by the normal loads and friction forces. Consequently, the friction forces and
the Poisson’s ratio are responsible for the non-symmetric properties of K and C. The
system matrix in the form of state space is
M*Z + K*z = o, (4.26)
where the Z is deï¬ned as [YEY]. Here the M“, K“ are deï¬ned by
C
M‘ = , K“ = . (4.27)
This eigenvalue problem should be solved by following the general eigenvalue prob-
90
lems, which are discussed in the previous section, since it has non-symmetric proper-
ties. When there is a negative SIOpe in friction characteristic (c3 > 0) the whole system
is destabilized by the resultant negative damping as the normal load increases. In
the compressive region, (17:,- — 22,..1 < 0), destabilizing effects are ampliï¬ed. Obviously,
external and internal damping have stabilizing effects as presented in (4.25).
Figure 4.6 shows the locus of the eigenvalues with varying normal loads when the
coefï¬cient of friction is expressed by equation (4.21). For small normal loads with
sufficient external and internal damping, the system is stable because the maximum
real parts of the eigenvalues is negative. When the normal load increases and produces
large compressive stress some eigenvalues placed in the left half plane approach and
cross over the imaginary axis. Then the system contains at least one positive real
eigenvalue, which indicates dynamic instability. The detailed transition steps of the
eigenvalues are shown in Figure 4.7 with varying the normal loads.
4.4 A Lumped-Parameter Model under a Periodic
Boundary Condition
4.4.1 A Stability Analysis of an Undamped System
The onset of self-excited oscillations in a continuous medium by linear instability has
been investigated by several researchers (Adams[24, 23], Martins et al.[25]). Accord-
ing to these studies a system with a periodic boundary condition becomes unstable
in the form of self-excited motions for any ï¬nite driving speeds even under a constant
91
friction coefï¬cient.
0
‘
v'|~/
Figure 4.8. A schematic diagram for the lumped-parameter model with a periodic
boundary condition.
Figure 4.8 shows a schematic diagram of the bushing system represented by
lumped elements. Assuming that any motions are expressed by circular coordinates
and any centrifugal effects are neglected (large radius), an equation of motion for the
undamped mass is
mix-(t) + k{—(1 + pu)x,-_1(t) + (2 + Vu)x,-(t) — $§+1(t)} + pNo = 0, (4.28)
where the friction coeflicient p is a constant. The periodic boundary conditions are
$1“) = $n+l(t)1 (4'29)
dxl (t) = dxn+1 (t)
dt dt ’
92
which represent the ring-shaped conï¬guration in Figure 4.8.
The system matrices consist of an identity mass matrix M and the stiffness matrix
' l
2+z/u —1 0 O —(1+uu)
—-(1+up) 2+1/ll —1 0 0
K: 0 -(1+Vp) 2+l/p 0 0 - (4.30)
—1 0 O —(1+Vu) 2+l/p.
:04“
32»
'3’
§°
:32.
Ed,»
0.1
—0.05
Friction coefficient 0 ’0-1 '
l1 Real parts of eigenvalues
Figure 4.9. The locus of eigenvalues with varying u for the undamped lumped-
parameter model under a periodic boundary condition. Here ,u is increased by 0.05.
Figure 4.9 shows the locus of eigenvalues with various values of ,u, where ,u is
assumed a constant with respect to relative speed. When ,u equals to zero, all eigen-
93
3 j Y 7 U) 3 T Y .'
§ ué0.0 3 g ; u=0.1 :
a 2 ......................................... a 2 ........... IWAW ,
> > . . _
c c . .
8,1,. 8,1. ....... ..
6 5 3 3 3 fr i
‘6 1r ‘6 1 air » rlr -
. ............................ . ......... ...... .. .
é ° * *3 ° ' :l: - 2:
o. a : :
z‘_1,.... 2‘" .......... ...*.....* .....
Q a 1 .
E V l i g» 2 3 U
E 3 E j E :
_3 i . i _3 A i .
—0.1 -0.05 O 0.05 0.1 —0.1 -0.05 0 0.05 0.1
Real parts of eigenvalues Real parts of eigenvalues
3 T Y T 3 f r
§ ; race ; a . u=03
a 2, _..mmg, % 2, "'****f**Â¥* ......
E I L . > . § , fl» .
'_1 ........... . ..
o i it t 1 '5 i : :1
‘6 air air. 3 all; ~ *-
m 0 .................... 4*“ . .. . a t o . ...* .
t iill : *1 111 *' 7 3*
a : * : : : ‘3- : 3 £3
Z._1§*, g._1 ......... * ......... *.
‘2 3 â€its. ? )1?" l .5 "all : 111*
g) _2 ................. ‘ ...... g .......... g _2 ‘ *1 **'.**¥* ,
. ‘ ; E ;
E ' —
_3 L i A _3 L A .
-0.1 -0.05 0 0.05 0.1 -O.1 —0.05 0 0.05 0.1
Real parts of eigenvalues Real pan of eigenvalues
Figure 4.10. The detailed presentation of the trajectories of the eigenvalues by in-
creasing the friction coefï¬cient.
values are located on the imaginary axis, which represents marginally stable pure
oscillations without increasing and decreasing motions. As the friction coefficient
11 increases, complex conjugate eigenvalues, which have positive and negative real
components, come into existence (Figure 4.10). Any non-zero value in the friction co-
efficient destabilizes the system. Comparing this result to those in Chapter 3, similar
trends are found. According to equation (3.6) in Chapter 3, the low frequency terms
(for small It) have a crucial role in destabilization since they have large imaginary com-
ponents in the characteristic equation. (Remember that in the analysis in Chapter 3,
the imaginary part of the characteristic solution determines overall system stabilities.
94
Refer to Figure 3.1 for the unstable solutions.) Figure 4.10 veriï¬es this trend by
showing the large positive real eigenvalue corresponding to the low frequencies. Here
the zero eigenvalue indicates the rigid body motion with no oscillations.
4.4.2 Addition of Damping
Consider the system again by including internal, external and general frictional damp-
ing. The equation of motion has the same form as equation (4.22). In order to ï¬nd
the linear stability, take the coordinate change with respect to the static equilibrium,
and followed the same procedures of the previous section. Then linearized equation
of motion with respect to the equilibria is
Mv+CY+KY=a am)
where M and K are identical to that of the undamped system. The damping matrix
is
c11 —7 0 0 —70~+vufl0)
-v(1 + 141(0)) 622 -v 0 0
C3: 0 —dun+uuan) am .n 0 0 (432)
95
where
Cï¬ 2' d + ’Y{2 + l/(Cl + 626—C3V)} + (0203e‘c3v)[u{:i:, — Ti_1} + NO/k].
3 - - 3 a .
8 Normal load = —10 N 8 Normal load = -40 N
3 3
2 — 2'
g â€'1?“ a. g â€in†alr
32 1 *i * air i 31 1 ‘ 4"all * alr
«1 fl. all. 3 I* 111*
L3 0* a at . {3 0L air air
q; 1. ‘l’ q; 1* f
a 'i *i g 1 *1! *i
?.1 » 4 _ .
g “i: all * g “at: all *
3221 «1* 822» #1
E .E
_3 a a _3 a .
-0.2 -O.1 O 0.1 -O.2 —0.1 0 0.1
Real part of eigenvalues Real part ol eigenvalues
3 . a 3 . -
8 Normal load = --70 N 3 Normal load = -100 N
3 2 . 3 2 . 1
‘3 â€a? all ‘3 â€4‘ air
c ** * . c * ï¬ . i
.31 1 ‘ all as i .31 1 ’ all alt
0 air all e all all
'5 all at *5 all all
m ‘* * a ... I
11 *§ ‘11! g 1 i"all .41 4
b-lr a - .
g {11:2 * g nil: alr *
g -2 t “ 3-2 1 fl‘
.5. E
-3 ‘ 2 —3 A ‘
-0.2 -0.1 0 0.1 -0.2 —0.1 0 0.1
Real part of eigenvalues Real part of eigenvalues
Figure 4.11. Trajectories of the eigenvalues for the damped lumped-parameter model
under a periodic boundary condition. Here 7 = 0.05, d = 0.05, CI = 0.1, c2 = 0.2, and
c3 = 5.0 are selected.
Figure 4.11 shows locus of eigenvalues with varying normal loads. When the
normal load is relatively small, the eigenvalues are placed on the left side of complex
plane, which indicates that the system is stable due to sufficient damping. However,
96
as the normal load increases the eigenvalues move toward the imaginary axis from
the left-half plane and cross over the imaginary axis at critical value of normal load.
This produces positive real eigenvalues and makes the system unstable.
4.5 Conclusion
In this chapter the discretized lumped-parameter model has been established. In the
lumped—parameter model, the stiffness matrix is non-symmetric due to the friction
force. The general eigenvalue problem, which dealt with the non-symmetric eigenvalue
problem, was reviewed and the linear stability was evaluated.
Under the ï¬xed boundary condition, the system is marginally stable when the
friction is a constant. This result is consistent with the exact results obtained in
Chapter 2. External and internal damping stabilize the system. On the other hand,
friction damping which has a negative slope in friction-speed relation destabilizes the
overall system. Under the periodic boundary condition the system becomes unstable
one even with a constant coefï¬cient of friction. This is also consistent with the results
obtained in Chapter 3.
The consistency of results from the lumped-parameter system suggest that such
a discretization of the non-symmetric problem converges to true solutions without
having contradictory results. Thus we can use this lumped-parameter model for
further non-linear studies. We will use this lumped-parameter model in Chapter 5 in
order to investigate non-linear phenomena.
CHAPTER 5
STICK—SLIP OSCILLATIONS
5. 1 Introduction
Investigation and characterization of dynamic responses of systems subjected to fric-
tion are made difï¬cult by the presence of stick-slip oscillations. Such stick-slip oscilla-
tions have been believed to be responsible for mechanisms of generating the noise and
vibrations. In explanations of such noise generating mechanisms most of the previous
researchers have devoted their efforts on system stabilities and characterizations by
using low-dimensional models.
Dynamic behaviors of multi-degree-of-freedom models including the stick-slip os-
cillations were investigated by Awrejcewicz and Delfs [89, 90]. They showed the
qualitative changes of equilibria by changing system parameters and explained nu-
merical integration techniques applicable to problems involving stick-slip oscillations.
Later, Pfeiffer [91] studied the turbine blades as a multi-dimensional stick-slip sys-
tem. Popp has explained and reviewed the previous stick-slip systems with various
97
98
examples [78, 79].
In addition, some works have focused on the characterization of stick-slip oscilla-
tions induced by friction. Stelter and Sextro [85] investigated the characterization of
one- and two-degree-of-freedom frictional systems and provided the bifurcation be-
haviors due to friction. Later, Galvalnetto et al. [86, 87] investigated the stick—slip
vibrations of two-degree-of-freedom mechanical model. They showed that the global
dynamics of the system can be characterized by the periodic, quasi-periodic, and
chaotic oscillations in presence of friction.
Most of the previous research, however, has dealt with the models having low de-
grees of freedom, and with simpliï¬ed friction models, such as the point-contact model.
A distributed friction system has been eluded in most of the previous studies since it
is hard to implement and analyze through numerical or experimental approaches.
In spite of these difï¬culties, several approaches using distributed contact systems
have been found in areas of the geophysics. The model consisting of blocks of masses
has been used to describe the earthquake fault phenomena and have explained the
dynamics of the multi-dimensional systems with stick-slip oscillations [95, 96, 97, 98].
Carlson and Langer [96, 97, 98] have incorporated stick-slip phenomena and pro-
vided mechanisms responsible for noise and the sequences of earthquake-like events.
Extended ideas to chaotic behaviors of earthquake events are found by Huang and
Turcotte [88].
Through experiment, the generation mechanisms for noise and vibrations in a
distributed contact friction system have been investigated. Vallette and Gollub [101]
studied the stick-slip oscillations of a spatiotemporal system by means of stretched
99
latex membranes in contact with a translating glass rod and measured the internal
displacement ï¬eld, u(:z:, t), with imaging techniques. They showed the experimental
behaviors of the stick-slip motions as propagating waves. Some other works related
to elastomeric friction systems are found in the works by De Togni et al. [99] and
Rorrer [100]. However, the analytical and numerical approaches for investigating
the mechanism responsible for noise and vibrations induced by friction need further
development.
Here are issues to be addressed through this study. The mechanisms responsible
for the generation of the stick and slip oscillations in elastic media subjected to a
distributed friction contacts are to be identiï¬ed. In addition, the system parameter
effects on the generation of such noise need to be investigated through this study.
In this chapter, a discretized multi-degree—of-freedom model is adopted to ana-
lyze the dynamic behaviors of the elastic media subjected to distributed friction.
We choose the lumped-parameter model established in Chapter 4, because its linear
stability is convergent and it is straight forward to simulate and analyze stick-slip
motions. The numerical techniques in handling the stick-slip oscillations are pre-
sented and the detailed explanations associated with the stick-slip oscillations are
provided. The contributions of parameters, such as a normal load, a driving speed,
the Poisson’s ratio, and friction characteristics, are studied and their influences on
the stick-slip noise generation are explained. In this numerical study we also choose
a particular set of initial conditions, which have to do with motivational topic of a
squeaky bushing. The initial conditions are chosen as stuck since that might be rep-
resentative of bushing at beginning of some maneuver. We also seek the possibility
100
of sustained stick-slip motions regardless of linearized stability in this chapter.
