Active Nonlinear Vibration Control of Engineering Structures of
Multiple Dimensions
A Thesis
Submitted to the Faculty of Graduate Studies and Research
in Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy
in Industrial Systems Engineering
University of Regina
by
Lin Sun
Regina, Saskatchewan
March, 2015
Copyright 2015: L. Sun
UNIVERSITY OF REGINA
FACULTY OF GRADUATE STUDIES AND RESEARCH
SUPERVISORY AND EXAMINING COMMITTEE
Lin Sun, candidate for the degree of Doctor of Philosophy in Industrial Systems Engineering, has presented a thesis titled, Active Nonlinear Vibration Control of Engineering Structures of Multiple Dimensions, in an oral examination held on March 27, 2015. The following committee members have found the thesis acceptable in form and content, and that the candidate demonstrated satisfactory knowledge of the subject material. External Examiner: *Dr. C. Steve Suh, Texas A&M University
Supervisor: Dr. Liming Dai, Industrial Systems Engineering
Committee Member: Dr. Adisorn Aroonwilas, Industrial Systems Engineering
Committee Member: **Dr. Amr Henni, Industrial Systems Engineering
Committee Member: Dr. Nader Mobed, Department of Physics
Chair of Defense: Dr. Andrei Volodin, Department of Mathematics & Statistics *SKYPE **Not present at defense
I
ABSTRACT
An active nonlinear mechanical vibration control strategy is developed in the
research of the author’s PhD program for the nonlinear vibration control of engineering
structures of multiple dimensions. The proposed control strategy has been applied in
several wildly applied typical engineering structures, including Euler-Bernoulli beams
and axially moving structures.
Nonlinear vibrations wildly exist in engineering structures, such as bridge, aircrafts,
micro-electro-mechanical devices, and elevator cables. Comparing to linear vibrations,
nonlinear vibrations may lead structure failures in short time, and chaotic vibrations
among the nonlinear vibrations features unpredictability.
Considering the damage and unpredictability of nonlinear vibrations, nonlinear
vibrations is ought to be controlled. However, most of the existing active nonlinear
vibration control strategies can only be applied to the nonlinear dynamic system of single
dimension, while multi- dimensional dynamic systems show the advantages over those of
single dimension in dynamic analysis.
Therefore, an active nonlinear control strategy has been proposed based on the
existing control strategy the Fuzzy Sliding Mode Control (FSMC) strategy, and has been
applied in the vibration control of the following engineering structures: Euler-Bernoulli
beams subject to external excitation; axially moving Euler-Bernoulli beam without
external excitation; retracting Euler-Bernoulli beam without external excitation; axially
translating cable; extending nonlinear elastic cable.
II
First of all, the nonlinear vibration and control of an Euler-Bernoulli beam subjected
to a periodic external excitation is given as an example to demonstrate how the active
nonlinear control strategy is developed and applied for a multi-dimensional nonlinear
dynamic system. Then, considering the two typical engineering structures modeled with
Euler-Bernoulli beams, the control strategy is applied in the nonlinear vibration control of
a micro-electro-mechanical system (MEMS) beam and a fluttering beam. After that,
corresponding to the attentions paid to the axially translating materials, the control
strategy is applied in the nonlinear vibration control of four typical axially moving
structures.
Applications of the proposed control strategy evidently show effectiveness and
efficiency of the active control strategy in controlling the nonlinear vibrations of typical
engineering structures.
III
ACKNOWLEDGEMENTS
The author would like to express his sincerest appreciation to his supervisor, Dr.
Liming Dai, for his guidance throughout the whole process of the author’s PhD program.
The advice, encouragement, and financial support from Dr. Liming Dai are of great
importance to the author’s research in the last four years. Without the supervision of Dr.
Dai, it would be impossible for me to implement this research.
The author would also acknowledge the financial support from Natural Sciences and
Engineering Research Council of Canada (NSERC), and the financial support from the
Faculty of Graduate Studies and Research of the University of Regina in the form of
Graduate Scholarships and Travel Funding Awards.
The author’s appreciation also goes to his current research team member Xiaojie
Wang and former research team member Lu Han for their suggestions and
encouragements during his PhD program.
IV
DEDICATION
Special thanks to my parents Huiyan Xing and Dayong Sun for their continuous
encouragement and unconditional support.
V
TABLE OF CONTENTS
ABSTRACT ......................................................................................................................... I
ACKNOWLEDGEMENTS .............................................................................................. III
DEDICATION .................................................................................................................. IV
TABLE OF CONTENTS ................................................................................................... V
LIST OF FIGURES .......................................................................................................... XI
LIST OF TABLES ......................................................................................................... XIX
CHAPTER 1 INTRODUCTION ..................................................................................... 1
1.1 Background ................................................................................................... 1
1.1.1 Euler-Bernoulli Beams .......................................................................... 2
1.1.2 MEMS Beams ....................................................................................... 3
1.1.3 Fluttering Structures .............................................................................. 4
1.1.4 Axially Translating Structures ............................................................... 5
1.1.4.1 Axially Translating Structures with Invariable Dimensions .......... 6
1.1.4.2 Axially Translating Structures with Variable Dimensions ............. 7
1.2 Vibration Controls ........................................................................................ 9
1.2.1 Nonlinear Vibration Control .................................................................. 9
1.2.2 Fuzzy Sliding Mode Control ............................................................... 10
1.3 Aims of the Research .................................................................................. 11
VI
1.4 Construction of the Dissertation ................................................................. 12
1.4.1 Engineering Structures Represented with Euler-Bernoulli Beam ....... 12
1.4.2 Engineering Structures Represented with Cable ................................. 15
CHAPTER 2 DEVELOPMENT OF ACTIVE NONLINEAR VIBRATION
CONTROL STRATEGY FOR MULTI-DIMENSIONAL DYNAMIC SYSTEMS ....... 17
2.1 Introduction ................................................................................................. 17
2.2 Equations of Motion ................................................................................... 18
2.3 Series Solutions ........................................................................................... 22
2.4 Active Nonlinear Vibration Control ........................................................... 24
2.5 Nonlinear Vibration Characterization ......................................................... 29
2.6 Active Nonlinear Vibration Control ........................................................... 37
2.7 Conclusion .................................................................................................. 40
CHAPTER 3 MEMS EULER-BERNOULLI BEAM SUBJECTED TO EXTERNAL
NON-PERIODIC EXCITATION ..................................................................................... 42
3.1 Introduction ................................................................................................. 42
3.2 Equations of Motion ................................................................................... 43
3.3 Series Solutions ........................................................................................... 48
3.4 Stability Analysis ........................................................................................ 51
3.5 Control Design ............................................................................................ 59
3.5.1 Active Control Strategy ....................................................................... 59
VII
3.5.2 Two-Phase Control Method ................................................................ 60
3.6 Application of the Control Method ............................................................. 63
3.6.1 Application of the First Control Phase ................................................ 63
3.6.2 Application of the Second Control Phase ............................................ 65
3.7 Conclusions ................................................................................................. 69
CHAPTER 4 FLUTTERING EULER-BERNOULLI BEAM SUBJECTED TO
EXTERNAL NON-PERIODIC EXCITATION ............................................................... 71
4.1 Introduction ................................................................................................. 71
4.2 Equations of Motion ................................................................................... 71
4.3 Series Solution ............................................................................................ 73
4.4 Control Design ............................................................................................ 75
4.5 Numerical Simulation ................................................................................. 76
4.6 Conclusion .................................................................................................. 86
CHAPTER 5 AXIALLY TRANSLATING EULER-BERNOULLI BEAM OF FIXED
LENGTH WITOUT EXTERNAL EXCITATION .......................................................... 87
5.1 Introduction ................................................................................................. 87
5.2 Equations of Motion ................................................................................... 87
5.3 Series Solutions ........................................................................................... 92
5.4 Control Design ............................................................................................ 95
5.5 Numerical Simulation ................................................................................. 97
VIII
5.6 Conclusion ................................................................................................ 107
CHAPTER 6 AXIALLY RETRACTING EULER-NOULLI BEAM WITHOUT
EXTERNAL EXCITATION .......................................................................................... 109
6.1 Introduction ............................................................................................... 109
6.2 Equations of Motion ................................................................................. 109
6.3 Series Solution .......................................................................................... 117
6.4 Control Design .......................................................................................... 120
6.5 Numerical Simulation ............................................................................... 121
6.6 Conclusions ............................................................................................... 128
CHAPTER 7 AXIALLY TRANSLATING CABLE WITHOUT EXTERNAL
EXCITATION .............................................................................................................. 130
7.1 Introduction ............................................................................................... 130
7.2 Equations of Motion ................................................................................. 130
7.3 Series Solutions ......................................................................................... 138
7.4 Control Design .......................................................................................... 139
7.5 Numerical Simulation ............................................................................... 140
7.5.1 Chaotic Vibration .............................................................................. 142
7.5.2 Amplitude Synchronization ............................................................... 145
7.5.2.1 0175.0rA ................................................................................ 145
7.5.2.2 015.0rA ................................................................................. 149
IX
7.5.2.3 010.0rA ................................................................................. 153
7.5.3 Frequency Synchronization ............................................................... 157
7.5.3.1 0553.0r ............................................................................... 157
7.5.3.2 1107.0r ............................................................................... 161
7.5.3.3 1660.0r ............................................................................... 165
7.6 Conclusions ................................................................................................... 169
CHAPTER 8 AXIALLY EXTENDING CABLE WITHOUT EXTERNAL
EXCITATION .............................................................................................................. 170
8.1 Introduction ............................................................................................... 170
8.2 Equations of Motion ................................................................................. 170
8.3 Series Solutions ......................................................................................... 177
8.4 Control Design .......................................................................................... 180
8.5 Numerical Simulation ............................................................................... 181
8.6 Conclusions ............................................................................................... 188
CHAPTER 9 CONCLUSIONS AND FUTURE WORKS ......................................... 190
9.1 Conclusion ................................................................................................ 190
9.2 Future Works ............................................................................................ 192
BIBLIOGRAPHY ........................................................................................................... 194
APPENDIX ..................................................................................................................... 203
X
PEER REVIEWED PUBLICATIONS OF THE AUTHOR........................................... 209
XI
LIST OF FIGURES
Figure 2.2 The chaotic vibration of w : (a) the wave diagram; (b) the 2-D phase diagram;
(c) the Poincaré map ................................................................................................. 33
Figure 2.3 The chaotic vibration of pw : (a) the wave diagram; (b) the 2-D phase diagram;
(c) the Poincaré map ................................................................................................. 34