5.2 Numerical Aspects of Stick-Slip Phenomena
When a system is modeled as a discretized multi-degree-of-freedom and expected
to undergo alternating stick-slip oscillations during a time of interest, a numerical
algorithm to simulate the system behavior needs a special attention in handling the
problem. Since the alternating stick-slip oscillations produce a time-varying degree of
freedom and a time-varying boundary condition, an analytical approach looking for
the system behaviors is not an easy job and is limited only for a low-degree—of-freedom
system.
Theoretical backgrounds for handling such problems have been presented by
several researchers, who have formulated the switching contact status in terms of
constraint indicator functions (Pfeiffer [111], Glocker and Pfeiffer [113], and Wiisle
and Pfeiffer [114]). For example, impulsive and stick-slip phenomena for percus-
sion drilling machines (Glocker and Pfeiffer [113]), assembling and mating processes
(Pfeiffer and Glocker [112]), and frictional damping in turbine blade (Pfeiffer and Ha-
jeck [91]) have been analyzed with the constraint indicator functions. The described
phenomena, such as the stick-slip motions, the sliding-stop motion device, and the
impulsive impact process, have a common fact that the beginning and end of any of
the changing status are always represented. by certain constraint indicator functions,
which are controlled by the dynamical process itself.
Let us consider the system of interest which includes distributed friction force.
101
Though the switching status, i.e., the changes from stick to slip, or vice versa,
influences the system’s degree of freedom and results in the variable multi-degree-
of-freedom systems, their motions are simply categorized by “stick†or “slip†state
dependent on their state variables. The undamped, n degree-of-freedom, lumped-
parameter model constructed in Chapter 4 is represented by
man-(t) + k{—$i_1(t) + 211%“) — $i+1(t)} '1' f1“) = 0, 2:1, 2, . . . , n. (5.1)
This can be rewritten as
man-(t) + g,(:r,-_1,:r,-,a:,~+1) + f1“) = 0, Z: 1, 2, . . . ,71, (5.2)
where f,-(t) and g,(t) are implicitly time dependent variables. The f,-(t) is the friction
force and g,(t) is the elastic force exerted to the 2,), mass at speciï¬c time t. (See the
system conï¬guration in Figure 4.1.)
By setting kinetic and kinematic constraint indicator functions, which signal the
beginning and end of each switching status, the status of the it), lumped mass is
determined as stick or slip. The kinetic constraint indicator function h,(t) is deï¬ned
by
me) déf lfr(t)l — lg.(t)l. (5.3)
which indicates a magnitude difference of elastic spring and friction forces. Similarly,
102
the kinematic constraint indicator function 3,-(t) is deï¬ned by
31(t) (grin-31,1 = 233' — V, (5.4)
where the 3,-(t) indicates the relative velocity between the it), mass and the driving
speed V.
The condition h,(t) Z 0 with zero relative velocity (5,-(t) = 0) implies the “stick
stateâ€, which means that the it), mass remains without relative motion with respect
to the moving rigid body. In this state friction force is sufï¬cient to counteract against
elastic force, so a mass can remain without having relative motion. The condition
of non-zero relative velocity (3,-(t) 96 0) represents the “slip stateâ€, which means the
it), mass has relative motion with respect to the moving rigid body at speciï¬c time
t. In between the two states, there always is an instant in which the signs of the
constraint indicator functions changes. At this moment, called an event, the kinetic
and kinematic constraint functions are always complementary. Thus their scalar
product is always zero (Pfeiffer [111]).
Detection of events during each integration time step in digital computation needs
highly accurate numerical techniques. Calculation of the accurate switching moments,
which determine the beginning and end of the friction events causing the time-variant
or unsteady topological behaviors, influences the quality of the solutions.
The concepts of adapted integration algorithms are presented as follows. Firstly,
an e-limit should be chosen for computational tolerance. Then integration take
place over a predetermined time interval [t0, t1], during which indicator functions are
103
checked in order to detect the switching time if any events occur in that time interval.
Let us check the kinematic indicator function for the it), mass. If the relation
8i(t0)'3i(t1) < 0 (5.5)
holds during the time interval [t0, t1], this indicates that the kinematics events hap-
pened during the interval. Then an adapted integrated time ts, is evaluated from the
equation
I 3i(t8l) I < 6:1 (5'6)
where the 15,, should be detected by several backstepping iterations to satisfy the
equation (5.5) within the kinematic accuracy limit 6?.
Similarly, the kinetic constraint indicator function is checked. If the relation
hi(t0)'hi(t1) < 0 1 (5-7)
holds during the time interval [t0,t1], an adapted integrated time th, is evaluated
based on the equation
I hi(thi) I < 6? (58)
in order to satisfy the kinetic accuracy limit 6?.
Considering a n degree-of-freedom model with 72 possible constraints, the time
104
instant for which a change of status ï¬rst occurs should be determined. The smallest
time step in kinematic event t is deï¬ned by
3min
t min (ts, | s,(ts,) = 0}, (5.9)
s - 2.
m'n 2:1,2,
and the smallest time step in kinetic event it), is also deï¬ned by
min
th = min {thsI hi(th,)=O}. (5.10)
mm i=l,2,...,n
Therefore, the ï¬nal smallest adapted time integration step t f is selected by
tf = min{t3min’ thmin}' (5'11)
After ï¬nding the ï¬nal adapted time integration step t f, the ending time is set to
t f instead of t1 in integration time interval on [to, t f].
5.3 Stick-Slip Oscillations with Fixed Boundary
Conditions
Numerical investigations for stick-slip oscillations using the lumped-parameter model
subjected to a distributed frictional contact are conducted to seek mechanisms related
to generating noise and vibrations. In this section descriptions of system conï¬gura-
tions and investigations related to stick-slip oscillations are presented.
105
5.3.1 Conditions of Numerical Simulations
In this study the adaptive step size Runge-Kutta—Fehlberg integration methods using
a fourth and ï¬fth pair, which have been proved to have high accuracies with small
time steps, are used in numerical integrations. In order to acquire the high quality
simulation results small predetermined integrations time steps are selected. Regarding
Normal Load Friction Mass Stiffness Poisson’s ratio Driving Speed
N0[N] c1 c2 c3 m [kg] k [N/m] V V [m/s]
-1.0 0.1 0.2 0.1 1.0 1.0 0.4 +1
Table 5.1. The typical system parameters selected for numerical simulations in Chap-
ter 5.
to the system conï¬guration, twenty blocks of lumped-masses connected with linear
springs are placed under evenly distributed normal loads. (Mass positions are assigned
from left to right direction in Figure 4.1.) In addition, ï¬xed boundary conditions
are imposed at both ends. Neither external or internal damping is included in this
development. The friction-speed relations, which have primary effects on the stability
and dynamic behaviors, are expressed by equation (4.21), i.e.,
u = Ma's.) = sign<V — one. + c. e'cs'l'rii'}. (5.12)
Since we have already investigated Coulomb friction effects on system stability we
choose the speed-related friction model in this chapter. The typical parameter values
106
selected in numerical simulations are summarized in Table 5.1. Throughout this
chapter, the same initial conditions are imposed for all numerical studies: zero relative
velocities and zero initial displacements to all masses. The set of initial conditions
might be a representative of bushing at beginning of some maneuver.
5.3.2 Investigations of Stick-Slip Oscillations
In this section stick-slip oscillations of the one dimensional multi-degree-of—freedom
system are presented and their dynamic characteristics are explained based on their
numerical results. Investigations based on displacements and velocity responses with
typical initial conditions in Table 5.1 are performed.
Analyses based on the Velocity Response
Figure 5.1 shows the time evolution of the mass velocities. At the beginning
of the time evolution, all masses move together with the moving rigid body. For
better presentation the velocities are placed on the negative velocity axis, so —1
[m / 3] indicates the driving velocity. The potential energies in the left and right ended
springs gradually increase as the rigid body moves. There are particular moments,
such that the increased spring forces are no longer resisted by the counting frictional
forces. At this moment the masses begin to slide on the driving rigid body, which
implies the stick to slip event.
The sudden changes in motions from the stick to slip state give the momentum
to the neighbor masses and sometimes can trigger series of events in a short time like
falling “domino blocksâ€. The slip motions are shown as the peaks in Figure 5.1.
107
N
4
_L
1
â€II “II lll‘I INN
1IIII I II' IIII'IFI II II(I
IiIII-‘I l’I.I‘ [I'IIII IIIIIIII ‘HII III4 I“ III] III â€â€˜0’? II."
(-) velocity [m/s]
0
N I 1I I I IIl II‘W“ .
WJJIIII“ IiI III IIII I II‘I’I II III I,“ "Iâ€I lIII "MI‘IIIIV‘IL'IIII SI'L‘III‘Q
‘1‘ IIIIIIIlI I M 'w. v'll’l' “llll‘l’†'ll’II‘l"
lid
—2>
20
150 200
1 0 100
50
position 0 0 time [sec]
Figure 5.1. Velocity responses of stick-slip oscillations for the lumped-parameter
model. (The selected parameters are in Table 5.1.)
Phenomena which are similar to that events are also found in nature, for examples,
avalanches in a pile of sand and an earthquake fault phenomena. When small amounts
of sand are added on a pile of sand very slowly, it is expected to exhibit avalanches
once a sandpile achieves steady state.
The earthquake fault phenomena have similar structures. Series of the events of
stick-slip motions, sometimes happened only in localized regions or over whole do-
mains of contact, are generated and ampliï¬ed, which result in the earthquake fault
phenomena (Huang and Turcotte[88]). Carlson and Langer [97, 98] and Carlson et
al. [96] investigated that possible sizes of events and slip wave instabilities for explana-
tion of catastropical events of earthquake faults by using the Burridge and Knoppoff
model, which consists of inï¬nite masses connected with springs. They showed that
108
1 00
time [sec]
Figure 5.2. A contour plot of the stick-slip response in velocity. (The selected pa-
rameters are in Table 5.1.)
such events from stick to slip are responsible for generations of the noise, which are
based on the nature of frictional characteristics (Carlson et al. [96]). Generally, the
difference between static and dynamic friction coefï¬cient (i.e., a condition of u, > pk)
produces sudden changes in accelerations. The changing forces can influence the
neighboring masses. When the systems have large discontinuities in the static and
kinetic friction characteristics, their effects on the neighbor masses are generally in-
creased. Detailed explanations about the stick-slip motions related to the friction
parameters are shown in the next section.
With the presence of the Poisson’s ratio in elastic materials, the stick-to—slip events
are more apt to occur in axial tensioned regions then compressive regions. Since in
the tensioned region the potentially countable static friction forces are reduced by
109
g5 T H H
8 4 _
s i
E 3 ~ ‘
g ‘ . u'A'.
82‘ , ¢,.:I‘7"""L‘§S‘::®~§ \
r, :~.i~:.r~"£'.'t"... â€N‘?»
g ggkw 20
o, N
O 10
()05 OJ
(115 0:2
' 0 position
frequency [Hz]
Figure 5.3. A power spectral density of the velocity responses. (The selected param-
eters are in Table 5.1.)
effects of the Poisson ratio, the stick-to-slip events are more likely initiated from
the axially tensioned regions. The stick-to—slip motions, called detachments [26], are
ï¬rst triggered in the axially tensioned regions, and propagate toward the compressive
regions. Thus the series of the detachment motions, which are like propagating waves
of detachments, travel over the contact domain. They may collide each other and
bounced back from the boundary conditions. Shallamach [26] has observed these
detachment waves in his experimental works by rubbing a rubber on a hard track.
The series of stick-to—slip events are ampliï¬ed from the local motions to the whole
scale motions. Such stick—to-slip phenomena over whole regions are believed to be
responsible for frequencies of the noise. The frequency dependency on the stick-slip
motions are investigated in the next section.
110
20w
—l
01
1
:’
:i’g
:’;;
i=5.‘
displacement [m]
C
1
3%.
Egiflflit
‘ - :4"
‘9
\
N
O O 01
v I
k
t
K:
{ii
I?â€
tit;
53‘
E
3F;::
“9 ‘L?
43
"
â€Ii.
200
10 150
1 00
50
position 0 0 time [see]
Figure 5.4. Displacement responses of stick-slip oscillations for the lumped-parameter
model. (The selected parameters are in Table 5.1.)
Figure 5.2 shows the contour presentation of the velocity responses. The series
of stick-slip events, which are shown as the crossing lines from the one boundary to
the other boundary, are observed distinctly. The high slipping velocities are observed
around the rear masses since the presence of the Poisson’s ratio increase the friction
forces for the rear masses. (Refer to a conï¬guration of the mass position in Figure 4.1.
The rear masses are deï¬ned the masses positioned at the end of driving direction.
Thus in Figure 4.1 the masses positioned at right hand side are the rear masses.)
The power spectral density diagrams (FFT) of the velocity responses are shown in
Figure 5.3.
111
strain
200
position
0 0 time [sec]
Figure 5.5. Strain presentations from the displacement responses. (The selected
parameters are in Table 5.1.)
Analyses based on the Displacement Response
Figure 5.4 shows the displacement responses of the stick—slip oscillations. All the
masses oscillate with respect to their static equilibria. Each mass experienced low
frequency stick-slip oscillations accompanied by high frequency oscillations. The stick—
slip oscillations are not distinguishable in Figure 5.4. By considering the strains,
deï¬ned by Ax,(t) = x,(t) — mi+1(t), the releasing strain energies are observed in
Figure 5.5.
The strains, which represent the potential energies stored in the connected springs,
are released abruptly under the stick-to-slip events in a short time interval. These
are seen as the propagating sharp waves of relieved strains. The waves sweeping over
the domains represent the series of relieved energy over the stick-to—slip events. The
112
..
it ‘
Q.
Q. ï¬ O in! -
em 0. ï¬ e I
m I. O O i —
on O. ï¬ 0 C O O
O Q C it C d. O -
O 4. Cr Ci. 0
O O CO. C C -I -
t O O -O
G lid. 9 O - -
.3
G I I -
C
. _
50 100 150 200
time [sec]
Figure 5.6. Sticking events versus time. The mark ‘*’ indicates “the stick stateâ€
and the others (the blanks) indicate “the slip state†for each mass. (The selected
parameters are in Table 5.1.)
propagating wave speeds are influenced by system parameters and are investigated
in the next section. According to the works by Carlson and Langer [97, 98], the
small spatial inhomogeneities in displacement are ampliï¬ed during the large scale
stick-to—slip event.
Figure 5.6 shows the time evolution of the sticking regions. The ‘*’ marks posi-
tions with sticking status in a speciï¬c time. The state-space (displacement-velocity)
presentations for 5th, 9th, 13th, and 17th positioned masses are shown in Figure 5.7.
The each mass experienced stick-slip oscillations with respect to its static equilibrium
position.
113
position = 5 position = 9
velocity [m/s]
velocity [m/s]
20
1 0 1 O
displacement [m] displacement [m]
Figure 5.7. State-space (displacement versus velocity) presentations for several po-
sitioned masses. (The 5th, 9th, 13th, and 17th positioned masses are shown.) The
selected parameters are in Table 5.1.
5.4 Parameter Effects on Stick-Slip Oscillations
Limited studies have been conducted to predict, analysis, and control the stick-slip
oscillations resulting the chattering and squealing noise and vibrations. Moreover,
experimental investigations of the previous studies have not clearly explained the
general mechanisms generating the stick-slip noise and vibrations induced by friction
since the experimental results and analyses have been closely related to the experi-
mental apparatuses and operating environmental conditions, such as contact surface
conditions and small geometrical misalignment of contacting materials. These may
generate non-repeatable responses even in the same operating conditions and may
cause difï¬culty in doing system analyses via systematic approaches. Explanations of
114
the stick-slip mechanisms using mathematical models have not been fully achieved so
far, especially for the parameter effects on distributed friction contacts, and system-
atic approaches for the analyses of the noise generation mechanisms have not been
conducted as well.
In this section, analyses and interpretations of numerical results and discussions
associated with system parameter effects, such as the normal load, driving speed,
Poisson’s ratio, and the friction characteristics, are provided.
5.4.1 Effects of Normal Loads
The experimental and analytical approaches in the previous studies have shown that
the normal load has a primary influence on the ability to generate noise, and the
frequency and intensity of the noise. Generally, the friction-induced noise are dis-
tinguished by the chattering and squealing noise according to their frequencies and
intensities of noise signals.
The squealing noise is characterized by the high frequency with small amplitudes
of oscillations, while the chatter is usually generated at low frequency with relatively
large amplitudes and much higher intensity than the squealing noise. For example,
in the experiment associated with a rubber-bearing noise investigation by Bhushan
[73], the noise which had low frequency characteristics, namely chatter, typically had
frequencies around 30 to 310 Hz with the maximum amplitude of 15 pm. On the
other hand, the high frequency squealing noise, which had 770 to 830 Hz, had the
maximum amplitude of 1.5 pm. Though these frequency values were not typical
115
frequencies representing the chattering and squealing noise, but they were totally
dependent on system parameters, especially the normal load. It should be noted that
the chattering noise had large amplitude with low frequencies, while the squealing
noise had relatively high frequencies and small amplitude from the real experimental
results. In the following discussions, the low frequency noise is termed “chatterâ€
and the high frequency noise is termed “squeal†for explanations of the noise signal
characteristics.
Figure 5.8 shows responses of displacements of each mass by increasing the normal
load with —1,—5, and —10 N. The static equilibria for the masses are changed by
the normal load. The static equilibria have unsymmetric shapes along the spatial
axis and their unsymmetries are ampliï¬ed as the normal load increases, as can be
recalled from the static equilibria for the continuous system in equation (2.6) and for
the lumped-model in equation (4.7).
Normal Load Mass position
N0[N] 5th mass 9th mass 13th mass 17th mass
-1 7 12 14 10
-5 12 17 17 I5
-10 25 40 45 25
Table 5.2. The approximate peak-to—peak amplitudes in displacement responses by
changing the normal load.
In Figure 5.9 trends in the velocity response with increasing normal load are shown.
The maximum velocity magnitudes are increased as the normal load increases. For
116
Normal Load Mass position
N0[N] 5th mass 9th mass 13th mass 17th mass
-1 1.5 2 1.7 1.8
-5 2.5 3 3 3.5
-10 3.5 5.5 9 7
Table 5.3. The approximate peak—to—peak amplitudes in velocity responses by chang-
ing the normal load.
an example, comparing the maximum velocity the velocities in the slipping state
reach about 2, 3, and 8 m/sec magnitude when the normal loads are —1 N, —5
N, and —-10 N, respectively. (Recall that (-) velocity responses are plotted in Figure
5.9.) Moreover, the amplitude of the stick-slip oscillations (measured by peak-to—peak
amplitudes during the stick-slip oscillations) of the displacements and velocities are
increased by the normal load increases. As shown in Figure 5.10 of the displacement-
velocity presentations of several masses (the 5th, 9th, 13th and 17th positioned mass),
the approximate stick-slip oscillating amplitudes are increased in high normal load.
Table 5.2 and 5.3 show the summarized approximate peak-to-peak amplitudes in
displacement and velocity responses, respectively.
With a closer look in Figure 5.9 and 5.10, small amplitude high frequency oscil-
lations, which are usually superposed on the low frequency responses, are observed.
Especially, under the high normal load condition, these small amplitude stick-slip
oscillations, called as “creep motions†by Carson and Langer [97 , 98], dominate in re-
sponses along with the low frequency responses. They attribute the high frequency of
stick-slip oscillations, which implies the partial relaxation of strain energy in the form
of “small-grouped motionsâ€. In this numerical response the localized small-grouped
117
motions are observed under the high normal load condition. Such oscillations are
responsible for the high frequency noise (the squeaking noise) under the high normal
load. These phenomena can be analyzed by using frequency analysis.
118
20
E1 5 "Qt-"Aft†wit)â€.
.. ,ciivrti'l'liillli’illi’l‘l’i’l’l'ii“
“ \ï¬â€˜ll' â€LS“! .l‘ll 1“ ‘i
5 iiiiï¬ikiiiggt.egiiyi'aiili’l‘it ‘
.4 «it, '~ '~. ~.
28 M \" “â€r,
200
position 0 0 time [sec]
(a)
so
E60 . . . -Io'*.‘v".":-\-,.
:"v.;’!92"1‘“.-<'é’r 9"
w u
20 III], "
0
20 200
Figure 5.8. The stick-slip displacement responses for normal loads of (a) N0 = —1 N,
(b) N0 = —5 N, (c) N0 = —10 N. The other parameters are in Table 5.1.
119
H VGIOCW [m/Sl
position 0 0 time [sec]
lull 1'
willllâ€ll"\“ll l i ll
,. .
â€Ni l‘.il(,,..w,li,w ll‘l ‘
(-l velocity lm/Sl
f â€HI
1, ““ ’
‘ilzl'l
10
50
position 0 0 time [sec]
(-) VGiOCiill [In/S]
position 0 0 time [sec]
(C)
Figure 5.9. The stick-slip velocity responses for normal loads of (a) N0 = —1 N, (b)
= —5 N, (c) N0 = -—10 N. The other parameters are in Table 5.1.
120
position = 5 position = 9
9
veliicily [m/s]
O 1 O 20
position a: 13
s‘ va'srv‘rv‘m
a ., twill
E‘ . i‘ It“ I
.8 —1 \ti‘ ‘\._l
2 tel/"J
’20 1 o 20 1 o
displacement [m] displacement [m]
(a)
position - 5 position — 9
velocliy [m/s]
velocllYlm/SI
_2 —2 r
O 20 4O 60 O 20 4O 60
position - 1 3 position — 17
1
E o»
A?
—1
s
O 20 4O 60 O 20 4O 60
displacement [m] displacement [m]
position - 5 position - 9
—~ 0 1 0—0 0 I
U)
3‘2 25-2 .
— i _6 >
s 6 ‘ s
—8 . —8
O 50 1 OO O 50 1 00
position a 13 position = 17
'Y
velocity [W5]
0 o L N O
50 1 OO O 50 1 OO
displacement [m] displacement [m]
(C)
Figure 5.10. Projections of state-space trajectories (displacement versus velocity) for
several positioned masses (the 5th, 9th, 13th, and 17th positioned masses) for normal
loads of (a) N0 = —1 N, (b) N0 = -—5 N, (c) N0 = —10 N. The other parameters are
in Table 5.1.
121
As shown in Figure 5.11 the intensity of the low frequency spectra, which are lower
than 0.05 Hz, increases as the normal load increases. This is a reason for the chattering
noise has low frequencies under the high normal load condition. In addition, the high
frequency terms are also affected by the normal load. The intensity of high frequency
spectra increases as the normal load increases because of the prevailing small-grouped
motions in stick-slip oscillations. (The lowest natural frequency of the linear model
is 0.0328 Hz.)
As the normal loads increases, the mass positioned in the middle domain (around
the middle positioned mass over the twenty masses in this response) experiences low
frequency stick-slip oscillations. On the other hand, the masses positioned around
the boundary ends show the high frequency oscillations even under the high normal
load condition. Note that this stick-slip phenomenon totally depends on the applied
boundary conditions. Thereby, if the boundary conditions are changed different stick-
slip oscillations are expected.
Consequently, as the normal load increases from —1 N to —10 N, the high normal
load contributes to both low and high frequency terms in the noise generated from the
stick-slip oscillations. In other words, as the normal load increases the low frequencies
of the signals, which are originated from the propagating waves of detachments, are
lowered (Dweib and D’Souza [55, 56], Nakai and Yokoi [72, 57], and Bhushan [73]).
On the other hand, the high normal load increase the high frequencies of signals,
which generally come from the stick—slip oscillations with the localized small-grouped
motions.
Strain distributions over the domain can give clear explanations about the strain
122
recoveries. Figure 5.12 shows that the speed of the sweeping strain relief waves are
lowered as the normal load increases. On the other hand, the creeping motions in
part (c), which are characterized by high frequency small-grouped motions, usually
can not give enough strain relief compared to the effects of propagating wave of
detachment represented by the crossing lines in Figure 5.12. The propagating waves
of detachments have been also observed in the work by Carlson and Langer [97]. They
claimed that such motions give large irregularities in strain of the elastic materials
after experiencing the stick—slip oscillations. The irregularities are ampliï¬ed after each
event, so the system may experience catastrophic events or chaotic behaviors under
the stick-slip oscillations.
By presentations in contour plots, shown in Figure 5.13, the high and low velocity
regions are clearly distinguished on the two-dimensional plots. The sweeping waves of
detachments are shown as the deflected lines reflecting off the boundary ends. Under
the high normal load conditions in Figure 5.13 (c), the transient motions from the
initial condition to the formation of the propagating waves of detachments are shown
in detail. At the beginning stage in Figure 5.13 (c), from 0 to about 100 seconds of
the simulation time, the small-grouped stick-slip motions are initiated and gradually
propagate toward the stick regions by repeating the stick and slip â€oscillations. Because
that repetition of the stick-slip oscillations can not release enough strain energies
the localized stick-slip motions around the both boundary ends are observed. Such
repetitive stick-slip oscillations, which has begun from both boundary ends, proceed
until both detachment waves meet each other. At around 110 seconds in Figure 5.13
(c), both proceeding waves of detachments collide and produce large strain reliefs
123
over the entire domain. After such transitions of strain reliefs, ï¬nally the large strain
relief motions are observed in the form of the waves of detachments over the entire
domain. Such phenomena can explain the generating mechanisms of the low frequency
chattering noise.
Effects of the normal load on the sticking area, i.e., the contact area, are shown
in Figure 5.14. The high normal loads make possible to produce the large sticking
areas, which are shown as the extend areas in Figure 5.14 (c). Under the light normal
load condition, shown in Figure 5.14 (a), the relatively small sticking areas localized
in the middle-positioned masses are observed. However, as the normal load increases
(—5 N, —10 N), the sticking areas are enlarged and almost extended to the boundary
ends (Figure 5.14 (b) and (c)). These phenomena can be explained by the fact that
high static friction force capacity which can hold the masses with no relative motions,
f,(t) in equation (5.2), is increased under the high normal load condition.
On the other hand, under the light normal load condition, stick-slip oscillations
are rarely observed since the kinetic constraint indicator functions in equation (5.3)
can not easily satisfy the sticking condition. Thus in this condition the oscillations
can not experience the stick-slip oscillations and the motions are observed under the
pure slipping motions with relatively high frequencies than the stick-slip oscillations.
These phenomena are conï¬rmed in Chapter 6 by ï¬nite element analysis.
5
&
1_.J
(n
1
L
WSP'gdelalWlY
0.05
124
I lit
‘9‘? -
slim.» W,
éw 2°
\ 1 o
. 0.2 0 position
frequency [Hz]
(a)
. iii
iiiiliiiitiii‘g‘i‘ is t. - .
A ‘.'7"|:; , , â€A
Mi. vi. «as: tsï¬ss‘adw 20
&W« C?’ k'ï¬Â»
WW 1 0
O. 1 0.1 5 ‘W'
02 0 position
frequency [Hz]
it...
. “V,“ ngti¥3§iiyw~ ‘ ..