Figure 2.4 The wave diagrams of the first three vibration modes before the application of
the active control strategy: (a) 1w ; (b) 2w ; (c) 3w ..................................................... 36
Figure 2.5 The wave diagram of pw with the application of the active control strategy . 38
Figure 2.6 The wave diagrams of the first three vibration modes with the application of
the active control strategy: (a) 1w ; (b) 2w ; (c) 3w ..................................................... 39
Figure 2.7 The comparison between the wave diagram of pw (the blue continuous line)
and the reference signal rw (the green dash line) ..................................................... 40
Figure 2.8 The control input U ......................................................................................... 40
Figure 3.1 The sketch of the MEMS beam ....................................................................... 43
Figure 3.2 The wave diagram of pw in the case of vV ac 14 ............................................. 54
Figure 3.3 The wave diagrams of the first three vibration modes in the case of vV ac 14 : (a)
1w ; (b) 2w ; (c) 3w ..................................................................................................... 55
XII
Figure 3.4 The wave diagram of pw in the case of vV ac 5.14 .......................................... 56
Figure 3.5 The wave diagrams of the first three vibration modes in the case of vV ac 5.14 :
(a) 1w ; (b) 2w ; (c) 3w ............................................................................................... 57
Figure 3.6 The wave diagram of pw in the case of vV ac 15 ........................................... 58
Figure 3.7 The wave diagrams of the first three vibration modes in the case of vV ac 15 :
(a) 1w ; (b) 2w ; (c) 3w ............................................................................................... 59
Figure 3.8 The comparison between the wave diagram of pw (the continuous blue line)
and the reference signal (the green dash line) in the first control phase ................... 64
Figure3.9 The wave diagrams of the first three vibration modes in the first control phase:
(a) 1w ; (b) 2w ; (c) 3w ............................................................................................... 65
Figure 3.10 The control input U in the first control phase .............................................. 65
Figure 3.11 The vibration of the second control phase: (a) the wave diagram of pw ; (b)
the comparison between the wave diagram pw (the continuous blue line) and the
reference signal (the green dash line) ....................................................................... 67
Figure 3.12 The wave diagrams of the first three vibration modes in the second control
phase: (a) 1w ; (b) 2w ; (c) 3w .................................................................................... 68
Figure 3.13 The control input U in the second control phase .......................................... 69
XIII
Figure 4.1 The sketch of the fluttering Euler-Bernoulli beam .......................................... 71
Figure 4.2 The wave diagram of pw before and after the application of the active control
strategy ...................................................................................................................... 78
Figure 4.3 The 2-D phase diagram of pw before the application of the active control
strategy ...................................................................................................................... 79
Figure 4.4 The wave diagrams of the first six vibration modes before the application of
the active control strategy: (a) 1w ; (b) 2w ; (c) 3w ; (d) 4w ; (e) 5w ; (f) 6w .............. 81
Figure 4.5 The wave diagram of pw after the application of the active control strategy . 82
Figure 4.6 The wave diagrams of the first six vibration modes after the application of the
active control strategy: (a) 1w ; (b) 2w ; (c) 3w ; (d) 4w ; (e) 5w ; (f) 6w .................... 84
Figure 4.7 The comparison between pw (denoted with the continuous blue line) and rw
(denoted with the green dash line) in wave diagram ................................................ 85
Figure 4.8 The control input U ........................................................................................ 85
Figure 5.1 The sketch of the axially translating Euler-Bernoulli beam ............................ 88
Figure 5.2 The wave diagram of pw before and after the application of the active control
strategy .................................................................................................................... 100
Figure 5.3 The 2-D phase diagram of pw before the application of the active control
strategy .................................................................................................................... 100
XIV
Figure 5.4 The wave diagrams of the first six vibration modes before the application of
the active control strategy: (a) 1w ; (b) 2w ; (c) 3w ; (d) 4w ; (e) 5w ; (f) 6w ............ 103
Figure 5.5 The wave diagram of pw after the application of the active control strategy 104
Figure 5.6 The wave diagrams of the first six vibration modes after the application of the
active control strategy: (a) 1w ; (b) 2w ; (c) 3w ; (d) 4w ; (e) 5w ; (f) 6w .................. 106
Figure 5.7 The comparison between pw (denoted with the continuous blue line) and rw
(denoted with the yellow dash line) in wave diagram ............................................ 107
Figure 5.8 The control input U ....................................................................................... 107
Figure 6.1 The sketch of the retracting Euler-Bernoulli beam ....................................... 110
Figure 6.2 The wave diagram of pw without the application of the active control strategy
................................................................................................................................. 123
Figure 6.3 The wave diagrams of the first three vibration modes without the application
of the active control strategy: (a) 1w ; (b) 2w ; (c) 3w .............................................. 124
Figure 6.4 The wave diagram of pw with the application of the active control strategy 125
Figure 6.5 The wave diagrams of the first three vibration modes with the application of
the active control strategy: (a) 1w ; (b) 2w ; (c) 3w .................................................. 126
Figure 6.6 The comparison between pw (the continuous blue line) and rw (the green
dash line) in wave diagram ..................................................................................... 127
XV
Figure 6.7 The control input U ...................................................................................... 128
Figure 7.1 The sketch of the axially translating cable with fixed-fixed ends ................. 131
Figure 7.2 The wave diagram of pw without the application of the active control strategy
................................................................................................................................. 142
Figure 7.3 The wave diagrams of the first three vibration modes without the application
of the active control strategy: (a) 1w ; (b) 2w ; (c) 3w ............................................. 144
Figure 7.4 The wave diagram of pw with the application of the active control strategy in
the case of 0175.0rA .......................................................................................... 146
Figure 7.5 The wave diagrams of the first three vibration modes with the application of
the active control strategy in the case of 0175.0rA : (a) 1w ; (b) 2w ; (c) 3w ...... 147
Figure 7.6 The comparison between pw (the continuous blue line) and rw (the green
dash line) in wave diagram in the case of 0175.0rA .......................................... 148
Figure 7.7 The control input U in the case of 0175.0rA ........................................... 149
Figure 7.8 The wave diagram of pw with the application of the active control strategy in
the case of 015.0rA ............................................................................................ 150
Figure 7.9 The wave diagrams of the first three vibration modes with the application of
the active control strategy in the case of 015.0rA : (a) 1w ; (b) 2w ; (c) 3w ........ 151
XVI
Figure 7.10 The comparison between pw (the continuous blue line) and rw (the green
dash line) in wave diagram in the case of 015.0rA ............................................ 152
Figure 7.11 The control input U in the case of 015.0rA .......................................... 153
Figure 7.12 The wave diagram of pw with the application of the active control strategy in
the case of 010.0rA ........................................................................................... 154
Figure 7.13 The wave diagrams of the first three vibration modes with the application of
the active control strategy in the case of 010.0rA : (a) 1w ; (b) 2w ; (c) 3w ....... 155
Figure 7.14 The comparison between pw (the continuous blue line) and rw (the green
dash line) in wave diagram in the case of 010.0rA ........................................... 156
Figure 7.15 The control input U in the case of 010.0rA .......................................... 157
Figure 7.16 The wave diagram of pw with the application of the active control strategy in
the case of 0553.0r ......................................................................................... 158
Figure 7.17 The wave diagrams of the first three vibration modes with the application of
the active control strategy in the case of 0553.0r : (a) 1w ; (b) 2w ; (c) 3w ...... 159
Figure 7.18 The comparison between pw (the continuous blue line) and rw (the green
dash line) in wave diagram in the case of 0553.0r ......................................... 160
Figure 7.19 The control input U in the case of 0553.0r ........................................ 161
XVII
Figure 7.20 The wave diagram of pw with the application of the active control strategy in
the case of 1107.0r ......................................................................................... 162
Figure 7.21 The wave diagrams of the first three vibration modes with the application of
the active control strategy in the case of 1107.0r : (a) 1w ; (b) 2w ; (c) 3w ...... 163
Figure 7.22 The comparison between pw (the continuous blue line) and rw (the green
dash line) in wave diagram in the case of 1107.0r ......................................... 164
Figure 7.23 The control input U in the case of 1107.0r ........................................ 165
Figure 7.24 The wave diagram of pw with the application of the active control strategy in
the case of 1660.0r ......................................................................................... 166
Figure 7.25 The wave diagrams of the first three vibration modes with the application of
the active control strategy in the case of 1660.0r : (a) 1w ; (b) 2w ; (c) 3w ...... 167
Figure 7.26 The comparison between pw (the continuous blue line) and rw (the green
dash line) in wave diagram in the case of 1660.0r ......................................... 168
Figure 7.27 The control input U in the case of 1660.0r ........................................ 169
Figure 8.2 The wave diagram of pw without the application of the active control strategy
................................................................................................................................. 182
XVIII
Figure 8.3 The wave diagrams of the first three vibration modes without the application
of the active control strategy: (a) 1w ; (b) 2w ; (c) 3w ............................................. 184
Figure 8.4 The wave diagram of pw with the application of the active control strategy 185
Figure 8.5 The wave diagrams of the first three vibration modes with the application of
the active control strategy: (a) 1w ; (b) 2w ; (c) 3w .................................................. 186
Figure 8.6 The comparison between pw (the continuous blue line) and rw (the green
dash line) in wave diagram ..................................................................................... 187
Figure 8.7 The control input U ...................................................................................... 187
XIX
LIST OF TABLES
Table 2.1 The fuzzy rule of fsU ........................................................................................ 28
1
CHAPTER 1 INTRODUCTION
1.1 Background
Mechanical vibrations can be found everywhere in industrial practices and most of
such vibrations are undesirable. The vibrations may significantly reduce the accuracy,
stability and operational life of the equipment used in disciplines such as mechanical,
civil, structural, automotive, aeronautical, and aerospace engineering. Vibrations could
lead to structural resonance which may cause mechanical failures of catastrophic nature.
Statistical studies in North America estimate that approximately 90% of mechanical
service failures are fatigue failures, which are closely related to mechanical vibrations. It
is estimated that the total cost of fatigue failures alone in developed countries is on the
order of 4% of the GNP. Vibration control and effective attenuation of mechanical
vibrations are therefore economically and practically significant.
Among all the engineering structures, beams and cables are the structures commonly
seen in industries and the vibrations together with the control of the vibrations of these
structures are the primary concerns of the engineers in practice. The engineering
structures such as beams and cables are therefore considered in the present dissertation.
Strictly speaking, almost all the vibrations of the engineering structures are nonlinear
vibrations which are unpredictable in nature and sensitive to initial and operation
conditions. However, the conventional studies on the vibrations are focused on linear
vibrations. As it is well known in the field, methodologies used to study the behaviors of
nonlinear vibrations are significantly different from that used for linear vibrations.
Moreover, most of the investigations on the vibrations of the structures are with single
2
dimensional approaches, though most of the structures such as beams and cables are
actually multi-dimensional and the approaches for analysing the behaviors of the
vibrations should also be multi-dimensional. Indeed, more and more researchers
recognize the importance and necessity of multi-dimensional approaches in studying the
vibrations of the engineering structures, especially when nonlinear or chaotic vibrations
are considered. The findings and conclusive results of the existing research works in this
field are described below corresponding to typical beams and cables.
1.1.1 Euler-Bernoulli Beams
In 2002, the chaotic vibrations of an Euler-Bernoulli beam with simply-supported
boundaries was investigated (Ng and Xu, 2002), and it was discovered that a variety of
dynamic behaviors including multi-periodic vibration and chaos exist in the single
dimensional system. It should be noticed: only the first vibration mode of the nonlinear
system of the Euler-Bernoulli beam is derived to facilitate the numerical simulations
presented in the investigation, while the assumption for the simplification of the system
rarely exists in the practical engineering field. Years later, the nonlinear vibrations of an
Euler-Bernoulli beam with switching cracks was investigated (Caddemi et al., 2010), and
the vibrations of an Euler-Bernoulli beam with different boundary conditions were
investigated.