' ï¬. 5 S O o \ :-\‘0:‘-‘§‘,. . _ A
‘ v ‘, KM“ , VOQW «read»
“ $V¢¢Q$W C%% 20
a. ~o W‘W
v . W 1 O
0'1 o 15 M5" ,
0'2 0 position
frequencv [Hz]
(C)
Figure 5.11. Power spectral density of the stick-slip velocity responses for normal
loads of (a) N0 = —1 N, (b) N0 = —5 N, (c) N0 = —10 N. The other parameters are
in Table 5.1.
125
100
position
50
O 0 time [sec]
100
position
0 0 time [sec]
100
position
0 0 time [sec]
Figure 5.12. Stick-slip strain responses for normal load of (a) N0 = —1 N, (b) No — —5
N, (c) N0 = —10 N. The other parameters are in Table 5.1.
An
‘v
18
16
14
Ne0
126
Nil 5.15â€"“t
l
l.'
"til it).
s ‘ . I Itit
. . . 14;“
l H ,l‘
n t .1
. \. t . , ~
\
i
o. anode ML... _ a
0 0
(b)
.Qï¬.
1 60
time [sec]
(C)
Figure 5.13. Stick-slip responses in contour plot of velocity for normal load of (a)
N0 = —1 N, (b) N0 = —5 N, (c) N0 = —10 N. The other parameters are in Table 5.1.
127
20
1 84. ea- _
«b e-
1 61- e. e - .- -
_-- o... e- o u c-
1 41†e. o- e o o —
'—. a... e- e e - - -
1 21— o e - e - «I. - ~
- ¢—. - o - ea. -
i1 04—. - - ..- - - - - _.
q—e . - - -e
81—- e on»- - e — T
4— C.
61- . fl - -l
1- -
4i. - —i
«b
2 ~ ~
0 5O 1 OO 1 50 200
time [sec]
(a)
20 f .
-- e-- - e eon- - ------ - -di
1 81D..- .- ...... - ...- - -... - e-
i... - â€......†...-o e e- ...- -o .. e
1 6cm... ...... ... .e .o o».-- i)
4—. e. ... .- -- - - ..- et
1 4 .... — - - e .- u
— _ .-
1 2 ...- ... o - -
. e o..- .. - o -
g1 C - e e.- - - - -- ..
== - .- - -
8‘—. e. c... - e - -
qm- e e e - -e e. «-
61b. - e o o o e —
«_- - e. e - e b
4‘.- e - - -
1p†-
2 - -
O 50 O 1 50 200
time [sec]
20 .
-- o - - o- -- - - -di
1 84p... . --- a... - .. ... m - ... e... e. - m
«b— . ...-e... ... e . e ... u .... e...- m e d
1 c - ...... ... -... -... ....ee... .. - q}
= - .- -... . - ...... one. a»
1 4 : -.. ...- .- - e - q»
- — m. .1»
1 2 - -p
. e .
E1 C — —- :p
.— . _e -
8 .- .. e.. m. .-
-- .- .. .. me e ‘i
61—. — - .- .o _- —
4.— - - - - _..
44.— .. - - e - - e .1
1... e e - e -
2 .. e - -l
O 50
1 00
time [sec]
(C)
Figure 5.14. Sticking events versus times for normal loads of (a) N0 = —1 N, (b)
N0 = —5 N, (c) N0 = —10 N. The other parameters are in Table 5.1.
128
5.4.2 Effects of Driving Speed
Figure 5.15 shows numerical results in the velocity of each mass as the driving
speed V increases. Parameter values are based on Table 5.1. In this numerical
analysis a negative slope friction model is adopted. As the driving speed increases
the maximum velocities are increases. For V = 0.1, 0.5, and 1.0 m/ sec, the magnitude
of maximum velocities of 0.2, 1.2, and 2.0 m/sec are obtained, respectively.
In the low driving speed (V = 0.1 m/ sec), the small-grouped stick-slip motions are
dominated as shown in Figure 5.16 (a). This means when the masses are driven by
the low driving speed, the slipping velocities are usually low since the kinetic energies
stored during the stick state are relatively low compared to the high speed driving
condition.
When kinetic energy is not enough to provoke the series of triggering events from
stick to slip motions to neighboring masses, this appears as intermittent stick-slip
oscillations in the form of small-grouped motions. They are shown as the creeping
motions in the beginning of simulation time in Figure 5.16 (a). Thus under the low
driving speed, the intermittent small-grouped stick-slip motions are expected to be
observed. Under very low driving speeds it is expected that the distinct low frequency
stick-slip motions expressed in the form of waves of detachments may not be obtained
since the small-grouped, high frequency, creeping motions are generated. In real
situations, having large material damping, this high frequency can easily be damped
out by the system internal damping, so the noise may not be a serious problem.
On the other hand, under a very high driving speed, the system can not experience
129
steady-state stick-slip motions, but it can only undergo pure slipping motions. In
this case the system experiences high frequency slipping motions generating high
frequency noise. The frequency spectra of the velocity responses and projections of
the state-space trajectories for selected masses are shown in Figure 5.17 and Figure
5.18, respectively.
Driving Speed Mass position
V [m/ sec] 5th mass 9th mass 13th mass 17th mass
0.1 1.0 1.1 2.0 2.0
0.5 3.7 7.0 8.0 6.0
1.0 7.0 14.0 17.0 11.0
Table 5.4. The maximum peak-to—peak amplitudes in displacement response by
changing the driving speed.
Driving Speed Mass position
V [m/sec] 5th mass 9th mass 13th mass 17th mass
0.1 0.15 0.2 0.25 0.3
0.5 0.8 1.0 1.3 1.5
1.0 1.7 2.1 2.7 3.1
Table 5.5. The maximum peak-to—peak amplitudes in velocity responses by changing
the driving speed.
According to this investigation, the high driving speed increases the amplitudes of
oscillations in stick-slip motions. In Table 5.4 and 5.5 the approximate peak-to—peak
displacement and velocity amplitudes in stick-slip motions are presented, respectively.
130
As the driving speed increases the amplitude of stick-slip oscillations are increases.
It should be noted that such responses are affected by the frictional properties. The
dependency on frictional properties are shown in the work by Martins et al. [118].
According to his work, as the driving speed increases the stick-slip amplitudes are
decreased when they adopted the positive slope friction model.
131
,. lzimi l, N
'j,-""v'.- 9M“
,,' '1... ill: 9 iv.,."'.ljau‘:'
(-) WNW [In/S]
O
N
A" "V - . . i , . I . , ‘- If."
0 r . I v . T ‘ ..l ‘ ‘3 “luv: i,nï¬lill‘ll‘lhy‘l'“linil-“lllilsltl’l‘uâ€
r u w‘w Hug; ‘ .. t14".:.Elt‘.-*‘.Li‘av'—‘-"'
-o.2 . â€1"\v'\V'.'\'..'". 3'13 r.“ -‘->‘ "
',i' '.“\ '4‘th 7“
20 W _ _ .
position 0 0 time [sec]
(-) VGIOCW lm/Sl
position 0 time [secl
(—) velocity [m/s}
position 0 0 time [sec]
Figure 5.15. The stick-slip responses for driving speeds of (a) V 2 0.1m/s, (b) V =
0.5m/s, (c) V = 1.0m/s. The other parameters are in Table 5.1.
132
-
I... A
q q q .mu m d _
...._ . .
.........
e
2:..:.
. . ....
. .. ...
....†......
.
4
2
1C
6
4
2—
1
1
200 300 400
time [sec]
100
4 — _ — _ A 4 d.
........
.....n...
...—...._ .
..........
..........
........ .
¢o
.: . .
n m. e. .-
..——_—. .-_.
.
200 300 400
time [sec]
100
-
.22... .
200
time [sec]
100
400
300
0.1m/s, (b) V = 0.5m/s,
Figure 5.16. Sticking events for driving speeds of (a) V
(0)1â€
1.0m/s. The other parameters are in Table 5.1.
133
$0.047
Milli
EOIJZJ “Klat§=:;_.‘.&‘= ‘ A
0 o. 1 o
O. 1 0 position
position
0.05
pOSitiOn
frequency [Hz]
Figure 5.17. Power spectral density of the stick-slip velocity responses for driving
speeds of (a) V = 0.1m/s, (b) V = 0.5m/s, (c) V = 1.0m/s. The other parameters
are in Table 5.1.
134
position - 5
position — 9
0.1 o_1 ï¬v
77:" '7?
E. 0' E 0’
g-OJ > §—0.1 ' l
9
—O.2 ‘ —O.2
O 5 1O 15 0 5 10 15
position - 13 position -= 17
0.1 0.1
'3' '2?
E 0* 3:. 0
.E' .E‘
gs —o.1 » _8 -o.1 l
0) CD
> >
—O.2 —O.2 <
5 1 O 1 5 O 5 1 O 1 5
displacement [m] displacement [m]
(a)
position — 5 position - 9
0.5 0.5
g—o 5 g—os
g —1 i g —1
“1'50 5 1o 15 ’1'50 5 1o 15
position - 13 position a 17
0-5 2: WM, y— 0-5 VV
75* i w ,
E ° ( 1W) E °
.e-o 5 \..&o:‘. ter. .5—0 5r
.3 w _
g —1 g —1
—1 .5 -1 .5
5 1 O 1 5
displacement [m]
position a 5
O 10 20
position - 13
1
E o
> _2(
O
1 O
displacement [m]
5 1 O 1 5
displacement [m]
position — 9
1 O 20
displacement [m]
(C)
Figure 5.18. Projections of state-space trajectories of the stick-slip response by chang-
ing the driving speeds of (a) V = 0.1m/s, (b) V = 0.5m/s, (c) V = 1.0m/s. The
other parameters are in Table 5.1.
135
5.4.3 Effects of the Poisson’s ratio
The Poisson’s ratio effects on the linearized system stability have been investigated
in the previous chapters. Here let us check Poisson effects on stick-slip oscillations
using the lumped-parameter model. The stick-slip regions for the several different
Poisson’s ratio are shown in Figure 5.19. For a case 1/ = 0.0 in Figure 5.19 (a),
the system is symmetric in the stick-slip oscillations. These phenomena can also be
conï¬rmed by checking the strains, as shown in Figure 5.20 (a). The shapes of stick
regions and their strains are symmetric, and this symmetry is preserved in time.
However, when there is a non-zero Poisson’s ratio, the system can not preserve
its symmetry in the stick-slip behaviors. It is reasonable to expect such unsymmetric
stick-slip behaviors because the eigenvectors have non-symmetric shapes along the
spatial axis in the lumped-parameter model. Strain proï¬les in Figure 5.20 show that
the strain evolutions are distorted as the Poisson’s ratio increases. The asymmetric
and propagating proï¬les are also found in Figure 5.21 presenting with contour plots.
Moreover, the Poisson’s ratio can contribute to the frequencies of stick-slip oscil-
lations. The frequencies of stick-slip oscillations are increased as shown in Figure 5.19
and Figure 5.22. Since the system’s modal frequencies are increased as the Poisson’s
ratio increases (refer to the result in section 2.3 of this study), it is obvious that
the stick-slip oscillation with non-zero Poisson’s ratio have relatively high frequencies
than the case of zero Poisson’s ratio.
State-Space presentations for the masses positioned at the 5th, 9th, 13th, and 17th
are shown in Figure 5.23 and their approximate peak-to-peak amplitudes in displace-
136
Poisson’s ratio Mass position
11 5th mass 9th mass 13th mass 17th mass
0.0 9.0 16.0 16.0 7.0
0.1 8.5 16.0 16.0 7.5
0.4 6.5 12.0 15.0 10.0
Table 5.6. The maximum peak-to-peak amplitude in displacement responses by
changing the Poisson’s ratio.
Poisson’s ratio Mass position
v 5th mass 9th mass 13th mass 17th mass
0.0 2.1 2.5 2.4 2.0
0.1 2.0 2.4 2.4 2.2
0.4 1.5 2.0 2.7 2.8
Table 5.7. The maximum peak-to-peak amplitudes in velocity responses by changing
the Poisson’s ratio.
ment and velocity are summarized in Table 5.6 and Table 5.7 versus the Poisson’s
ratio. For the front positioned masses (in this example the 5th and 9th positioned
masses) the peak-to—peak amplitude of displacement and velocity are decreased by in-
creasing the Poisson’s ratio. However, for the rear positioned masses (17th positioned
masses) the peak-to-peak amplitudes of displacement and velocity are increased by
increasing the Poisson ratio. The asymmetry due the Poisson’s ratio changes the re-
sultant friction force and apparently increases the friction force at the rear positioned
masses.
137
20 v . m
1 a» 1' 1
c. "
1 6‘- - .. _,
4— - e. -
1 44—- e. e. e. e. u»
q—. .. .. .- .— «I
1 21—. - .- - - -
- 4— - e- - — -
g1 04— - e- - - -
4-. - .- - -
84—. e. .e e. .. 4.
(—- e. e. .. as. up
64- e e. - -«
q- - ..
44l- "' "
1P "’
2 ~ . r
O 50 1 00 1 50 200
time [sec]
(61)
20
1 an»
1.
1 61- - as. W»
{—0 e e .. - -
1 44—- e. .e .. -e -
1—. .. .. ... -- Gd!
8 1 21—. - .. .- -- .—
l_ - e- - - d
1 o:— - .- .. -.. .4
<—. - .- . ... - «i»
84—- e. .. .- up
1—e e .e .. -- -
6‘- - .1
I- - O I»
41. - -
1) * J
2 ..
O 50 1 OO 1 50 200
time [sec]
0))
20 v
1 sq. e- —4
(p e-
1 61- O- D - -e -4
{_-- e... -- e e e
1 41p†1... e- e e e -«
cm. a... -- e e - - e
1 21— e e - e - an. - -<
c—- - e - --- -
i1 04—- - . ..- - - - e —
c_. e - - --
81—. e «be- - e - —
l— ..
61- e . e -<
l- C
41. - A
1D
2 - .
O 50 1 OO 1 50 200
time [sec]
(C)
Figure 5.19. Sticking events versus time for Poisson’s ratios of (a) V = 0.0, (b) V = 0.1,
(c) V = 0.4. The other parameters are shown in Table 5.1.
138
100
position
0
0 0 time [sec]
position
0 0 time [sec]
50
0 0 time [sec]
(C)
position
Figure 5.20. The stick—slip response in strains for Poisson’s ratios of (a) V = 0.0, (b)
V = 0.1, (c) V = 0.4. The other parameters are shown in Table 5.1.
139
4‘ l I ‘ .
m ' . â€(i ’f .
5 ‘ y z -! i v
a - cl ‘ \ ill/l) lipï¬lkï¬l
'lAAf‘J’ :._-“'~u’ VJL‘LLQUQ
.' llj'uv
u .
00
time [sec]
(C)
Figure 5.21. The stick-slip responses in contour plot of velocity for Poisson’s ratios of
(a) V = 0.0, (b) V = 0.1, (c) V = 0.4. The other parameters are shown in Table 5.1.
140
position
WWWW
M ii
00
position
#01
4N
00
position
frequencv [H2]
Figure 5.22. Power spectral density presentations of the stick-slip responses of velocity
for Poisson’s ratios of (a) V = 0.0, (b) V = 0.1, (c) V = 0.4. The other parameters are
shown in Table 5.1.
141
position — 5 position 9
1 O 1 0
displacement [m] displacement [m]
(a)
position — 5 position - 9
1 1
'3‘ '3'
E 0’ B. or .
E†.3;
g —1 l g _1 l
> 9
“go _go
1
75‘ "a?
E. E 0'
.é' .é'
-§ 35—11 4
9 9
10 "go 10
displacement [m] displacement [m]
position — 5 position - 9
1
7
E. 0)
E -1
9
‘00 1o 20
position - 13
_... 'ï¬i’i"!"’â€Â«â€œâ€œi‘~'*‘~i
. .. (W
E? i ’\'k'
- \,9 t l
2 1 ‘Q‘l/lï¬"
’20 1o 20 1o
displacement [m] displacement [m]
(C)
Figure 5.23. Projections of state-space trajectories for Poisson’s ratios of (a) V = 0.0,
(b) V = 0.1, (c) V = 0.4. The other parameters are shown in Table 5.1.
142
5.4.4 Effects of Friction Characteristics
From the analysis of the single degree-of-freedom model a negative slope in the
friction-speed relations destabilizes the system as a result of apparent negative damp-
ing effects. This results also can be applied to the lumped parameter model having
negative frictional characteristics. As shown in the previous chapter, the negative
slope in lumped multi degree-of-freedom model make the whole system unstable based
on the linear stability criteria. In this section the dependence of stick-slip oscillations
on the friction characteristics are investigated.
If the static and dynamic friction coefï¬cients are constants and equivalent to each
other, i.e., n, 2 uk condition, the steady state oscillations do not experience the
stick-slip oscillations under the ï¬xed boundary conditions. Figure 5.24 (a) shows the
transient stick-slip oscillations subjected to the speciï¬c initial condition. After the
transient oscillations are damped out the steady-state oscillations have no stick-slip
oscillations in their responses (which is not shown).
A case of discontinuous coefficient of friction characteristics, for example ii, =
0.3 and m, = 0.1, the stick-slip oscillations are observed after transient responses are
damped out, as shown in Figure 5.24 (b). When the system has a negative slope in
the friction-speed relation the system can experience the stick-slip motions, as shown
in Figure 5.24 (c). Since that condition indicates the unstable motions in slipping
state, the motions generate more stick-slip oscillations. The state-space presentations
are shown in Figure 5.25 for the speciï¬c masses.
As mentioned earlier, since the system stability is closely related to friction-
143
speed characteristics, the peak-to—peak amplitudes during stick-slip oscillations (Fig-
ure 5.25) and the dynamic behaviors during the stick-slip oscillations are influenced
by the operating driving speed (V) as well.
144
20 .
1 a»
<-
1 6m- “
1— e - .
1 41— e e -
1— e e e
1 21— .- .. _,
Ii 1— e . . . _J
1 o«— - - - t
I- O "
84- e e ...
4. e -
61- O ‘ “j
1-
44. 'l
«D
2 - . ‘*
O 50 1 OO 1 50 200
time [sec]
(3»)
20
1 8L _
<-
1 61- e - _,
‘- - - - C
1 44— - - - -‘
i— e - -
1 21— -- e. e e -
i— - - - - -
1 04— e- e - - -(
1— e- e e e
84— - - -(
1— - e e
61- e e - e —
- O
41. e -
1b
2 ~ -
O 50 1 OO 1 50 200
time [sec]
20
1 8 D In» _
1. O.
1 64- e. e e on» -
q—ee e... -- e e e
1 44-.- e.. e- e e e -
I—. I... e- e e - - e
1 21— e e e e e e. e -
- i—- e - - -e- -
. 1 01—- - . ..- - - - e -
{—0 e - - -e
84b- . cue- - - - .
4- ..
61- e e - ~
1- C
4‘. O -1
1b
2 ._ —4
O 50 1 00 1 50 200
time [sec]
(C)
Figure 5.24. Sticking events versus time for the following friction-speed relations:
(a) a discontinuous function, ii, = pk = 0.1, (b) a discontinuous function, u, = 0.3,
[1,, = 0.1, (c) p = cl + Cge‘c3l‘fi‘vl, where c1 = 0.3, 02 = 0.2, and c3 = 0.1. The other
conditions are in Table 5.1.
velocity [m/s]
V6l00lly [m/s]
at
N
J
9
d
l
10
position - 5
9
145
O 5
displacement [m]
position - 5
5
displacement [m]
position — 5
20
\\ .
l
my, "’9 “19,“!
limp)
&. «‘l‘ll'“
WV
1 O
displacement [m]
VC'OCllY [W8]
velocity [m/s]
(C)
l
N
I
..i
I
'0
position = 9
C ‘
-‘-‘<v\~
‘/
‘9'
k
‘
O
O
49
‘: 9 -
/ ’ “V " 9 ‘k‘\
W\ ‘
»‘\\_A .14- ‘ ’\$ ’393'/
5
position = 17
1O
O
l
.i
d
O
5
displacement [m]
position a: 9
1O
1 O
displacement [m]
Figure 5.25. Projected state-space trajectories for the following friction-speed rela-
tions: (a) a discontinuous function, u, = M = 0.1, (b) a discontinuous function,
u, = 0.3, m, = 0.1, (c) u 2 c1 + C2e-Cali’—V|, where c1 = 0.3, 62 = 0.2, and c3 = 0.1.
The other conditions are in Table 5.1.
146
5.4.5 Effects of Other Parameters
Numerical results for different values system stiffness k are shown in Figure 5.26.
As the stiffness It increases the stick-slip oscillation frequencies increases. When the
stiffness is high, the system can not satisfy the kinematic constraint function for the
sticking condition. Figure 5.27 shows the state-space presentation for several speciï¬c
masses.
As investigated in Chapter 3, the boundary condition plays an important role in
system stability. In this section the investigations are thoroughly devoted to the ï¬xed
boundary conditioned model, which is on the neutrally stable state in the stability
under a constant coefï¬cient of friction. If periodic boundary conditions are used, the
system is linearly unstable from the result of Chapter 3 and is able to experience
more nonlinear stick-slip oscillations, as will be seen in section 5.6.
It has been reported that the surface roughness is able to increase a apparent
coefï¬cient of friction and may contribute to stick-slip oscillations in real situations
(Nakai and Yokoi [72]). Thus the ï¬nal treatment for contacting surface affects the
stick-slip conditions and the generation of noise induced by friction.
Small geometric misalignment in system may generate stick-slip motions. Accord-
ing to the experimental work in compliant rubber bearings in ships (Bhushan [73]) the
shape of elastic materials between the channels was an important factor for generating
stick-slip oscillations. The divergent shape of elastic materials along the sliding direc-
tion can more easily generate the stick-slip oscillations than the convergent shaped
materials. Dweib and D’Souza [55, 56] and Tworzydlo et al.[52] investigated the in-
147
fluence of the angle of attack of the contact on the stability. In the experimental
work, the small misalignment of the angle of attack in the sliding contact caused
the coupling of the normal and rotational mode. This result conï¬rmed a well known
sensitivity of stability to the angle of attack, which is one of the reasons for the poor
reproductivity of the results of various frictional experiments.
148
..
{_-- -- C G
i
'i
0
‘01.
i"
,1
C
I
i
0
i
12".
0 so 160 150 200
time [sec]
2o - . .
181
16‘
14:
.
......O
*:il|“'
A
J J
.9- -QJQ
Thiflllll'll'"v
1 00 1 50 200
O 50
time [sec]
20
C. C Q
18» - - «
1- O
1 61. e e - e. -«
1- - e e an - - -
1 41- a. e e . 1|. .9 - -i
1— e e e - e - e.
1 21— 1:. e e e - .9 .1» - e -<
1— e e e e - - - - - e
1 01- e - - - - - a. - -
1- e - e . e. - on» .
81b 1. e - e -4
1. - .-
61. 1. ~
1-
4m -
1b
2 - _
O 50 1 OO 1 50 200
time [sec]
(C)
Figure 5.26. Stick events versus time for stiffnesses of (a) k = 1, (b) k = 2, (c) k
The other conditions are in Table 5.1.
position = 5
149
position - 9
g": m
E -1 .
93
'20 1o 20
position — 13
1 . . 1
'5‘ (ï¬ll 9" “' ‘1" i“, '7:
E, 0» Q, m J E, OJ
5 . ii s
_1 \'A‘\ I _1 . .
s ( ‘6‘â€! s
'20 1o 20 "'20 1o 20
displacement [m] displacement [m]
(a)
position =— 5 position - 9
1 ’,~ '
4!
s. o. if."
54 l “
2
,I.
9' \t’f’i‘:§‘.
' (if.§79 ‘ "’ ‘5
1i» ill
‘ Q5312w' ‘ ’
‘ ‘, -‘I ,
1O
1O
5
displacement [m]
position a 5
1O
5
displacement [m]
Figure 5.27. Projected state-space trajectories for stiffness values of (a) k
k = 2, (c) k = 3. The other conditions are in Table 5.1.
5 1
displacement [m]
position = 9
5 1 O
displacement [m]
(C)
150
5.5 Stick-Slip Oscillations: Modal Projection
Method
In the previous sections we have used the lumped-parameter model to show multi-
dimensional behaviors of stick-slip oscillations. In this section, by applying the modal
projection method to the continuous system a possible alternative method in handling
the low-dimensional stick-slip oscillations is proposed.
du(x,t)/dt // // stick region
R
l
\ \ \ \1 l\ slip region
Position x
Figure 5.28. A possible velocity proï¬le showing the stick-slip motions in a continuous
one-dimensional system: High dimensional model.
We may encounter some difï¬culties in analyzing the continuous system experi-
encing stick-slip phenomena since a system may have high frequency oscillations and
discontinuous properties within their domains. For an example, as shown in the Fig-
ure 5.28, a possible stick-slip conï¬guration at speciï¬c instant in time has numerous
stick and slip regions within the domain. There are inï¬nitely many possible stick-slip
151
du(x t)/dt stick region
slip region
1 l k
BI 82 L Position x
Figure 5.29. A possible velocity proï¬le showing the stick-slip motion in a continuous
one-dimensional system with bounded boundary condition model: Low dimensional
model.
conï¬gurations, making it difï¬cult to formulate analytical or numerical solutions.
On the other hand, when we consider low frequencies in stick-slip motions only,
we can simplify the stick-slip system and are able to overcome the difï¬culty. Possible
typical subregions are shown in Figure 5.29. We divide the whole system (a: 6 [0, L])
into three subregions of low dimensional systems—a slipping region in the front (:1: E
[0,B1]), a sticking region in the middle (:1: E [81, B2]), and a slipping region in the
rear (:1: 6 [32, L]). The low dimensional characteristics of each region are assembled
to approximate the whole system behaviors.