Corresponding to the demands in aeronautics and astronautics, the variation of the
nonlinear dynamic behaviors of an Euler-Bernoulli beam in response to different
boundary conditions has been investigated in 2011 (Awrejcewicz et al., 2011). In the
study, four transition scenarios have been discovered when the vibration of the
3
investigated Euler-Bernoulli beam varies from a periodic one to a chaotic one. Two years
later, the chaotic behaviors of an Euler-Bernoulli beam was further studied (Awrejcewicz
et al., 2013), and a dynamic behavior transition of the beam from chaos to hyper-chaos
was discovered. In the numerical simulation, the first two vibration modes of the
transverse vibration were taken considered to guarantee the reliability of the numerical
results.
It has been clearly figured out in the previous studies: the single dimensional
dynamic system of an Euler-Bernoulli beam can only be applied in the investigation of
the primary resonance (Askari and Esmailzadeh, 2014), and it will not be available to
represent the vibration in the case of internal resonance (Alhazza et al., 2008), which
requires at least a two-dimensional dynamic system. Attempts have been made to
determine the number of dimensions required to approximate the nonlinear vibration of a
rotating Euler-Bernoulli beam (Kuo and Lin, 2000) and the vibration of a Euler-Bernoulli
beam with clamped-clamped boundaries (Weeger et al., 2013), while so far only few
efforts has been made to develop an active control strategy available for the nonlinear
vibration control of a beam of multiple dimensions.
1.1.2 MEMS Beams
In the application of micro-electro-mechanical systems (MEMS), beams including
Euler-Bernoulli beams have been implemented as one of the fundamental models of
various MEMS devices, such as resonators (Ghayesh et al., 2013), actuators (Choi and
Lovell, 1997; Nayfeh and Younis, 2004; Tusset et al., 2012), sensors (Zhang and Meng,
2007; Guerrero-Castellanos et al., 2013; Ramezani, 2013), and radio frequency switches
4
(Zhang et al., 2002; Patton and Zabinski, 2005; Guo et al., 2007). The existence of
nonlinear behaviors, such as chaos has been discovered in a number of MEMS devices
(Wang et al., 1998; Azizi et al., 2013). A nonlinear resonant microbeam represented with
an Euler-Bernoulli beam was systematically analyzed (Younis and Nayfeh, 2003) with
accurate dynamics predictions, which linear models fails to explain. It is interesting to
notice that in this study a two-dimensional dynamical system was derived in order to get
a reliable model of the microbeam and to better analyze the microbeam. In addition to a
series of theoretical studies on microbeams, the experimental investigation has also been
found in the literature. The influence of a resonant microbeam on its dynamical behavior
was studied (Mestrom et al., 2008). In this study, a fixed-fixed microbeam represented
with an Euler-Bernoulli beam was investigated both experimentally and numerically, and
a quantitative match between the numerical simulation and the experiments was reported.
Through both the numerical simulations and the experimental investigations, the
nonlinear behavior of an electrostatically driven microbeam was studied (Alsaleem et al.,
2009) in a series of experiments, and numerous nonlinear behaviors, such as dynamic
pull-in and jumps, were demonstrated.
From the previous studies on the nonlinear dynamics of MEMS devices, which are
generally represented with Euler-Bernoulli beams, it can be learned that a multi-
dimensional nonlinear dynamic system of the MEMS structure should be considered, and
at least a two-dimensional system is required for studies on the internal resonance
(Younis and Nayfeh, 2003).
1.1.3 Fluttering Structures
5
With the development of panels applied in aerospace, civil structures, beams, as well
as plates, have been implemented as fundamental models in the nonlinear dynamic
studies of fluttering panel. In 2001, the nonlinear fluttering phenomena of a panel
subjected to thermal loads were studied, and a Timoshenko beam theory (Oh and Lee,
2001) was implemented in the model establishment. It was reported in the study that
periodic vibrations and chaotic vibrations was discovered in numerical simulation, and it
was concluded nonlinear large-amplitude vibrations of a fluttering panel may lead to a
fatigue failure.
Considering the extensive application of supersonic flight vehicles and space
shuttles, the nonlinear vibration of graded plate in supersonic airflow was investigated
(Haddadpour et al., 2007). To guarantee the accuracy of the results derived in numerical
simulations, two multi-dimensional systems of the graded plate were applied, including a
four-dimensional system and a six-dimensional system. After the verification of the
numerical results, the six-dimensional system was selected and it was discovered that the
stress distribution along the thickness of the panel is nonlinear.
It should be noticed that a six-dimensional nonlinear dynamic system of a fluttering
panel has not been only applied in the recent works, but also in the works published
decades ago. A six-dimensional nonlinear dynamic system of a fluttering panel with
simply-supported boundary conditions was implemented to investigate the buckling
effect (Dowell, 1966; Dowell, 1967). That is, a multi-dimensional system is necessary in
the nonlinear dynamic analysis of fluttering panel.
1.1.4 Axially Translating Structures
6
The investigations on axially translating structures, such as axially translating beam
(Tabarrok et al., 1974; Chang et al., 2010; Zhao and Wang, 2013), cable (Le-Ngoc and
McCallion, 1999; Tang et al., 2011; Sandilo and Horssen, 2014), and plate (Luo and
Hamidzadeh, 2004; Zhou and Wang, 2007; Ghayesh and Amabili, 2013), have been
conducted extensively in the last several decades. The axially translating structures can be
generally divided into two classes depending on whether or not the dimension along the
translating direction is variable.
1.1.4.1 Axially Translating Structures with Invariable Dimensions
Corresponding to designing the aerospace and aeronautical structures, an axially
travelling plate, of which the length along the travelling direction is a constant, were
investigated (Luo and Hamidzadeh, 2004). In the study, the analytical solutions of high-
speed travelling plates, as well as the buckling stability, have been derived. In addition to
axially moving plates, axially moving beam, of which the length is invariant, has drawn
scholars’ attention as well. For the applications (Carrera et al., 2011), in which the effects
of shear deformations cannot be neglected, an axially moving Timoshenko beam was
introduced (Ghayesh and Amabili, 2013) to investigate the internal resonance of the beam.
In the study, a twenty-dimensional nonlinear dynamic system of Timoshenko beams with
invariant length, has been derived, and bifurcations as well as chaos have been discovered
from the established system.
The effects of the translating speed of the power transmission chains, aerial
cableways, textile fibers and paper sheets on their vibration features were also
investigated by researchers and engineers. In 1992, the nonlinear behavior and translating
7
speed of axially translating Euler-Bernoulli beam and string were quantitatively studied
(Wickert, 1992). The author even claimed that higher-order equilibria of the translating
beam can help complete the dynamic investigation of the investigated Euler-Bernoulli
beam. In 2009, the energy transfer in a moving belt system was investigated, and a
nonlinear eight-dimensional dynamic system was implemented (Hedrih, 2009). String-
drive system also draws attentions from scholars, and the nonlinear vibration of an axially
translating string with invariant length was investigated (Ghayesh and Moradian, 2011).
In the study, a two-dimensional nonlinear system of a string was applied and it was
discovered that the frequencies of the first two vibration modes would be significantly
influenced by the foundation length.
1.1.4.2 Axially Translating Structures with Variable Dimensions
The class of the axially moving structures with varying length, has also been
investigated wildly for its various applications in areas of engineering, and Euler-
Bernoulli beam has been taken as one of the fundamental model in the previous studies.
Coming from the dynamics of spacecraft antenna, of which the length is varying with
time, the equation of motion of a cantilevered beam was established and an Euler-
Bernoulli beam assumption was implemented (Tabarrok et al., 1974). In the study, a
sixteen-dimensional system of an Euler-Bernoulli beam with fixed-free boundary
conditions has been derived, and a decreasing frequency has been discovered with respect
to the increasing length of the beam. Corresponding to hydraulic and motor driven
systems, four nonlinear axially moving cantilevered beam models considering a tip mass
have been derived, including Euler-Bernoulli beam, Timoshenko beam, simple-flexible
beam and rigid-body beam (Fung et al., 1998). In the model establishment of the study, it
8
is pointed out the rigid-body motion and the flexible vibration of the beam are
nonlinearly coupled and there exists Coriolis forces in the system. Motivated with the
application of axially translating media in elevator, the linear dynamics of the class of
arbitrary varying length cable with a tip mass was investigated (Zhu and Ni, 2000). In
2013, a five-dimensional system has been derived for the investigation of the energetics
and stability of the cable. In the quest for understanding the fluid-structure interactions, a
cylindrical cantilevered beam axially immersing in fluid was investigated (Gosselin et al.,
2007). A four-dimensional system has been derived and it is reported the system presents
a phase of decaying oscillation with increasing amplitude and decreasing frequency. With
the interests in self-spinning tethered satellites, an Euler-Bernoulli beam was
implemented to represent a tether with varying length (Tang et al., 2004). The
deployment process of two-self spinning tethered satellite systems have been successfully
simulated through a newly proposed hybrid Eulerian and Lagrangian frame work. In 2014,
the geometrically nonlinear kinematics of a two-dimensional extensible Euler-Bernoulli
was investigated, to demonstrate the effects of realistic load on the dynamics of thin-
walled structures, such as ships and bridges (Kitarovic, 2014).
From the previous works on the dynamics of axially translating structures, a multi-
dimensional system of an Euler-Bernoulli beam is usually preferred (Tabarrok et al.,
1974; Zhu and Ni, 2000; Gosselin et al., 2007; Tang et al., 2004) in approximating the
transverse displacement of the structures, and an increasing-amplitude vibration has been
reported in the case of beams and strings with varying length (Zhu and Ni, 2000;
Gosselin et al., 2007).
9
1.2 Vibration Controls
1.2.1 Nonlinear Vibration Control
Primarily, strategies of vibration control fall in two categories: passive and active
vibration controls. Though widely used in engineering fields, passive vibration control
comes at the cost of added weight and size and proves inadequate in moderate to high
frequency regimes. Active vibration control has therefore increasingly attracted attention
from researchers and engineers due to its advantages of self-adaptation and high
efficiency in practice. Theoretically, a proper active vibration control system may bring
no resonance and no amplification of mechanical vibrations at any frequency. Among all
the mechanical vibrations, most are actually nonlinear vibrations which are unpredictable
and sensitive to initial conditions and may lead to very large amplitudes with random like
frequencies and variations. The analytical tools widely used in linear vibration studies,
such as linear superposition, are not valid for nonlinear vibration analyses. Control of
nonlinear vibrations is hence a challenge facing researchers and engineers in physics,
engineering and industries. Although numerous investigations have been carried out and
a considerably large number of control strategies have been reported, most of them are
for linear systems. Among the nonlinear vibration control strategies, only a few are
dealing with active nonlinear vibration control (Hong et al, 2014; Qin et al, 2013) which
are all limited to systems of single dimension. Few control strategies are available in
literature for controlling nonlinear vibrations of multi-dimensional systems, though most
engineering structures are multi-dimensional and analyses with multi-dimensional
approach are more accurate and reliable.
10
1.2.2 Fuzzy Sliding Mode Control
Corresponding to the nonlinear and chaotic vibrations and large-amplitude vibrations
in beams and cables, a number of active control strategies were proposed and
implemented both theoretically and experimentally (Zakerzadeh et al., 2011), but most of
them are with single dimensional approach. An active vibrations control strategy named
the sliding modes control (SMC) was proposed as one of the active nonlinear vibration
control strategy (Utkin, 1992). Based on SMC, a control strategy considering the external
uncertainties of nonlinear systems was developed through the application of fuzzy logic
theories and hence named as fuzzy sliding mode control (FSMC). FSMC has been
applied for controlling the nonlinear vibrations existing in engineering systems, and its
applicability has been demonstrated in nonlinear vibration controls (Yau et al., 2006; Kuo,
2007; Yau et al., 2011).