Assuming that the system has a conï¬guration shown in Figure 5.29, the indicator
function for the kinetic constraint equations at the stick-slip boundaries (2; = Bl, or
82) are
H(2:, t) “é‘ |F(:z:, t)| — |G(:z:,t)|, (5.13)
152
A slipping region, a: 6 [0,81] A slipping region, a: E [82, L]
Equation 5%{6—025 6†: e—dngiuf FMWe-‘dx Bu : e—QI%
Boundary u,(0 t): (0, t) = 0, u(L, t) = -g—';( ,t) = 0,
condition %:(B,, t)- _. V 33%, t) = V
Solution u(,z t)- — v(x,t) + 79‘1th u,(:r t)- — w(z, t) + LL_ ’2; Vt
— -}’°E =1¢°B’($)7‘j(t) + éVt — °°1¢B’L(CC )8 1'“) + 33—53, W
Projected i'j +3 w} jrj- — 0, s',- + (d2 J'SJ‘: 0,
motion rj(0)— — -2- $031 (x)v( ,0)d:r, 31(0): -L—-——_ 2232 fBgiZ) wB’L(x)w($,0)dx
19(0): 3131, w°Bl(a:)v(x 0)dx s'J-(O):33,13.¢¢3:L(x)w(x,0)dx
Table 5.8. Equations of motion, boundary conditions, and projected motions on
modal coordinates for the stick-slip oscillations with ï¬xed boundary conditions. Here
‘5 - in 1:—
â€â€” t‘ = t/,/..E, 453% >= a sum—1 " . ...)“ wm )= 33,334.11),
65 :
n-m
153
where
02110:, t) ,,
F($,t) = W, (0.14)
_ 00 u 011(33, t)
C(x’t) _ “4 AE A 8.11 }’
And the indicator function for kinematic constraint equation is
def 3U
S(;r,t) = bit-(x, t) — Vl. (5.15)
The F (:r,t) represents the stiffness forces at stick-slip boundary a: and time t. The
function G (:17, t) express the frictional forces including the Poisson effect of the ma-
terials at the boundary. The condition of H (:13, t) 2 0 with zero relative velocity
(S (as,t) = 0) is necessary for the stick status. On the other hand, non-zero rela-
tive velocity (S (3:, t) 75 0) yields a slipping status. These constraint formulations are
similar to those of the lumped-parameter models in the previous study.
Considering the system which can be divided into the slip and stick regions (Fig-
ure 5.29), characteristics for each slipping region (a: 6 [0,31], and :r E [32, L]) are
summarized on Table 5.8. Table 5.8 presents the equations of motion, boundary
conditions, the modal projected motions and solution for each slipping region.
Within the sticking region, which is placed on between those two slipping regions,
all motions are constrained. Thus it is represented as
—(:c, t) = V, (5.16)
I‘l 1“ 4 I‘ll 1‘ ‘ 1‘ 1‘ If.‘
1‘ L I‘ u
l‘l 1P ‘
154
where a: 6 [B1, Bo].
By taking its adjoint function 1/1,(:c), the motion in both slipping regions are ob-
tained by casting to the modal coordinates rj(t) and sj(t), respectively. (Refer to
Table 5.8.) Note that the obtained equations of motion are only valid until the ki-
netics and kinematic conditions do not change their sign at their boundaries.
A slipping region, as E [0, L]
_c?_ «'1ng _ —6x62u
Equatlon of mot1on a: 372
Boundary condition u(0, t) = 9%(0, t) = u(L,t) = %(L,t) = 0
Solution u(a:, t) = 3:1 <1)?!†($)(Ij (t)
Projected motion qJ-(O) = % If 11;†(a:)u(a:, 0)d:1:,
«11(0) = 313 «13L(z)u(x, 0)dx
Table 5.9. The equation of motion, boundary condition, and projection to modal
coordinates for pure sliding oscillations under the ï¬xed boundary condition.
When the system is in a pure sliding state, i.e., the state that does not have
any stick regions inside of the domain a: E [0, L], the whole system is represented in
Table 5.9. The integration should continue as long as the constraint equations (5.13)
and (5.15) do not change their sign during its integration period. In each iteration
the boundary conditions must be updated from the system state. In doing so its
155
(Emil/(h du(x,t)/dt
growing stick
V ____________ v _ _ _ 2*:— “':"_ _ _ _
(I) x (b) x
(1069/!!! du(x,t)/dt
- shrinking stick
V ___:“1" *—_:—_.__*_ v -__._ _____
(d) x (c) x
Figure 5.30. Schematic diagrams showing stick-slip oscillations. (a) a pure sliding
stage, (b) a growing sticking stage, (c) an enlarged sticking stage, and (d) a shrinking
sticking stage.
motion is de-projected to have real conï¬guration and projected again to integrate its
states in time. The growing and shrinking of the sticking regions are determined by
checking the constraint boundary equations (5.13) and (5.14). In this simulation we
discretized the domain :5 E [0, L] with 20 divisions and applied a difference method to
equation (5.13) in order to determine and update the sticking boundaries 31 and 32.
Schematic diagrams of stick-slip oscillations are shown in Figure 5.30. Figure 5.30
(a) represents a schematic diagram showing a pure sliding stage. In Figure 5.30 (b)
and (c) a sticking region is grown and extended. A shrinking sticking stage is shown
in Figure 5.30 ((1).
Figure 5.31 shows a displacement proï¬le of a numerical simulation and Figure
5.32 shows the stick events on the contact surface. (All stuck initial conditions are
156
displacement [m] u(x,t)
position 0 0 time [sec]
Figure 5.31. Stick-slip responses in displacement by applying the modal projection
method. Here 118 = 0.3, and #1: = 0.1. (Displacement variations with respect to static
equilibria are shown.)
selected.) Comparing the results obtained in Figure 5.8 (a) and Figure 5.14 (a)
these results reveal qualitative agreement in the analysis of low dimensional stick-slip
motions.
Note this approach still has some limitations in its application to all parameter
conditions. If actual frequencies of bushing squeak noise are too high and the fre-
quency of the low-dimensional model suggested in this study is too low then we may
need to use high-frequency models. However many other applications can be analyzed
with low-frequency models. This algorithm only valid as long as the system can keep
the conï¬guration in Figure 5.29. Alternatively the high dimensional dynamic motions
can be obtained by ï¬nite element analysis, presented in Chapter 6.
157
20
18P~ ..
I- l .
12—~- vfl- — _ ------ — .1
1o_-- - .". ------- _ — ---------- — J
— — — : — _-
8—» :- — ...— — -1
b — : - x—
6—-- 1- .... 1- -1
- .
I, .
2...... ,2, .1
o A P i
O 50 100 150 200
timelsoc]
Figure 5.32. Sticking region versus time by applying the modal projection method.
158
5.6 Stick-Slip Oscillations under Periodic Bound-
ary Conditions
Let us return to the motivation of this study—the bushing model. The simpliï¬ed
model and its stability have been investigated in the previous chapters. In this section
stick-slip oscillations with periodic boundary conditions are investigated. The model
in Figure 4.8 and boundary condition (4.29) in Chapter 4 are adopted for numerical
studies. The equation of motion is shown in (4.22), and small damping which can
suppress several unstable modes are added.
Figure 5.33. A contour presentation of the velocity response for the model with
periodic boundary conditions. Here (1 = 0.01,'y = 0.01, 113 = 0.3, and 11k = 0.1. The
other parameters are in Table 5.1.
Figure 5.33 shows a contour plot of stick-slip responses in velocity with small ex-
ternal ( = 0.01) and internal (7 = 0.01) damping terms. With sufï¬cient damping,
159
20-
. .- -.----ccccccccoco
m» o no. .0. ......u......
o c .- . -.-------- .0.
m-- o .- ..........-...0.
-0 o- a. o ----------O-.- o
M-- .. 0.. ......o.-..... u
o o o -0 ......c..-.... a
m a . ...-..-ccccucooc . o
g o ......c............. 0
gm— o. a a. ------------ a
.
-..-
w -0 a
W a
m a- a a- ......c....-.....
M a. o o- ......c..........
a. c .- ......c.........-
m o o. - -.-------ccuuccuu
———L++A—.—.—.&..d.....*...*l
O 50 100 1 50 200 250 300 350 400
time [sec]
Figure 5.34. Stick events versus time for the periodic boundary condition model.
such that the overall system is stable based on the linear stability criteria, it is ob-
served that the oscillations are damped out, so that the system can not experience any
sustained stick-slip oscillations under the condition of a constant coefï¬cient of friction
(which are not shown in a ï¬gure). However including “small†damping, such that
suppress several unstable modes are suppressed, the system experiences sustained
stick-slip oscillations, as shown in Figure 5.33. Stick-slip oscillations are detected as
series of detachments and shown as propagating waves around the contact surface.
Figure 5.34 shows the stick events versus time. The series of detachments, like
as falling dominos, are observed distinctly. The state-space presentations for selected
masses are shown in Figure 5.35. Under this condition with small damping, the
responses seem to have steady state sustained stick-slip oscillations. In this numerical
simulation the system is slightly modiï¬ed by adding small spring stiffness (1 / 10 of the
connected spring stiffness k) between each mass and the ground. Random velocities
160
position =2 5 position = 9
.. «n1 . - t
\ ‘ . , - \‘ \I/ l I/ ,
93 \‘%;’/// \S
_2 , . _2 .
‘3 o 5 1 o _3 o 5 1 0
position - 1 a 1 position = 1 7
5 ° «Mfg/W â€
. -1 \\\‘\.\2;~;/ZV/
2 _2 . \\;—/
_3 5 1 o _3 o 5 1 o
displacement displacement
Figure 5.35. A state space diagram for several positioned masses under a periodic
boundary condition model.
are selected as the initial conditions.
Note that since the model used in this section is a simpliï¬ed one-dimensional model
the responses obtained from numerical analysis may not express the whole dynamic
behaviors for the actual bushing squeaking phenomena. Detailed mathematical mod-
eling, which includes coordinate couplings such as the interactions between the radial
and circumferential motions and radial and coriolis acceleration effects, is needed for
future study. (The influence of the interactions of two coordinates on stick-slip oscil-
lations are shown in Chapter 6 for the two-dimensional system through ï¬nite element
analysis.)
161
5.7 Conclusion
In order to explain the stick-slip oscillations in the distributed contact system,. the
discretized, lumped-parameter model was established and the numerical techniques
for handling such systems—the variable degree of freedom and the variable boundary
conditions—were presented. Since it was conï¬rmed that linear stability of the lumped-
parameter model is convergent in the previous chapter, we adopted that model for
simulations and analyses of stick-slip oscillations.
Detailed dynamic behaviors in presence of distributed friction are presented and
the parameter effects on system responses were investigated. According to the nu-
merical responses of the stick-slip oscillations, the generation of noise and vibrations
originated from the mechanisms of series of detachments on the contact surface [26].
The characteristics of series of detachments slips were influenced by various system
parameters. The high normal loads had two effects: decrease the frequencies of de-
tachements slips and increase the frequencies of small-grouped stick-slip oscillations.
Moreover, the sticking regions were enlarged by increasing the normal loads. The
driving speed and the characteristics of friction were closely related in responses of
stick-slip oscillations. The Poisson’s ratio increased the stick-slip frequencies and
broke the symmetry of system responses. The friction characteristics had effects on
the stick-slip oscillations and the discontinuous friction model (113 > #1:) produced
the steady state stick-slip motions in the multi-degree—of-freedom model.
By applying the modal projection method to the continuous system a possible
alternative method in handling the low-dimensional stick-slip oscillations was pro-
162
posed. Numerical algorithms in handling such a system were presented. The qualita-
tive agreement in the analysis of low-dimensional stick-slip oscillations was obtained.
By using the modal projection method we simulated the low-dimensional system in
less computational time.
Under periodic boundary conditions—the same boundary condition of the bushing
model—the system with small damping underwent sustained stick-slip oscillations
even with a constant friction coefï¬cient. Stick-slip oscillations were detected as series
of detachments. With sufï¬cient damping, such that the overall system was stable,
oscillations were damped out and sustained oscillations were not observed.
CHAPTER 6
FINITE ELEMENT ANALYSIS
6. 1 Introduction
Finite element analysis has became the dominant system analysis method as a result
of the continual developments in computer technologies. In recent years, a rapid
development of the digital computer has made the ï¬nite element approach to nonlinear
contact problems possible, and its applications have become numerous in engineering
practice. Some applied examples include a shrink-ï¬tted shaft in a gear (Okamoto
and Nakazawa [116]), rail-wheel contact problems (Schneider and POpp [115]), and
cutting systems (Marusich and Ortiz [119]).
However, there have been limited studies for investigation of friction-induced vi-
brations of elastic media with distributed contact. Oden and Pires [44] have formu-
lated contact problems in elasticity and investigated the dynamic behaviors due to
the distributed contact. Later, Oden and Martins [117] established an elastodynamic
model and developed the computational methods for dynamic friction phenomena.
163
164
In this chapter a formulation of frictional elastodynamics in the form of ï¬nite ele-
ment analysis is performed in order to verify and conï¬rm the results obtained in the
previous chapters. The comparisons between the computational results obtained from
the lumped-parameter method and ï¬nite element analysis are shown. By adopting a
smooth approximation to the nonsmooth friction characteristic, responses of the con-
tinuous system under the distributed contact are analyzed. One— and two-dimensional
elastic systems are numerically investigated by using ï¬nite element analysis.
6.2 One-Dimensional System
6.2.1 Formulation and Algorithm for Nonlinear Finite Ele-
ment Analysis
This section is devoted to a ï¬nite element formulation of the one-dimensional time
dependent problem and its numerical technique for solving a nonlinear problem. An
equation of motion for the model of interest is written as
021.1 Bu 0211
—‘a—+“3=o?
Orr? 01: (61)
where the a is a velocity dependent parameter, a = oz(,u(1'1)), and the 6 depends on
the normal load and friction coefficient, [3 = [3(No,11(1'1)). Here the ï¬xed boundary
conditions, which are expressed as u(0, t) = u(L, t) = 0, are applied.
The algorithm for solving the discrete dynamical system is based on the schemes
in nonlinear structural dynamics calculations. Let us begin with the presentation of
165
a Newmark-type algorithm that has been proved to be effective for many problems in
computational studies. Assuming that the velocities and accelerations at time tk are
expressed as functions of the displacement, velocities and accelerations at time tk_1
and displacement at time tk, the following relations are obtained.