Although the FSMC strategy has been successfully employed in controlling the
nonlinear and chaotic response of a micro-electro mechanical system (MEMS) (Haghighi
and Markazi, 2010), the established FSMC strategy is merely applicable for the
dynamical system of single dimension. From the literatures presented previously, it is
clearly demonstrated that a multi-dimensional nonlinear dynamic system should be
implemented rather than a single dimensional one, in the investigations on chaotic
vibration or large-amplitude vibrations.
Therefore, a thorough and systematic study of active nonlinear vibration control is
therefore needed, and a theoretically and practically sensible active vibration control
11
technique needs to be developed for controlling the multidimensional nonlinear
vibrations extensively observed in industrial practices.
1.3 Aims of the Research
The aims of the research of this dissertation include the development of an active
nonlinear vibration control strategy with which the nonlinear mechanical vibrations of the
typical engineering structures including beams and cables can be effectively controlled.
With the advantages of the sliding mode control strategy (FSMC) and its considerations
of the uncertainties of nonlinear systems, the control strategy is to be developed on
modifying the existing FSMC strategy. Significantly, the active nonlinear vibration
control strategy to be developed is aimed to control the multi-dimensional engineering
structures. With the availability of such active vibration control strategy, it is anticipated
that the nonlinear vibrations of the engineering structures can be effectively and reliably
controlled via a multi-dimensional approach.
In properly applying the control strategy to be developed for controlling the
nonlinear vibrations of engineering structures of various types and of multi-dimensions,
different approaches may have to be taken corresponding to the structures of different
geometries, responses and materials of the structures and the characteristics of the
physical and mathematical models used for governing the responses of the structures.
Numerous types of typical engineering structures are to be taken into considerations.
Modeling of the dynamical systems of each of the structures, characteristics of the
vibratory responses of the systems and the specific conditions and restrictions in applying
the control strategy to be developed will be studied in details. The applicability and
12
reliability of the control strategy to be developed are also to be investigated for the typical
engineering structures such as beams and cables.
1.4 Construction of the Dissertation
To systematically describe the development of the active nonlinear vibration control
strategy desired and to express the characteristics and considerations in applying the
control strategy for controlling the nonlinear vibrations of the typical engineering
structures list above, this PhD dissertation is constructed in such a way that the control
strategy development and the applications of the control strategy together with the
particular requirements and conditions are emphasized specifically for each of the typical
engineering structures, which are represented with beams and cables. The following
chapter is mainly for presenting the development of the active nonlinear vibrations
control strategy, namely the modified fuzzy sliding mode control strategy for nonlinear
and multi-dimensional dynamic systems. In correctly applying the control strategy to be
developed, one may have to bear in mind the uniqueness of each of the typical
engineering structures considered. The chapters from Chapter 3 to Chapter 8 are therefore
structured for describing the applications of the control strategy developed together with
the structure modeling, solution developments, degree of the dimensions desired, and the
requirements and conditions essential for the applications, corresponding to a specific
engineering structure considered.
1.4.1 Engineering Structures Represented with Euler-Bernoulli Beam
13
As indicated above, development of the active nonlinear vibration control strategy is
described in Chapter 2. A general Euler-Bernoulli beam is utilized to demonstrate the
development and application of the control strategy. Accordingly, the active nonlinear
vibration control of an Euler-Bernoulli beam subjected to a sinusoidal external excitation
is presented, to investigate the vibration control of engineering structures such as bridge
(Wu and Law, 2012). The importance of multi-dimensional dynamic system is
demonstrated, and then an active control strategy for the nonlinear vibration control of a
multi-dimensional system is proposed for stabilizing the discovered chaotic vibration.
The sinusoidal external excitation, which is evenly distributed on the upper surface of the
beam, differentiates the active vibration control in Chapter 2 from those in the rest of the
chapters. The features of the application of the control strategy developed in frequency
synchronization, due to the evenly distributed periodic external excitation, are described
in this chapter.
In Chapter 3, the active nonlinear vibration control of a MEMS Euler-Bernoulli
beam is presented for the vibration control of MEMS resonators (Mestrom et al., 2008).
The engineering structure investigated in Chapter 3 features an external excitation in the
form of an electro-static force. It should be noticed that the electro-static force is a non-
periodic force and hence different from the external sinusoidal excitation in Chapter 2. In
the application of the active control strategy proposed in Chapter 2, difficulties have been
found in the selection of proper control parameters, and therefore a new control method
named as two-phase control method is proposed. In numerical simulation, the two-phase
control method shows its advantage in facilitating the process of the active nonlinear
vibration control.
14
In Chapter 4, the active nonlinear vibration control of a fluttering Euler-Bernoulli
beam in supersonic airflow is presented for the vibration control of fluttering panels (Oh
and Lee, 2001). The engineering structure investigated in Chapter 4 is similar to that in
Chapter 3, since both the structures feature non-periodic external excitation. However,
the external excitation applied on the fluttering Euler-Bernoulli beam is an aerodynamic
load and approximated with the first-order piston theory. In the numerical simulation, it is
discovered that the contributions from higher vibration modes are less than those from
lower mode, and thus a six-dimensional nonlinear system is applied instead of a three-
dimensional one that has been applied in Chapter 2 and Chapter 3.
In Chapter 5, the active nonlinear vibration control of an axially translating Euler-
Bernoulli beam with pinned-pinned boundaries is presented. The engineering structure
investigated in Chapter 5 features an axial moving velocity and no external excitation.
Besides, comparing to the chaotic vibrations discovered in the three-dimensional system
established in Chapter 2 and Chapter 3, a nonlinear dynamic system of higher dimensions
is found necessary to accurately describe the chaotic vibration of the axially translating
Euler-Bernoulli beam and therefore the first six vibration modes of the axially translating
beam is implemented. The numerical results prove the applicability of the proposed
active control strategy in the nonlinear vibration control of the established six-
dimensional system.
In Chapter 6, the active nonlinear vibration control of a retracting Euler-Bernoulli
beam is presented for vibration control of robotic arms (Chang et al., 2010). The
retracting Euler-Bernoulli beam investigated in Chapter 6 is different from the one in
15
Chapter 5 the, since it features a decreasing axial dimension instead of a constant one. A
large-amplitude vibration is discovered from the multi-dimensional dynamic system
established in Chapter 6, and the effectiveness of the proposed active vibration control
strategy is demonstrated in stabilizing the discovered large-amplitude vibration.
1.4.2 Engineering Structures Represented with Cable
In Chapter 7, the active nonlinear vibration control of an axially translating cable is
presented for the vibration control of power transmission belts (Le-Ngoc and McCallion,
1999). The importance of a multi-dimensional dynamic system of the engineering
structure is demonstrated with the large-amplitude vibration discovered from the system.
The active control strategy proposed in Chapter 2 is then applied, and it is discovered that
the active nonlinear vibration control of a structure without bending moment would
significantly enhance the applicability of the proposed control strategy, since both the
frequency synchronization and the amplitude synchronization of the axially translating
cable .are found more easily in the selection of the control parameters, low control cost
for continuous control, and various synchronizations corresponding to different desired
reference signals.
In Chapter 8, the active nonlinear vibration control of an extending nonlinear elastic
cable is presented for the vibration control of elevator cable (Zhu and Ni, 2000). The
engineering structure investigated in Chapter 8 is different from the one in Chapter 7,
considering that the axial dimension of the cable will increase with respect to time. The
increasing axial dimension of the investigated engineering structure features no chaotic
vibrations existing in the vibration of the structure, but a significantly large-amplitude
16
vibration. The application of the active control strategy of the large-amplitude vibration
of the cable features a very well synchronization since the difference between the actual
vibration of the cable and that of the desired reference signal can be barely observed, and
also an almost zero control cost once the synchronization is achieved.
17
CHAPTER 2 DEVELOPMENT OF ACTIVE
NONLINEAR VIBRATION CONTROL STRATEGY
FOR MULTI-DIMENSIONAL DYNAMIC SYSTEMS
2.1 Introduction
In this chapter, the equations of vibration of a general Euler-Bernoulli beam are to be
established based on the Hamilton’s principle. In physics, the Hamilton’s principle
considers the energies of a dynamic system and the actual path of the system followed is
that which minimizes the time integral of the difference between the kinetic and potential
energies.
A multi-dimensional system is to be derived for modeling the vibration of the beam,
via the non-dimensionalization and discretization. With the concepts of the existing
FSMC design, an active control strategy will be developed corresponding to the
governing equations of the beam. The chaotic vibration of the Euler-Bernoulli beam is to
be discovered and determined via the PR method (Dai and Singh, 1997; Dai, 2008), and
the effects of each of the vibration modes on the vibrations of the beam will be
emphasized and compared with that of a single dimensional system. The active control
strategy will be applied in controlling the large-amplitude chaotic vibrations of the beam
at a selected point via a single controller, rather than multiple controllers in the previous
work (Dai and Sun, 2012). The effectiveness of the control strategy to be proposed will
be emphasized in this chapter.
18
2.2 Equations of Motion
The Euler-Bernoulli beam with simply-supported boundaries investigated in this
chapter is sketched in Fig. 2.1. The governing equations of motion of the beam are to be
derived based on the Hamilton’s principle. As can be seen from Fig. 2.1, the length of the
beam is given as l , the width of the beam is b , and the thickness of the beam is h . The
x axis is along the axial direction of the beam. The displacement of any point of the
beam along the x- and z- axes are designated with u and w .
Figure 2.1 The sketch of the Euler-Bernoulli beam
Starting from the origin of the beam, a position vector, r
, of any point zx, of the
beam without any deformation is given as
kzixr
,
where i
and k
are the unit vectors of the fixed Cartesian coordinate shown in the Fig.
2.1.
The deformation is given in the following,
kwix
wzu
0
0
0
.
Thus, the displacement field of the beam can be derived as,
19
kwzix
wzuxkwiurR
0
0
0
,
where 0u and 0w are the displacement components along the x- and z- directions
respectively, of a point on the beam.
Taking the total differentiation of R
with respect to the time t, the following can be
obtained,
k
dt
txdwi
dt
txdu
dt
dx
dt
Rd
,, 00
.
Hence, the kinetic energy of the beam is expressed as,
dxdzdt
Rd
dt
RdbdV
dt
Rd
dt
RdT
h
h
l
V
2
20 2
1
2
1
, (2.1)
where ρ denotes the density of the beam.
The von Karman-type equations of strains of large deflection associated with the
displacement field, normal to the cross section of the beam along the x direction, can be
given by,
2
0
22
0011
2
1
x
wz
x
w
x
u
.
Therefore, the total strain energy of the beam can be given by,
2
20
1111111111112
1
2
1h
h
l
VdxdzQbdVQU , (2.2)
20
where 11Q represents the elastic coefficient in the same direction with 11 .