, 9 a . 0 ..
n, = ï¬z(uk—uk-1)+(l—g(f-)uk_1+At(1—56%)uk_1, (6.2)
" — —1——( — )——1—a —<—1——1)a (63)
“I: — 01At2 W 7111—1 Blt k—l 261 k—I, -
where 00 and 01 are called as Newmark parameters (usually those parameters are
used as [3 and 'y in ï¬nite element analysis, respectively), and At(= tk — tk_1) denotes
the interval of time length for integration time in the whole time domain of [0, T]
with t0(= 0), t1, . . . , tk, . . . , tM(= T). The parameter 00 and 01 are selected based on
the stability schemes in numerical convergence, thus 00 = % and 01 = i are selected
since the Newmark-type algorithm has been proved to be an unconditionally stable
algorithm, which corresponds to the constant-average-acceleration method.
The equation of motion (6.1) is put in the operator form of deï¬ned residual Rk(uk),
then
Rk(uk) = ilk — 112+ an), — C15. (6.4)
From the weighted residual method, the solution 11,, is approximated by setting
the integral of the weight residual of the approximations over the domain to zero,
166
that is
< Rk(uk),v > = < ilk — 112+ an;c —— aï¬w > = 0. (6.5)
Let Kk(uk) = DRk(uk) be the derivative of the R), at 11],. Then the Newton-
Rapson iteration technique for solving the weighted residual becomes an iteration
process seeking for the solution.
Using the standard ï¬nite element procedure, the system can be constructed in
ï¬nite-dimensional subspaces. For each of a certain mesh h, the nodal values the
h
displacements 11", velocity 1')", and accelerations, ii are expressed in the form of
vh(:r,t) = ZN-zr( )v,(t), (6.6)
1')"(:c,t) = ZNi(x)ii,-(t),
where Ne denotes the number of nodes of the elements, Ni(a:) is the element shape
test function associated with the node 2'.
Given the starting value of “[0), successive approximations of the solution 11,, are
obtained by using the recurrence formula
u(‘+1)—u("— R—k(uk)
uk “ k K]: (—:u(i)) (6'7)
where (2') is the iteration counter. The termination of iterative procedure at each
167
time k can be checked by a convergence ratio in relative displacement conditions.
The convergence ratio is deï¬ned as
[éulmax _ [Auli'l'll "' Aâ€(immax
— . , 6.8
[dulmax lAu("|max ( )
convergence ratio =
where léulmaz and [dulmax denote the maximum displacement change and maximum
displacement increment in each iteration, respectively. In this numerical analysis
(MARC/MENTAT [120]) the relative displacement tolerance is set as 0.1 and the
maximum iteration number is set as 30.
6.2.2 Eigenvalue Comparison
In this section the system eigenvalues calculated from both models—the ï¬nite
element model and the lumped-parameter model—are compared under the same pa-
rameter conditions. The system domain selected in the ï¬nite element analysis is an
elastic medium of 10 inches x 1 inch with the unit thickness (1 inch). The continuum
is assumed to be in plane strain. It has ï¬xed boundary conditions at both ends and
under goes a compressive stress by means of a preload on top of the elastic medium.
The elastic material selected in this simulation is polyethelen, which has a properties
of E = 2.0 x104 lb/in2, Poisson’s ratio V = 0.45, and the mass density p = 0.033
1b/in3 (28.55 x10‘5 lb-secz/in"). Refer to the system description in Figure 6.1. Here
front and rear nodes are deï¬ned the nodes positioned in the left- and right-hand sides,
respectively.
The equivalent discrete system is obtained by dividing the material into 71 equal
168
jllllillllfllllllllll
l 5 7 9 11131517192123252729313335373941 2
’.
Figure 6.1. A schematic diagram of the system used in ï¬nite element analysis. The
model is composed of twenty elements. The top rigid body is stationary without
friction. The lower rigid body moves at 1 inch/sec to the positive 2: direction. There
is friction between the elastic material and the lower moving body. Selected nodes
are shown.
segments, lumping the mass of the segment in the center and regarding the each
lumped mass M as being connected by springs of equivalent stiffness k, where k
is selected such that the springs undergo the same elongation as the corresponding
material segment would under identical loading. Thus the each lumped mass has the
value of m = pAL/n and the spring constant is k = nEA/ L, where L, A is the length
and the area of cross section of the material, respectively. The normal load on each
mass in the distributed loading condition can also be discretized by N.- = aoAL/n,
where the 00 denotes the normal stress.
Table 6.1 presents the approximated eigenvalues with a = 0, which is a condition
without friction forces. The left column in Table 6.1 shows the exact eigenvalues
169
Exact Freq. 10 elements 20 elements 40 elements
x 103 [H z] x 103 [H z] Error[%] x 103 [H 2] Error[%] x 103 [H z] Error[%]
1 0.7647 0.767 0.41 0.765 0.10 0.764 0.02
2 1.5294 1.555 1.67 1.536 0.43 1.531 0.10
3 2.2941 2.380 3.74 2.315 0.91 2.299 0.03
4 3.0588 3.262 6.67 3.109 1.67 3.071 0.42
5 3.8236 4.216 10.26 3.922 2.57 3.848 0.63
6 4.5883 5.246 14.33 4.759 3.72 4.631 0.93
7 5.3530 6.322 18.10 5.625 5.08 5.421 1.26
8 6.1177 7.348 20.11 6.523 6.62 6.219 1.65
1 9 6.8824 8.132 18.15 7.458 8.36 7.026 2.08
Table 6.1. The approximate modal frequencies by applying ï¬nite element analysis.
The numerical results including 10, 20, and 40 elements are presented with the exact
frequencies.
Exact Freq. 10 masses 20 masses 40 masses
x 103 [H z] x 103 [Hz] Error[%] x 103 [Hz] Error[%] x 10T[Hz] Error[%
1 0.7647 0.5718 25.2 0.6546 14.0 0.7099 7.17
2 1.5294 1.1339 25.8 1.3164 13.9 1.4188 7.23
3 2.2941 1.6766 26.9 1.9662 14.2 2.1258 7.33
4 3.0588 2.1907 31.0 2.6060 14.8 2.8297 7.49
5 3.8236 2.6673 30.3 3.2324 15.4 3.5298 7.68
6 4.5883 3.0983 32.4 3.8424 16.2 4.2249 7.92
7 5.3530 3.4761 35.0 4.4329 17.1 4.9141 8.19
8 6.1177 3.7965 37.9 5.0008 18.2 5.5963 8.52
9 6.8824 4.0479 41.1 5.5432 19.4 6.2708 8.88
Table 6.2. The approximate modal frequencies from the lumped-parameter model.
The numerical results including 10, 20, and 40 masses are presented with the exact
frequencies.
170
of one-dimensional axial motions with natural frequencies of w, = ing/5%, where
7‘ = 1,2,. . ..9. From this table it is obvious that with high numbers of nodes, i.e.,
with ï¬ne meshes in the ï¬nite element model, the more accurate approximate solutions
which approach the exact solutions are guaranteed. Table 6.2 shows the numerical
solution for the lumped-parameter model in evaluating the approximate frequencies.
The system eigenvalues of the lumped-parameter model are lower than the exact
solutions calculated from the continuous model. The reason is that, although the total
mass is the same in both systems, in the discrete model the mass is shifted toward the
center of the system instead of being uniformly distributed (Meirovitch [19]). This
tends to increase the effect of the system inertia relative to its stiffness, resulting in
lower natural frequencies. Of course, accuracy can be improved by increasing the
number of degrees of freedom of the lumped-discrete system.
It should be noted that when the friction forces are large, i.e., large value a in the
equation of motion in (6.1) it is expected that the ï¬nite element method may be poor
in accuracy. The argument is quite general. The Galerkin’s approximate method
may be unsatisfactory when the odd-derivative term is of signiï¬cant size (Strang and
Fix [121]). Since a strongly influences the eigenvalues and eigenfunctions in the PDE
solution as [a] increases to 00, the ï¬rst order term dominates the second derivative,
and the system model is a boundary-layer problem. Thus at the far end there should
be a rapid variation in :1: in order to satisfy the boundary condition, and an extremely
ï¬ne mesh is required to satisfy good approximations in such case. In this study
extremely large values a of are excluded for good convergence.
171
6.2.3 Numerical Results
0.4 I 1 f f
0.3“ '
.°
N
1
friction coefficient 11
O
-O.1]- Ii ..
l
—O.2~ ’1 -
—O.3 ----~—-""T" . q
_O.34 _é _é _: 6 1 2 5 4
relative speed (V - du/dt)
Figure 6.2. The friction coefficient versus relative velocity. 11 = sign (V -
1'1),uk%arctan(JLE—i‘1), where C = 0.1 for dotted line and C = 0.01 for solid line.
In order to formulate and analyze the friction system we need to model the friction
characteristics in a mathematical form. The friction function used in the previous
chapter, which is a discontinuous function, may be inappropriate for ï¬nite element
analysis. Instead of using the discontinuous friction function a smooth function having
a steep variation of friction coefï¬cient is usually adopted in ï¬nite element analysis.
When we choose a steep slope and a small integration time step we can expect this
smooth function can represent the discontinuous properties well. Strictly, stick-slip
motion no longer exits, but approximate or near stick-slip behavior takes place (F eeny
and Moon [107]). The analytical model for the friction coefï¬cient used in this study
172
is expressed as
IV—ul
C
2
11 = sign(V — 1'1) 11],; arctan( ), (6.9)
where the parameter C determines the slope of the dynamic friction coefï¬cient at
zero relative velocity. Figure 6.2 shows typical continuous friction model used in
ï¬nite analysis. For a small value C a steep slope is generated around the relative
velocity of zero. Thus a small integration time step is needed to satisfy the dynamics
around near sticking conditions. For a large value C an gentle slope is generated, but
near stick-slip dynamics may not be seen. In this study C = 0.01 and 111: = 0.3 are
chosen.
The ï¬nite element meshes used in this analysis consist of twenty-four—node isopara-
metric quadratic elements, illustrated in Figure 6.1. The total simulation time is set
as T = 0.005 sec, and iteration steps during the time are 5,000, which make the time
step for the integration is set as At = 1 x 10‘6 sec. The initial conditions applied to
this numerical analysis are u‘(t) = 0, and 11i(t) = +1 inch/sec for all nodes 1'.
In the presence of Poisson’s ratio, in this study V = 0.45 , the distributions of
the friction forces, which are not symmetric along the nodes on contact, are shown in
Figure 6.3. Since the near sticking motions are associated with the steep slope with
respect to relative velocity, the friction forces around the sticking regions can have a
value between —|11k022| to +|pk022|, where the normal stress 022 is not a constant,
but a variable value. Higher friction forces are observed around the rear nodes than
the front nodes. The friction forces traveling on the contact surface are observed like
the sliding rubber mechanisms performed by Schallamach [26].
173
The axial strain on changes the normal stress and also influences the friction forces
since the system is under a constraint in y direction. The variations in axial stress
011 are shown in Figure 6.4. The front and rear nodes are under tensile (011 > 0) and
compressive (011 < 0) stress conditions, respectively. After the stick to slip transition
the traveling stress waves propagate back and forth within the medium.
Figure 6.5 shows a velocity response of nodes 19. Since the friction model used in
this simulation has identical static and dynamic friction coefficients (113 = 11),), “pure
sliding oscillations†are observed, which agrees with results obtained in the previous
chapters. (See Figure 5.24 (a) for details.) Quasi-harmonic oscillations which carry
high frequencies are observed. From the investigation by De Togni et al. [99] in
the bushing-squeak system, similar oscillations, which have sawtooth waveforms, are
detected in their experimental studies. Usually they have higher frequencies and
smaller amplitudes in pure sliding oscillations than the stick-slip oscillations.
Figure 6.6 represents the friction force versus velocity presentations for various
nodal responses. At velocity near +1 inch/sec, which represents the near sticking
state, the steep slope in the friction characteristics is apparent. Since the normal
forces are changed by displacements the friction forces exerted on each nodes are
not only a velocity dependent, but also position dependent. Large friction forces
are observed at rear nodes. Figure 6.7 shows the friction force versus displacement.
After some transient motions, which are shown as sparse trajectories, each mass has
steady state responses represented as dense trajectories. At node 35 the friction force
increases as the displacement increases. On the other hand, at node 11 and 19 the
friction forces are decreased linearly with respect to the displacement. At node 27 it
174
apparently looks as if the friction forces are independent of the displacement, shown
as flat trajectories. These phenomena result from the coupled effect of the Poisson’s
ratio between friction force and linear strains. These results are veriï¬ed by the results
in equation (2.2). Figure 6.8 shows the displacement and velocity relation for node
19. The results obtained in this study have qualitatively similar behaviors as those
worked by Oden and Martins [117].
175
Friction Force x
1.185
lcasei
0
(l
0
-0.324
0.01 5
Time (x.001)
*-‘-* Node 11 +-+-+ Node 19
i-dk-I Node 27 e—-4>—-e Node 35
Figure 6.3. The distributions of friction forces versus time at nodes on contact.
177
lcasei
Velocity x Node 19
1.023
ll
’1
_J
-0.794
Tine (x.001l
Figure 6.5. Velocity response at node 19.
178
lcasei
Friction Force 11
1.185
its
3.1—:31
MN
1
0
-0.324
-0.963
Velocity x
-—o—-Node 11 +—i—+Node 19
x—I—itlode 27 e—o—ctlode 35
1.021
M
Figure 6.6. Friction force versus velocity at nodes 11, 19, 27, and 35.
179
Friction Force x
1.185
lcasei
4H
...
-0.324
--—~Node 11
r-ae-iNode 27
Displacement x (x.0001)
+—-+-+Node 19
H—e‘lode35
2.547
ill
Figure 6.7. Friction force versus displacement at nodes 11, 19, 27, and 35.