The virtual work done by the external excitation force is given below,
dxqwbWl
0
0 . (2.3)
In the following analysis, the Hamilton’s principle will be employed to obtain the
nonlinear equations of motion for the beam. The mathematical statement of the
Hamilton’s principle is given by,
02
1
2
1
dtWdtLt
t
t
t , (2.4)
where the total Lagrangian function L is given by,
UTL . (2.5)
From Eq. (2.4), the nonlinear governing equation of an Euler-Bernoulli beam can be
derived in the following form,
2
2
dt
xdh
2
0
2
dt
udh
2
0
2
11x
uhQ
0
2
0
2
011
x
w
x
whQ , (2.6-a)
2
0
2
dt
wdh
2
0
2
2
23
12 dt
wd
x
h
x
w
x
uhQ
0
2
0
2
11
2
0
2
011
x
w
x
uhQ
2
0
22
011
2
3
x
w
x
whQ
4
0
4
11
3
12 x
wQ
h
0 q . (2.6-b)
21
Based on the reference (Abou-Rayan et al., 1993; Younis and Nayfeh, 2003) and Eq.
(2.6-a), it can be obtained as
22
1
2
12
2
110
2
0
2
00 lx
dt
xd
Qdx
x
w
lx
w
x
u l . (2.7)
With the substituting of Eq. (2.7) into Eq. (2.6-b), the nonlinear differential
governing equation of the beam in z direction is derived as
02
12
0
2
0
2
0112
0
2
x
wdx
x
wQ
ldt
wd l
. (2.8)
To validate the governing equation Eq. (2.8) and facilitate the numerical simulations
in the consequent sections, the following non-dimensional variables are introduced,
4
0
11
3
12
1
blI
Qbh
tt t , l
xx ,
h
ww 0
0 . (2.9)
Introduce the non-dimensional variables shown in Eq. (2.9) into Eq. (2.8), the non-
dimensional governing equation of the investigated nonlinear Euler-Bernoulli beam can
be expressed as,
2
0
2
td
wdA
2
0
2
2
2
td
wd
xB
1
0 2
0
22
0
x
wxd
x
wC
4
0
4
x
w
0 q . (2.10)
where,
2
2
12l
hA ,
24
2
11
2 l
hQB ,
24
11
2
12 l
QhC ,
22h
qq .
22
2.3 Series Solutions
Based on the Galerkin method of discretization, the transverse displacement 0w is
expanded in a series form, in terms of a set of comparison functions as,
1
0
n
nn twxφw . (2.11)
Corresponding to the boundary conditions of the Euler-Bernoulli beam, xφn can
be given as follows,
xnxn sin . (2.12)
Substitute the series solution of Eq. (2.12) into Eq. (2.11), and to assist the following
presentation, replace n , nw , nw , nw , and q for )(xφn , twn , td
wd n , 2
2
td
wd n , and q
respectively, and
1,11 ww , 1,22 ww , 1,33 ww ,
2,11,1 ww , 2,21,2 ww , 2,31,3 ww .
Therefore, with the application of the Galerkin method at 1n and 3n , the
discretized governing equations of the Euler-Bernoulli beam can be obtained in the
following,
0002,1
2,11,1
w
ww
, (2.13-a)
23
3332,3
2,31,3
2222,2
2,21,2
1112,1
2,11,1
w
ww
w
ww
w
ww
, (2.13-b)
where,
A20
2
1
2
1
1
,
3
1,1
4
1,1
4
04
1
2
1BwCw ,
q20 ,
A21
2
1
2
1
1
,
A22
22
1
1
,
A23
2
9
2
1
1
3
1,1
42
1,31,1
4
1,1
42
1,21,1
4
14
1
4
9
2
1BwwBwCwwBw ,
2
1,11,2
42
1,31,2
4
1,2
43
1,2
4
2 984 wBwwBwCwBw ,
2
1,11,3
43
1,3
4
1,3
42
1,21,3
4
34
9
4
81
2
819 wBwBwwCwBw
,
q21 , 02 ,
3
23
q .
24
2.4 Active Nonlinear Vibration Control
In the literatures (Yau et al., 2006; Kuo, 2007) available to the authors, the general
nonlinear dynamic system of single dimension, to which the application of the existing
FSMC strategy can be applied, is given as
tw
ww
ww
n
ii
,,1
1,1,1
2,11,1
W
,
where, Tnii wwwww ,11,1,12,11,1 W represents the variables of the single
dimensional system, and t,W is the specific expression of the governing equation of
the system.
The application of the existing FSMC strategy in controlling the multi-dimensional
nonlinear dynamic system in Eq. (2.13-b), which is derived via Eq. (2.11) when 2n ,
would render the limitation of the existing control strategy: The nonlinear dynamic
systems of multiple dimensions considering higher vibration modes 1,nw ( 2n ), is not
compatible with the existing FSMC strategy. That is: although the established multi-
dimensional dynamic system in Eq. (2.13-b) is going to prove necessary in approximating
the vibrations of an Euler-Bernoulli beam subjected to the evenly distributed external
sinusoidal excitation, the exiting FSMC strategy cannot be applied in controlling the
chaotic vibration discovered in such a system. Therefore, an active nonlinear control
25
strategy is demanded, which can be applied in the nonlinear vibration control of a
nonlinear dynamic system of multiple dimensions.
With the developed governing equations, boundary conditions and the solutions of
the governing equations in Section 2.2 and 2.3, an active vibration control strategy can be
developed. Based on the previous works (Utkin, 1992; Kuo, 2007; Haghighi and Markazi,
2010; Dai and Sun, 2012), the proposed active nonlinear control strategy is developed for
the nonlinear vibration control of dynamic systems of multiple dimensions, such as the
one given in Eq. (2.13-b).
For a nonlinear governing equation in the following general form,
twww ,, , (2.14)
If U is given as the control input and wwF , as the unknown external disturbance
applying on the beam, the governing equation Eq. (2.14) for the nonlinear Euler-
Bernoulli beam with the control input and the external disturbance can be given by,
wwFUtwww ,,, . (2.15)
With the application of the control to be fully developed in the following, it is
expected that the nonlinear vibration of an engineering structure of multiple dimensions
can be controlled.
If the nth
Galerkin method is applied in the discretization of the governing equation
given in Eq. (2.15), a series of 2nd
-order ordinary differential equations considering the
26
control input U and the unknown external disturbance wwF , will be derived as
follows,
tfutw
ww
tfutw
ww
tfutw
ww
tfutw
ww
innn
nn
iiii
ii
,,
,,
,,
,,
2,
2,1,
2,
2,1,
2222,2
2,21,2
1112,1
2,11,1
WW
WW
WW
WW
, (2.16)
where ti ,W , iu , and tfi ,W represent the expressions of tww ,, , U , and
wwF , after the application of the Galerkin discretization. With the Galerkin
discretization, the column vector W in Eq. (2.16) is given below,
Tnnii wwwwwwww 2,1,2,1,2,21,22,11,1 W .
Considering the response of a point on the engineering structure, based on Eq. (2.16)
and the expression in Eq. (2.11), the non-dimensional response of the selected point pw
can be given as,
1n
npnp twxφw , (2.17)
where px denotes the location of the selected point.
For a desired reference vibration expressed as,
27
twr , (2.18)
the control input U can be given as,
req UUU , (2.19)
where eqU and rU are expressed as,
ppeq wwU , fsfsr UkU . (2.20)
In Eq. (2.20), designates the control parameter governing the sliding surface, fsk
is given as RkwwF fs, , and the value of fsU depends on the fuzzy rule shown in
the table below, in which PB, PM, PS, ZE, NS, NM and NB represent 1, 2/3, 1/3, 0, -1/3,
-2/3, and -1 respectively. The introduction of the fuzzy rule will increase the robust of the
proposed controls strategy by taking into consideration of the external uncertainty
wwF , . Besides, it also reduces the time consumption involved in the control input
calculation: instead of calculating the exact value corresponding to the point defined by
eqU and dt
dU eq, the approximated value of fsU can be derived as per the table shown in
Table 2.1, so long as the point falls into the area determined by PB, PM, PS, ZE, NS, NM
and NB.
With the active nonlinear control strategy developed in Eq. (2.15) ~ Eq. (2.20), the
active nonlinear vibration control of the Euler-Bernoulli beam expressed with the general
governing equation Eq. (2.14) is to be realized.
28
Table 2.1 The fuzzy rule of fsU
fsU
eqU
PB PM PS ZE NS NM NB
dt
dU eq
PB NB NB NB NB NM NS ZE
PM NB NB NB NM NS ZE PS
PS NB NB NM NS ZE PS PM
ZE NB NM NS ZE PS PM PB
NS NM NS ZE PS PM PB PB
NM NS ZE PS PM PB PB PB
NB ZE PS PM PB PB PB PB
29
Take the Euler-Bernoulli beam governed by Eq. (2.10) as an example. Use the
control strategy developed and apply the control input as shown in Eq. (2.15), the
governing equation with the control input for the Euler-Bernoulli beam can be given by
the following expression,
2
0
2
dt
wdA
2
0
2
2
2
dt
wd
xB
1
0 2
0
22
0
x
wdx
x
w
C4
0
4
x
w
q U 0, 00 wwF . (2.21)
With the application of the 3rd
-order Galerkin method, Eq. (2.21) may have the
following form,
tfuw
ww
tfuw
ww
tfuw
ww
,
,
,
333332,3
2,31,3
222222,2
2,21,2
111112,1
2,11,1
W
W
W
, (2.22)
where 1u , 2u , and 3u are derived as follows through the 3rd
-order Galerkin method,
Uu 83.01 , Uu 02 , Uu 36.03 .
2.5 Nonlinear Vibration Characterization
The vibrations of a selected point on the Euler-Bernoulli beam considered are
investigated in this section. With the numerical simulations performed, chaotic vibrations
of the selected point are discovered, and the comparison is provided between the
30
vibrations of the established single-dimensional dynamic system and the multi-
dimensional dynamic system. To facilitate efficient numerical calculation in this research,
the fourth-order P-T method (Dai, 2008) is implemented.
It should be noticed, to evaluate the nonlinear characteristics of the system
considered, a characteristic diagnosing method named Periodicity Ratio (PR) method is
employed. The PR method (Dai and Singh, 1997; Dai, 2008) has shown great advantages
in diagnosing the nonlinear behavior such as periodic, quasi-periodic and chaotic
vibrations for a nonlinear system. The PR criterion is applied based on Poincaré section
of the nonlinear system considered. The PR value is a criterion for analyzing the
nonlinear dynamic behavior with the considerations of the overlapping points in
comparing with the total number of points derived with Poincaré section. The Periodicity
Ratio (PR) is defined as,
n
NOP
n lim .
NOP in the equation above denotes the total number of periodic points that are
overlapping points and n represents the number of all the points forming derived with
Poincaré section. NOP can be obtained by the formula shown below:
1NOP
n
k
K2
P
1
1
K
L
KLKL XX
31
In the equation above, K represents the number of points overlapping the Kth
point via the Poincaré section, and ∏ is the symbol for multiplication. KLX , KLX and P
are functions expressed as,
LTtXKTtXX KL 00 ,
LTtXKTtXX KL 00
,
1
0P
0
0
if
if,
where, 0t is a given time, and T is a period of a periodic loading.
With the criterion such defined, if the vibration is perfectly periodic, equals
one; if value approaches zero, the vibration is then quasi-periodic or chaotic. When
falls between 0 and 1, theoretically, the vibration is neither periodic nor chaotic. With this
single value criterion, the dynamic behavior of a nonlinear dynamic system can be
conveniently characterized.
The parameters of the Euler-Bernoulli beam are given as those from the work,
PaQ 9
11 10127 , ml 2 , 37800 mkgρ ,
and the non-dimensional external peroidic excitation is,
tq 90.5sin58.5 .
32
The non-dimensionalized initial conditions, corresponding to the displacements
described by Eqs. (2.13) after the implementation of the 3rd
-order Galerkin method, are
taken as,
01.001,1 w , 05.002,1 w , 005.001,2 w , 025.002,2 w ,
003.001,3 w , 02.002,3 w .
pw , the transverse displacement at a selected point p located at 1.7 meter from the
origin of the Cartesian coordinates shown in Fig. 2.1, is expressed corresponding to Eqs.
(2.13) as below,
1,1
1
1
1, 454.0 wwxwn
npn
, (2.23-a)
1,31,21,1
3
1
1, 988.0810.0454.0 wwwwxwn
npnp
. (2.23-b)
A chaotic vibration of the Euler-Bernoulli beam occurs as shown in Fig. 2 for w
and Fig. 3 for pw , while the developed active control strategy is not applied. The wave
diagram, 2-D phase diagram, and Poincaré map are shown respectively, in Figs. 2.2 (a)
and 2.3 (a), Figs. 2.2 (b) and 2.3 (b), and Figs. 2.2 (c) and 2.3 (c) . With the utilization of
the PR method, the typical chaotic cases shown in Fig. 2.2 and Fig. 2.3 can be determined
with the PR value approaching to 0 in both cases.
33
(a)
(b)
(c)
Figure 2.2 The chaotic vibration of w : (a) the wave diagram; (b) the 2-D phase diagram;
(c) the Poincaré map
34
(a)
(b)
(c)
Figure 2.3 The chaotic vibration of pw : (a) the wave diagram; (b) the 2-D phase diagram;
(c) the Poincaré map
In Fig. 2.2 and Fig. 2.3, one may notice in Fig. 2.2 (a) the maximum amplitude of the
chaotic vibration of the beam is less than 2, and w is derived from Eq. (2.13-a), which
35
shows the nonlinear dynamic system of the beam in single dimension. However, in Fig.
2.3 (a) it can be discovered that the nonlinear dynamic system of three dimensions given
in Eq. (2.13-b) presents that the maximum amplitude is close to 2.5, which is much larger
than the one in Fig. 2.2 (a) by almost 25%.
Besides, from Fig. 2.4 (a), Fig. 2.4 (b) and Fig. 2.4 (c), it can be learned: although
the first vibration mode in Fig. 2.4 (a) is larger than those of the other two vibration
modes as shown in Fig. 2.4 (b) and Fig. 2.4 (c), the amplitudes of the other two vibration
modes are obviously not negligible. Actually it can be found that the other two vibration
modes also significantly contribute to the actual vibration of the selected point pw .
36
(a)
(b)
(c)
Figure 2.4 The wave diagrams of the first three vibration modes before the application of
the active control strategy: (a) 1w ; (b) 2w ; (c) 3w
37
Thus, the development of a multi-dimensional dynamic system is necessary for the
accurate prediction of the dynamics of the Euler-Bernoulli beam subjected to an external
periodic excitation.
2.6 Active Nonlinear Vibration Control
Considering that the displacement shown in Fig. 2.3 (a) is non-dimensional and the
amplitude is actually 2.5 times the thickness of the beam, the maximum amplitude
showing in Fig. 2.3 (a) is may lead to structure failure. Therefore, the large-amplitude
chaotic vibration of the beam requires to be suppressed. The proposed active control
strategy is found not only effective in reducing the amplitude of the vibration, but also
synchronizing the vibration to the given frequency of the desired reference signal.
As shown in Fig. 2.5, the proposed control strategy is applied at 20t , and the
control parameters and the unknown external disturbance take the following values,
twr 90.5sin5.1 , 10 , 500fsk , )sin(01.0, pwwwF .
As can be seen from Fig. 2.5, the maximum amplitude of the vibration of the beam is
reduced significantly from about 2.2 to 1.5. The synchronization of the vibration of the
beam also makes the chaotic vibration at the selected point become a periodic one.
38
Figure 2.5 The wave diagram of pw with the application of the active control strategy
From Fig. 2.6 (a), Fig. 2.6 (b) and Fig. 2.6 (c), it should be noticed that the vibrations
of the first three vibration modes of the beam are indeed affected with the application of
proposed active control strategy.
39
(a)
(b)
(c)
Figure 2.6 The wave diagrams of the first three vibration modes with the application of
the active control strategy: (a) 1w ; (b) 2w ; (c) 3w
Fig. 2.7 is presented to fully demonstrate the effectiveness of the proposed control
strategy. In Fig. 2.7, the difference is small between the actual vibration of the beam and
40
that of the reference signal, and the synchronization between the actual vibration of the
beam at the selected point and the reference signal shows the significant effectiveness of
the proposed control strategy
Figure 2.7 The comparison between the wave diagram of pw (the blue continuous line)
and the reference signal rw (the green dash line)
In Fig. 2.8, the control input required in the vibration control of the selected point on
the beam is given.
Figure 2.8 The control input U
2.7 Conclusion
An active control strategy is developed in this chapter for controlling the nonlinear
vibrations of an Euler-Bernoulli beam subject to external periodic excitation. A nonlinear
41
multi-dimensional dynamic system via the higher order Galerkin method is developed for
modeling the nonlinear vibrations of the beam. Development of such control strategy is
significant, and in fact there is no control strategy found in the literature available for
controlling the chaotic vibrations of a multi-dimensional dynamic system. With the
findings of the research presented in this chapter, the following can be concluded:
First of all, to better and accurately analyze the vibrations of an Euler-Bernoulli
beam subject to external periodic excitation, the contributions of the higher order
vibrations are significant especially when chaotic vibrations need to be taken into
consideration. As shown in this chapter, the effects of the first three vibration modes must
be considered in approximating the vibrations of the nonlinear beam system.
Secondly, the active control strategy developed in this chapter is effective in
controlling the large-amplitude chaotic vibrations of the Euler-Bernoulli beam subject to
external periodic excitations.
Thirdly, a continuously applied control input is needed for stabilizing the vibration
of the selected point on the beam. Besides, the magnitude of the required control input ,
which is decided by both the control parameters and the difference between the actual
response of the beam and the reference signal, does not decrease significantly after the
vibration of the beam is stabilized.
42
CHAPTER 3 MEMS EULER-BERNOULLI BEAM
SUBJECTED TO EXTERNAL NON-PERIODIC
EXCITATION
3.1 Introduction
In this section, the active vibration control strategy developed in Chapter 2 is to be
applied for controlling and stabilizing the nonlinear vibration of a MEMS Euler-Bernoulli
beam with a non-periodic electro-static excitation. The equations of motion of the beam
are established based on the von Karman-type equations. In developing the solutions of
the beam’s vibrations, the equations in the forms of partial differential equations are non-
dimensionalized and transformed into three ordinary differential equations via the 3rd
-
order Galerkin method. The importance of the established multi-dimensional dynamic
system is demonstrated via the stability analysis of the MEMS beam, and a chaotic
vibration is discovered. Corresponding to the multi-dimensional dynamic system derived,
the active control strategy developed in Chapter 2 is applied. In enhancing the efficiency
of the active control strategy, a practical method, namely the two-phase control method is
proposed. The method divides the control progress into two control phases of controlling
process, and the proposed active control strategy will be applied in each of the control
process. The chaotic vibration of the MEMS beam is suppressed and stabilized in the
control with the control method, and the effectiveness of the developed control method,
as well as the active control strategy, is to be demonstrated. The vibration control method
43
in this chapter provides the availability for controlling the nonlinear vibrations of MEMS
beams of multiple dimensions.
3.2 Equations of Motion
The MEMS Euler-Bernoulli beam considered in this Chapter is sketched in Fig. 3.l.
The equations of motion of the beam are to be derived based on the Hamilton’s principle
and von Karman-type equations. As can be seen from the figure, the beam with fixed-
fixed boundaries is placed between two electrodes connected to electrical sources, the
length of the beam is given as 0l , the area of the rectangular cross section of the beam is
constant with the width b and thickness h . The x axis is along the mid-plane of the
beam. The displacement of any point of the beam along the x- and z- axes are designated
as u and w .
Figure 3.1 The sketch of the MEMS beam
Starting from the origin at the left support of the beam, a position vector, r , of any
point zx, of the beam without deformation is given as,
kir zx , (3.1)
44
where i and k are the unit vectors of the fixed Cartesian coordinate shown in the Fig. 3.1.
Thus, the displacement field of the beam can be derived as,
kiR txwzx
txwztxux ,
,, 0
00
, (3.2)
where txu ,0 and txw ,0 are the displacement components, along the x- and z-
directions respectively, of a point in the mid-plane ( 0z ). It should be noticed that the
displacement field described considers the vector position and displacement of a point.
Taking full differentiation of R with respect to time t, one may obtain,
ki
R
dt
txdw
dt
txdw
xz
dt
txdu
dt
d ,,, 000
, (3.3)
Hence, the kinetic energy of the MEMS beam over a volume V of the beam is
expressible as,
dxdzdt
dρbdV
dt
dρT
h
h
l
V
2
2 0
220
2
1
2
1 RR, (3.4)
where ρ denotes the density of the material of the beam.
The von Karman-type equations of strains of the beam’s large deflection associated
with the displacement field, normal to the cross section of the beam along the x direction,
in Eq. (3.2) can thus be given by,
2
0
22
0011
,,
2
1,
x
txwz
x
txw
x
txu
. (3.5)
45
Therefore, the total strain energy of the beam can be given by,
022 2
11 11 11 112 0
1 1( )
2 2
h l
V hE Q dV b Q dx t dz
, (3.6)
where 11Q represents the elastic coefficient in the same direction with 11 .
The virtual work done by the electro-static force, 0wFe , is given below,
l
e dwwFbW0
00, (3.7)
where 0wFe , based on the work (Mestrom et al., 2008), can be expanded as below,
toh
d
w
d
w
d
w
ftVV
ftVVdCwF
acdc
acdc
e ..43212sin
2sin
2
13
0
3
0
2
0
2
0
0
0000 2
2
, (3.8)
and toh .. denotes higher order terms; 0C is the capacitance over the gap when 00 w ; 0d
is the corresponding initial gap width as shown in Fig. 3.1; dcV is the bias voltage, and
acV and f are the amplitude and frequency of the ac voltage, respectively.