180
1 1
“99 331111116
Velocity x Node 19
1.023
0 II I
V
-0.966
0.001 2.547
Displacement x Mode 19 (x.0001)
Figure 6.8. Displacement versus velocity at node 19.
181
6.3 Two-Dimensional System
We investigated the two-dimensional elastic model with periodic boundary conditions
in Chapter 3. In this section a two-dimensional elastic medium under distributed
contact with ï¬xed boundaries is numerically investigated by applying ï¬nite element
analysis.
(0,5) (10,5)
(0,0) (10,0)
47 102 146 190
>
Figure 6.9. A schematic diagram for a two-dimensional elastic medium under dis-
tributed contact. The lower rigid body moves at 1 inch / sec to the positive a: direction.
Selected nodes are shown.
Figure 6.9 shows the schematic diagram for a meshed block. The polyethelen
block with dimensions of 10 inches x 5 inches x 1 inch is divided into a 20 x 10
mesh. The lower rigid body moves toward to positive a: direction at 1 inch/ sec. Here
front and rear nodes are deï¬ned the nodes positioned in the left- and right-hand
182
sides, respectively. The govern equation of motions for the elastic medium are shown
in equation (3.12). The boundary conditions are represented as
11(0, y) = u(10, y) = 0, for 0 < y < 5, (6.10)
v(0, y) = 11(10, y) = 0, for 0 < y < 5,
11(17, 5) = 0, for 0 < :1: < 10,
v(:r, 5) = 0, for 0 < :1: < 10.
The boundary conditions on the contact surface at y = 0 are
v(:r,0,t) = 0, (6.11)
0yx(:r,0,t) = payy(.1:,0,t),
where p is a coefï¬cient of friction, which is represented in Figure 6.2. The stress-strain
relations are 0y; = C(g—Z + g—Z), and 0W 2 A-g—‘x‘ + (A + 2G)g—;. (Stability analysis of
this two-dimensional system in pure sliding motions will be a furture work.)
Given the sticking initial conditions for all contacting nodes at y = 0, the nu-
merical integrations are performed by following the same procedures in the previous
section with small time step of At = 1 x 10‘6 sec.
Figures 6.10 shows that the axial stress distributions at speciï¬c nodes on contact.
The nodes at 47 and 102 oscillate under tensile stresses and the nodes 146 and 190
are under the compressive stresses. Since in two-dimensional system we consider
183
the normal directional motions and include that oscillations to the normal stress in
equation (6.11), the system responses are more complicated and strongly coupled by
the y-directional motions. The resultant friction forces are shown in Figure 6.11.
The stick-slip oscillations are onbserved in the two-dimensional system even un-
der a condition with a constant coefficient of friction. They are easily expressed
by state-space presentations, shown in Figure 6.12. The stick motions are shown
as nearly flat trajectories in the state-space presentations. Based on the results of
the one-dimensional system studied the previous section the nodes oscillate with
quasi-harmonic pure sliding motions without having any stick-slip oscillations under
a constant friction coefï¬cient model.
The friction forces versus displacement are shown in Figure 6.13. The responses
in friction-displacement are similar to the responses in one-dimensional system in
Figure 6.7. However, in the two-dimensional model the variations of friction froces
with displacement do not have constant slopes, but include normal oscillations. The
importance of the normal vibrations in the stick-slip oscillations have been investi-
gated by several researchers (Tolstoi [48], Sakamoto [49, 50], Tworzydlo and Becker
[51], Tworzydlo et al [52], Pires and Oden [53]).
According to these numerical responses the near stick-slip oscillations can be ob-
served in the two-dimensional model even under a constant coeflicient of friction
model.
184
lcasel
Comp 11 oi‘ Stress
2.878
I /\1\
/ ' / \ "'\ l
L
01
-3.363 W
0.001
Time (x.001)
—-—Node47 +—-i—+Node102
I-—I-—-§Node146 o—o—otlodel90
2.998
Figure 6.10. Stress distributions (0“) versus time for nodes at the contact.
185
Friction Force x (x.1)
lcasel thc
2.998
7.66
o
9
0
ll
" J
0
i
l1
1
1
l
l
I
ll
0.053 I
0.001
Time (x.001)
--——— Node 47 +—-+-—+ Node 102
l-iP-i Node 146
°-0-0 Node 190
Figure 6.11. ï¬iction forces versus time at nodes on contact.
186
icasei
Velocity x Node 102
1
1
7
/ [ l
/ // \
l9
[11])V. . 11 g
[[11 11V ,
1,111 ,
1 ,3/
Displacement x Node 102 (x.0001)
3.693
Figure 6.12. Displacement versus velocity at node 102.
187
icasei an
WM11110
Friction Force x (x.1)
7.66
0.053 _
0.01 3.693
Displacement x (x.0001)
i—-l-iNOdE 146 o—-o—-eNode 190
Figure 6.13. Friction force versus displacement at nodes 146 and 190.
188
6.4 Conclusion
In this chapter we investigated one- and two-dimensional elastic systems under the
distributed frictional sliding contact by applying ï¬nite element analysis. The system
eigenvalues were evaluated and compared to those obtained by the previous studies.
Numerical analyses show that the one-dimensional system undergoes pure sliding
oscillations under the condition of same static and dynamic coeflicient of friction
(p, = 11),). The relation between friction and displacement shows that the system
has properties which depend not only on velocity but also on displacement. Such
relations are determined by the Poisson’s ratio and the positions of the nodes in the
system. The stress and strain have the form of traveling waves in the continuum.
In the continuous model the system undergoes quasi-harmonic oscillating motions
including high frequency signals.
Through the ï¬nite element analysis we demonstrated the possible stick-slip os-
cillations in a two-dimensional system. According to this investigation it is possible
to generate stick—slip oscillations in the two-dimensional system under a constant
coefï¬cient of friction condition due to oscillations in the normal degrees of freedom.
CHAPTER 7
CONCLUSIONS AND FUTURE
WORKS
In order to investigate dynamic behaviors of an elastic medium under the distributed
friction forces we constructed mathematical models and analyzed their dynamic sta-
bilities. The stick-slip phenomena of friction-driven systems were investigated and
veriï¬ed by ï¬nite element analysis, lumped-parameter method, and modal projection
method. This study yielded the some mechanisms responsible for vibrations, and
presumably noise, in the elastic materials and revealed the instability mechanisms of
the continuous system under distributed friction.
In Chapter 2, a continuous elastic medium with ï¬xed boundary conditions sub-
jected to the distributed friction was introduced, mathematically modeled, and its
exact solutions were provided. The friction made the system non-self-adjoint. The
approaches in handling such problems were provided as well. By using the adjoint
operator the problem was treated pr0per1y. In addition, by projecting through the
189
190
proper inner product the system was seen to truly be self-adjoint. A contradictory
result between the exact solution and an assumed modes approximation in evaluating
the eigenvalues was presented as a cautionary example. Consequently, under a con-
stant coefï¬cient of friction (11, = #1:) the system was shown to be marginally stable.
Thus an instability mechanism does not exist under the ï¬xed boundary conditions.
In Chapter 3, the effect of boundary conditions on the system stability was ex-
amined. Under periodic boundary conditions the one-dimensional system was desta-
bilized even under a constant friction (11, = 1111)- The destabilizing phenomenon
occurred in the form of an unstable traveling wave propagating in the direction of the
slider velocity. The effect of internal and external damping were evaluated and they
played stabilizing roles in the overall system stability. For a two-dimensional system
under periodic boundary conditions the system was destabilized under a condition of
a constant coefï¬cient of friction when the system coordinates are strongly coupled by
Poisson’s effect.
In Chapter 4, the lumped-parameter method was applied to discretize the con-
tinuous model which was investigated in the previous chapters. It was shown that
the system stability was closely related to the friction characteristics. The negative
slope in the friction-velocity curve had a destabilizing effect. By using the lumped-
parameter model the previous investigations were veriï¬ed. Internal and external
damping were also proven to be stabilizing factors. Based on consistent results from
the lumped-parameter system it was veriï¬ed that the lumped parameter model pro-
vided a convergent discretization.
In Chapter 5, by using the lumped-parameter model which was analyzed in the
191
previous chapter, the stick-slip oscillations under the distributed contact were inves-
tigated. For a background study the numerical algorithm for solving the nonlinear
stick-slip oscillations of the multi-degree-of-freedom model was provided. Using the
typical stick-slip responses the system dynamic characteristics were explained. Stick-
slip motions were observed in the form of propagating waves of detachments. The
analyses based on the velocity and displacement data were provided. We observed
two distinct motions in the stick-slip oscillations: the series of detachment waves
sweeping over the whole domain and small-grouped localized stick—slip motions. The
detachment waves were ground for the low-frequencies stick-slip oscillations generated
on the contact surface. The small-grouped localized stick-slip motions influenced the
high frequencies of stick-slip oscillations. Several parameter effects on the stick-slip
motions were evaluated. Under high normal loads the frequencies of the sweeping
detachment waves were lowered. In addition, high-frequency small-grouped motions
prevailed. The high Poisson’s ratio increased the natural frequencies of the linear sys-
tem and stick-slip frequencies as well. It was shown that the Poisson’s ratio breaks
the symmetry in the stick-slip motions and ampliï¬ed the irregularity over the stick-
slip motions. It was proved that the driving speed was closely related to amplitudes
of stick-slip oscillations. The friction characteristics, which had a primary effect on
stick-slip motions, were considered. It was observed that the steady state stick-slip
oscillations were prevalent when the system is linearly unstable. It is also shown that
the system having a discontinuous friction model (11, > 11],) generates steady state
stick-slip oscillations. By applying the modal projection method a possible alterna-
tive method in handling the low-dimensional stick-slip oscillations was proposed. The
192
sustained stick-slip oscillations were observed as series of detachments in the busing
system.
In Chapter 6, by adopting the finite element analysis the continuous system under
distributed friction was numerically analyzed. In case of 11, = 11],, the one-dimensional
system underwent steady state pure slipping motions with quasi-harmonic oscilla-
tions. On the other hand, for the two-dimensional system it was possible to sustain
the stick-slip oscillations even under the 11, = #1: condition. The influence of the
normal directional motions on the stick-slip oscillations were presented .
Conclusions related to the bushing system are listed as follows. Under periodic
boundary conditions the bushing system was unstable even with a constant friction
coefï¬cient due to the unstable traveling waves (Chapter 3). In addition, the negative
slope in friction-speed relation destabilized the system (Chapter 4). The instability
led to nonlinear stick-slip oscillations, observed as series of detachments (Chapter
5). Coordinates couplings of the two-dimensional elastic system induced stick-slip
oscillations even with a constant coefï¬cient of friction (Chapter 6). These results
were veriï¬ed through the lumped-parameter method (Chapters 4 and 5), the modal
projection method (Chapter 5), and ï¬nite element analysis (Chapter 6) along with
the exact solution (Chapter 2).
The analytical and numerical investigation in this dissertation dealt with compli-
cated friction-induced vibration phenomena. Listed below are the additional speciï¬c
problems to be investigated in future, which include analytical, numerical, and ex-
perimental studies.
193
0 Material and Friction related Issues
In this investigation there is an assumption that the material has linear prop-
erties. However under high normal loads the elastic material can undergoes
relatively large amplitude stick-slip oscillations. In such case the large oscilla-
tions may not result from the linear material properties. The nonlinear material
properties in stick-slip oscillations need to be investigated for the one- and two-
dimensional systems. When we consider ï¬nite radius bushing models coupling
effects between radial and circumferential motions need to be investigated.
All materials have roughness on the surface. Especially in an elastic material,
such as rubber, the system roughness and noise generation are closely related
(Soom and Kim [42, 43]). Thus including the contact surface roughness is
required in the system modeling for more accurate analysis.
For friction related issues, it has been reported from the experimental studies
that the friction force is not a single function dependent on relative velocities.
It has hysteresis effects caused from the material intrinsic properties (Martins
et al.[118]). Moreover friction models including the properties dependent on the
load and displacement need to be investigated (Dweib and D’Souza [55, 56]).
When the friction is involved heat is generated on the contact surface. In
real operating conditions huge amounts of heat can not be neglected in system
modeling. The system analysis based on the friction-heat relations need to be
investigated (Nakai and Yokoi [72]).
0 Load and Boundary Condition related Issues
194
It has been reported that when friction is involved, results may not be repeatable
even under apparently the same conditions. As investigated by Dweib and
D’Souza [56] geometrical misalignments can generate unpredictable results and
small mistunings can produce undesirable noise (Bhushan [73]). Sometimes
evenly distributed load conditions can not implemented in real situations. In the
bushing system uneven load conditions may occur in some operating conditions
due to a thrust force on the rotating bar. Effects including uneven loads and
external periodic forcing on a system stability need to be investigated.
Stability analysis of pure sliding motions for a two-dimensional elastic system
with ï¬xed ends boundary conditions (the model studied in Chapter 6) remains
as a future work.
System Analysis and Characterization
This dissertation has been devoted to the system analysis through analytical
and numerical approaches. Experimental veriï¬cations should be conducted.
However there may be difï¬culties in sensing the motions and handling huge data
sets from the continuous elastic media. The challenging part of the experimental
studies would be presentations of motions of the materials caused by the stick-
slip oscillations. The ï¬nite element analysis including issues described above
will also be needed in numerical approaches. The characteristic behaviors of
such non-linear phenomena, for example a relation between large fluctuations in
slipping group size and spatial self-similarity, called as self-organized criticality
(Carlson and Langer [97, 98]), would be a great challenge in analysis of stick-slip
195
oscillations induced by friction.
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