Next, the Hamilton’s principle is employed to obtain the nonlinear equations of
motion for the MEMS beam. The mathematical statement of the Hamilton’s principle is
given by
02
1
dtWLt
t , (3.9)
where the Lagrangian function L is given by,
L T E . (3.10)
46
Substitute Eqs. (3.4), (3.6), and (3.7) into Eq. (3.10), and neglect the higher terms
toh .. in Eq. (3.8), the nonlinear equations of motion of the MEMS beam can be derived
in as follows,
02
0
2
02
0
2
0112
0
2
11
dt
udI
x
w
x
wA
x
uA , (3.11-a)
2
0
2
04
0
4
112
0
22
0112
0
2
0112
0
2
011
2
3
dt
wdI
x
wD
x
w
x
wA
x
w
x
uA
x
u
x
wA
04321
2sin
2sin
2
13
0
3
0
2
0
2
0
0
0
00 2
2
d
w
d
w
d
w
ftVV
ftVVdC
acdc
acdc
, (3.11-b)
where,
hQA 1111 , hI 0 , 3
2 hI , 11113
Qh
D3
. (3.12)
Associated with the nonlinear dynamic equations and the boundary conditions
(Abou-Rayan et al., 1993; Younis and Nayfeh, 2003), the strain can be obtained as,
0
0
2
0
0
2
00
2
1
2
1 l
dxx
w
lx
w
x
u. (3.13)
Substitute Eq. (3.13) into Eq. (3.11-b), and the nonlinear differential governing
equation of the beam in the z direction is derived as,
4
0
4
112
0
2
0
2
011
2
0
2
2
2
22
0
2
02 x
wD
x
wdx
x
w
l
A
dt
wd
xI
dt
wdI
l
47
04321
2sin
2sin
2
13
0
3
0
2
0
2
0
0
000 2
2
d
w
d
w
d
w
ftVV
ftVVdC
acdc
acdc
. (3.14)
To validate the governing equation Eq. (3.14) and facilitate the numerical
simulations in the following sections of this chapter, the following non-dimensional
variables are introduced,
ttblI
IQt
4
00
11 , 0l
xx , (3.15)
and,
h
ww 0
0 , dt
dw
htd
wd 00 1
,
2
0
2
22
0
21
dt
wd
htd
wd
,
00
1l
hl . (3.16)
With the non-dimensional variables shown in Eqs. (3.15) and (3.16) introduced into
Eq. (3.14), the non-dimensional governing equation of the MEMS beam can be expressed
as,
3
03
2
02010
4
0
4
2
0
21
0
2
0
2
0
2
2
2
2
0
21
wHwHwHH
x
wG
x
wxd
x
wF
td
wd
xB
Atd
wd, (3.17)
where,
h
IA
0 ,
h
I
lB
2
2
1
, 4
0
2
11
2
1
lρ
hAF
,
4
0
2
11
lρh
DG
,
48
22
0
00
12sin2sin
2
122
ht
fVVt
fVV
d
CH acdcacdc
,
22
00
01
122sin2sin
2
122
hd
ht
fVVt
fVV
d
CH acdcacdc
,
222
0
2
0
02
132sin2sin
2
122
hd
ht
fVVt
fVV
d
CH acdcacdc
,
223
0
3
0
03
142sin2sin
2
122
hd
ht
fVVt
fVV
d
CH acdcacdc
.
(3.18)
3.3 Series Solutions
Based on the Galerkin method of discretization, the transverse displacement 0w is
expanded in a series form, in terms of a set of comparison functions as,
1
0
n
nn twxφw . (3.19)
Corresponding to the fixed-fixed boundaries of the beam, xφn can be given as
follows,
xxxxx nn
nn
nnnnn
sinsh
sinsh
coschcosch
. (3.20)
49
Substitute the series solution of Eq. (3.19) into Eq. (3.17), and for the sake of clarity
and simplification in expression, replace n , nw , nw , nw , and t for )(xφn , twn , dt
dwn , 2
2
dt
wd n ,
and t respectively, and,
1,11 ww ,
1,22 ww , 1,33 ww ,
2,11,1 ww ,
2,21,2 ww , 2,31,3 ww .
Therefore, with the application of the Galerkin method at 3n , the discretized
governing equations of the beam with the fixed-fixed boundaries can be obtained as the
following,
1221
1331
2,3
2,31,3
1
22,2
2,21,2
1221
2332
2,1
2,11,1
w
ww
w
ww
w
ww
, (3.21)
where,
AB 3.121 ,
B73.92 ,
50
3
1,3
3
1,2
3
1,1
2
1,31,2
1,3
2
1,2
2
1,31,1
2
1,21,1
1,3
2
1,11,2
2
1,11,31,21,1
1,111,13
14200005.461641212
3699918000046786
48612681301
501
wwwww
wwwwww
wwwwwww
FwHGw
3
1,3
3
1,2
3
1,1
2
1,31,2
1,3
2
1,2
2
1,31,1
2
1,21,1
1,3
2
1,11,2
2
1,11,31,21,1
3
0714.00000467.085.10000152.0
54.108.333.3
95.100025.000178.0
wwwww
wwwwww
wwwwwww
H
2
1,31,31,2
2
1,2
1,31,11,21,1
2
1,1
20899.0000452.0
582.0000105.033.1831.0
wwww
wwwwwHH ,
BA 461 ,
2
1,31,31,2
2
1,2
1,31,11,21,1
2
1,1
21,2120000215.046.10000000242.0
000452.099.10000527.0
wwww
wwwwwHwH
1,2
3
1,3
3
1,2
3
1,1
2
1,31,2
1,3
2
1,2
2
1,31,1
2
1,21,1
1,3
2
1,11,2
2
1,11,31,21,1
3803
216175000533.0673000
57206.151000
41.7230706.47
Gw
wwwww
wwwwww
wwwwwww
F
3
1,3
3
1,2
3
1,1
2
1,31,2
1,3
2
1,2
2
1,31,1
2
1,21,1
1,3
2
1,11,2
2
1,11,31,21,1
3
000260.069.10000835.034.3
00105.00000152.000014.0
000888.033.308.3
wwwww
wwwwww
wwwwwww
H ,
B73.91 ,
AB 9.982 ,
51
1,311,33 14619 wHGw ,
3
1,3
3
1,2
3
1,1
2
1,31,2
1,3
2
1,2
2
1,31,1
2
1,21,1
1,3
2
1,11,2
2
1,11,31,21,1
14500009.55487412381
37600014200036998
49543205960
wwwww
wwwwww
wwwwwww
F
3
1,3
3
1,2
3
1,1
2
1,31,2
1,3
2
1,2
2
1,31,1
2
1,21,1
1,3
2
1,11,2
2
1,11,31,21,1
3
0714.00000467.085.10000152.0
54.108.333.3
95.100025.00000303.0
wwwww
wwwwww
wwwwwww
H
2
1,31,31,2
2
1,2
1,31,11,21,1
2
1,1
20582.00000429.0731.0
8.1000452.0291.0364.0
wwww
wwwwwHH .
3.4 Stability Analysis
The influence of the designed parameters on the stability of MEMS beams has
already been investigated extensively, and the focus has been primarily put on the effect
of the AC voltage acV (Younis and Nayfeh, 2003; Mestrom et al., 2008; Alsaleem et al.,
2009; Haghighi and Markazi, 2010; Yau et al., 2011; Azizi et al., 2013), which may lead
to stable or unstable vibrations of the MEMS beam, such as periodic vibration (Haghighi
and Markazi, 2010), chaotic vibration (Yau et al., 2011), and dynamic pull-in (Alsaleem
et al., 2009). It should be noticed: in most of the previous studies (Mestrom et al., 2008;
Alsaleem et al., 2009; Haghighi and Markazi, 2010; Yau et al., 2011; Azizi et al., 2013),
it is the stability of a single dimensional dynamic system that has been investigated, while
in this section the effect of the AC voltage acV is briefly discussed, on the stability of the
multi-dimensional nonlinear dynamic system of the MEMS beam expressed in Eq. (3.21).
52
Via the stability analysis, the necessity of the application of a multi-dimensional
nonlinear dynamic system of a MEMS beam is proved, and hence the demand for a
vibration control strategy available for the multi-dimensional system is demonstrated.
In the following analysis, a MEMS beam subject to external non-periodic electro-
static force is specified with the geometric dimensions and system parameters given
below,
PaQ 10
11 106641.7 , ml 6100 , mb 40 , mh 2.2 ,
31460 mkgρ .
The non-dimensionalized initial conditions, corresponding to the vibrations
described in Eqs. (3.21), are taken as,
2.001,1 w , 5.002,1 w , 08.001,2 w ,
002,2 w , 001,3 w , 002,3 w .
Considering the vibration of a point at the position m5.457 along the x-axis of the
beam, the non-dimensional vibration of this point pw can be derived as follows,
3
1
1,
n
nnp wφw 86316310.01,1w 4449174.1
1,2w 370982.1 1,3w . (3.22)
By employing the governing equations established, the responses of the beam can be
numerically simulated with the conditions and parameters specified. To facilitate the
numerical simulation, the fourth-order P-T method (Dai, 2008), is implemented in the
numerical calculations of the simulations of the research.
53
The numerical results in the case of vV ac 14 are shown in Fig. 3.2 and Figs. 3.3.
From Fig. 3.2, the vibration of the MEMS beam at the selected point gives a multi-
periodic one, and its maximum amplitude is around 0.18. Therefore the multi-
dimensional dynamic system given in terms of 1w , 2w and 3w is in a stable state in the
case of vV ac 14 . In Figs. 3.3 (a) and (c), the vibrations of 1w and 3w show a multi-
periodic vibration, while the vibration of 2w is negligible due to its small maximum
amplitude 0.00006. Considering that the maximum amplitudes of 1w and 3w are about
0.15 and 0.06 respectively, a multi-dimensional dynamic system of the MEMS Euler-
Bernoulli beam is necessary in approximating pw since the contribution from the higher
vibration mode is significant comparing with that of 1w , in the stable state in the case of
vV ac 14 .
54
Figure 3.2 The wave diagram of
pw in the case of vV ac 14
(a)
(b)
55
(c)
Figure 3.3 The wave diagrams of the first three vibration modes in the case of vV ac 14 : (a)
1w ; (b) 2w ; (c) 3w
The numerical results in the case of vV ac 5.14 are shown in Fig. 3.4 and Figs. 3.5.
From Fig. 3.4, the vibration of the beam at the selected point still shows a multi-periodic
one, while its maximum amplitude increases from 0.18 to 0.2. Thus, the multi-
dimensional dynamic system given in terms of 1w , 2w and 3w remains in a stable state in
the case of vV ac 5.14 . In Figs. 3.5 (a) and (c), 1w and 3w show a multi-periodic
vibration, while the vibration of 2w can still be neglected due to its small maximum
amplitude 0.018. However, Figs. 3.5 shows: though the maximum amplitudes of 1w
and
3w remain 0.15 and 0.06 respectively in spite of the increase in acV , the contribution of
2w to
pw increases significantly since the maximum amplitude of 2w has increased
drastically from 0.00006 to about 0.018. That is in the case of vV ac 5.14 : in the stable
state, a multi-dimensional dynamic system of the beam is still required since the
contribution from the higher vibration mode is still significant. Besides, it should be
noticed the increase in acV results in a great increase in the contribution of 2w to
pw .
56
Figure 3.4 The wave diagram of
pw in the case of vV ac 5.14
(a)
(b)
57
(c)
Figure 3.5 The wave diagrams of the first three vibration modes in the case of vV ac 5.14 :
(a) 1w ; (b) 2w ; (c) 3w
The numerical results in the case of vV ac 15 are shown in Fig. 3.6 and Figs. 3.7.
From Fig. 3.6, the vibration of the MEMS Euler-Bernoulli beam at the selected point
turns to be a chaotic one, and its maximum amplitude increases greatly from 0.2 to 1.
Thus, the multi-dimensional dynamic system expressed in terms of 1w , 2w and 3w enters
into an unstable state in the case of vV ac 15 . In Figs. 3.7, it can be learned: in this case,
all the three vibration modes 1w , 2w and 3w
present chaotic vibrations. Besides the
amplitude of 1w increases greatly from around 0.15 to 0.80, while the maximum
amplitude of 3w just increases from about 0.06 to 0.1. In the case of chaotic vibration, it
should be noticed the maximum amplitude of 2w has again increased drastically from
about 0.018 to 0.3, and hence in this case the maximum amplitude of 2w is larger than
that of 3w . Therefore, in the unstable state, a multi-dimensional dynamic system of the
beam should be established, since each of the vibration model make significant
contributions to pw .
58
Figure 3.6 The wave diagram of
pw in the case of vV ac 15
(a)
(b)
59
(c)
Figure 3.7 The wave diagrams of the first three vibration modes in the case of vV ac 15 :
(a) 1w ; (b) 2w ; (c) 3w
From Figs. 3.2~3.7, it can be learned the increase in the AC voltage acV will make
the vibration of the multi-dimensional nonlinear dynamic system of the beam gradually
vary from a stable multi-periodic vibration to an unstable chaotic one. Furthermore, the
contribution of 3w to
pw should be considered comparing with that of
pw in either a
stable state or an unstable state. Especially in the discovered chaotic vibration in the case
of vV ac 15 , none of the first three vibration modes can be omitted for the vibration
prediction of a MEMS Euler-Bernoulli beam, since the contribution of 2w increase
greatly and thus becomes significant in comparing with the contribution of 1w and 3w .
3.5 Control Design
3.5.1 Active Control Strategy
With the previously established governing equations and series solutions, the active
vibration control strategy developed in Chapter 2 can be applied for controlling the
vibrations of the three-dimensional nonlinear dynamic system expressed in Eq. (3.21).
60
With the control input given in Eqs. (2.15) ~ (2.20), the control of a MEMS beam
governed by the governing equation in Eq. (3.17) becomes readily available as below,
wwFU
wHwHwHH
x
wG
x
wdx
x
wFw
xB
Aw
,
1
3
03
2
02010
4
0
4
2
0
21
0
2
002
2
0
. (3.23)
With the application of the 3rd
-order Galerkin discretization, Eq. (3.23) may have the
following form,
tfuw
ww
tfuw
ww
tfuw
ww
,
,
,
33
1221
1331
2,3
2,31,3
22
1
22,2
2,21,2
11
1221
2332
2,1
2,11,1
W
W
W
, (3.24)
where 1u , 2u , and 3u are derived as follows through the 3rd-order Galerkin method,
Uu 83086868.01 , Uu 02 , Uu 3637565114.03 .
The active control strategy developed can now be applied in controlling the
nonlinear vibrations of the MEMS beam. It can be demonstrated with a numerical
simulation that the actual vibration of the MEMS beam at a given point can be well
synchronized to a desired reference signal in the case that the coefficients of 2u is zero.
3.5.2 Two-Phase Control Method
61
To enhance the efficiency of the active control strategy developed, a two-phase
control method is proposed for practically applying the active control strategy in
controlling the nonlinear vibrations of the beam considered. The control process
employing the control strategy developed is divided into two phases. In each control
phase, the control strategy is applied with a set of specific values and expression assigned
to , fsk and rw . In the process of controlling and stabilizing the nonlinear vibration of
the investigated MEMS beam, the first control phase is applied. Once the vibration of the
second mode of the beam is under control, the control strategy of the second control
phase is then applied. Specifically, the second control phase begins when the vibration of
the second mode, of which the coefficient of 2u is zero, gradually enters into a stabilized
state. In the second control phase, the vibrations of the MEMS beam are finally
synchronized to the desired reference signals. The process of the first control phase is
therefore considered as a transition between the vibration without control and that under
complete control.
In the first control phase, the active control strategy developed is applied with the
specific values and expression of , fsk and rw as follows
1 ,
1
fsfs kk , twr
1 ,
where 1 , 1
fsk and t1 represent the values and expression assigned to ,
fsk and
rw in the first control phase.
62
In the second control phase, the active control strategy developed in Chapter 2 is still
applied but with another set of specific values and expression of , fsk and rw given in
the following form
2 ,
2
fsfs kk , twr
2 .
The advantages of applying the first phase control strategy are obvious: the second
mode vibration of the MEMS beam is stabilized and the vibration of the beam is
controlled corresponding to a reference signal given in the first control phase. More
significantly, the vibration amplitude of the beam is reduced in the first control phase
such that the second control process is made available. With the contributions of the first
control phase, the amplitude of beam can be further reduced to the targeted level
corresponding to the desired reference signal of the second control phase. It is the second
control phase that finalizes the control of the nonlinear vibration of the beam, with the
synchronization of the vibration corresponding to the desired reference signal. It should
be noticed that the control parameters assigned in the second control phase are much
smaller than those in the first control phase, and therefore result in a lower control input
that implies a lower control cost. In the next section, this two-phase control method
shows its effectiveness and efficiency in synchronizing the nonlinear vibration of the
MEMS beam subject to an external non-periodic excitation at a low control cost.
63
3.6 Application of the Control Method
To demonstrate the application of the control method, it is applied to control the
chaotic vibration of the MEMS beam and the results are shown in Figs. 3.8~3.13. The
unknown external disturbance wwP , .is given below,
pwwwP sin05.0, .
3.6.1 Application of the First Control Phase
In applying the first control phase as discussed previously, the control parameters
take the following values,
6.01 , 101 fsk , twr 7188.6sin6.01 . (3.25)
Starting from the non-dimensional time 28t until 39t , the active control
strategy developed in Chapter 2 is continuously applied. In the first control phase, the
vibration of the beam is shown in Fig. 3.8, corresponding to the time period considered.
As shown in Fig. 3.8, the nonlinear vibration of the beam is indeed controlled with not
only the significantly reduced amplitude but also the stabilization of the motion of the
beam, after the application of the active control strategy in the first phase. This implies
that the vibration of the beam is being controlled and stabilized from a chaotic vibration
into an almost periodic one. To illustrate the effectiveness of the first control phase, Figs.
3.9 show the vibration of the first three modes after the application of the first control
phase.
64
Figure 3.8 The comparison between the wave diagram of
pw (the continuous blue line)
and the reference signal (the green dash line) in the first control phase
(a)
(b)
65
(c)
Figure3.9 The wave diagrams of the first three vibration modes in the first control phase:
(a) 1w ; (b) 2w ; (c) 3w
The control input in the first control phase is shown in Fig. 3.10. As can be seen
from Fig. 3.10, the control input may go up to about 400 in the process. Once a stable
state is reached, the control input will decrease to about 100.
Figure 3.10 The control input U in the first control phase
As shown in Fig.3.9 (b), after the application of the first control phase, the vibration
of the second vibration mode of the MEMS Euler-Bernoulli beam becomes stable and
gradually becomes periodic. As discussed in the previous section, this indicates that the
application of the second control phase is readily available.
3.6.2 Application of the Second Control Phase
66
In the second control phase, the control parameters take the following values:
1.02 , 12 fsk , twr 7188.6sin18.02 . (3.26)
Starting from 39t until 150t , again, the active control strategy developed in
Chapter 2 is continuously applied. With these conditions and parameters, the vibration of
the beam is shown in Figs. 3.11. As can be seen from Fig. 3.11 (a), the nonlinear
vibration of the selected point on the beam is controlled with further reduced amplitude
and the vibration of the beam is furthermore stabilized, after the application of the second
control phase. As shown in Fig. 3.11 (b), the chaotic vibration of the beam is finally
controlled with high stability and good synchronization to the desired reference signal.
Notice that the maximum amplitude is about 1.0 in the chaotic vibration, the finally
reduced amplitude of the MEMS Euler-Bernoulli beam is around only 0.18. The
reduction in amplitude is significant.
67
(a)
(b)
Figure 3.11 The vibration of the second control phase: (a) the wave diagram of pw ; (b)
the comparison between the wave diagram pw (the continuous blue line) and the
reference signal (the green dash line)
To demonstrate the effectiveness of the second control phase, the vibrations of the
first three modes are shown in Figs. 3.12, respectively.
68
(a)
(b)
(c)
Figure 3.12 The wave diagrams of the first three vibration modes in the second control
phase: (a) 1w ; (b) 2w ; (c) 3w
69
As can be seen from Figs. 3.12, the vibrations of the first three vibration modes are
all further stabilized with lower amplitudes, in comparing with that in the first control
phase.
The control input of the second control phase is shown in Fig. 3.13. As shown in the
figure, the control input required to finalize the control is around 50. In comparison with
that of the first control phase, in which the control input is about 100 in the stabilized
state. This implies that the input cost is reduced by 50% in the second control phase.
Figure 3.13 The control input U in the second control phase
With the employment of the control method, the amplitude of the chaotic vibration
of the beam is significantly reduced and the vibration of the beam is highly stabilized
with a lower control input.
3.7 Conclusions
The active control approach proposed in Chapter 2 is applied in this chapter to
control and stabilize the nonlinear vibration of a three-dimensional MEMS Euler-
Bernoulli beam, to which the existing FSMC cannot be applied. Although it has been
reported from the previous work (Younis and Nayfeh, 2003) that nonlinear multi-
70
dimensional dynamic systems contributes to enhancing the reliability of resonant MEMS
beams, nonlinear dynamic systems of multiple dimensions has not been wildly employed
in the research works available in the literatures covering the nonlinear dynamics of
MEMS beams, let alone the development of an active control strategy to control the
vibrations of such systems. In this chapter, the governing equation of the geometrically
nonlinear MEMS beam subjected to nonlinear electro-static forces is converted into a
three-dimensional nonlinear dynamic system. Corresponding to the three-dimensional
dynamic system, a stability analysis is conducted and a chaotic vibration has been
controlled with the proposed active control strategy. In enhancing the efficiency of the
control strategy, a two-phase control method is proposed in this research. With the
application of the control method, following are found significant.
First of all, the control strategy developed in Chapter 2 is suitable for controlling the
nonlinear vibrations of a MEMS Euler-Bernoulli beam described in the form of a
multiple dimensions.
Secondly, with the employment of the control method, the amplitude of the chaotic
vibration of the beam can be significantly reduced. The amplitude of the case presented is
reduced from 1.0 to 0.18.
Thirdly, the vibration is highly stabilized with the control method, and the vibration
of the beam at the selected point becomes nearly periodic.
Lastly, the input required for controlling the nonlinear vibrations of the beam is low,
with the application of the two-phase control method.
71
CHAPTER 4 FLUTTERING EULER-BERNOULLI
BEAM SUBJECTED TO EXTERNAL NON-
PERIODIC EXCITATION
4.1 Introduction
In this chapter, the active vibration control strategy proposed in Chapter 2 is to be
applied for controlling the large-amplitude chaotic vibration of a multi-dimensional
fluttering Euler-Bernoulli beam in supersonic airflow. The commonly applied non-
dimensional model of a fluttering panel is represented with an Euler-Bernoulli beam and
the beam is to be converted into a multi-dimensional system through the 6th
-order
Galerkin method. With respect to the derived multi-dimensional dynamic system, the
active control strategy previously proposed in Chapter 2 is applied, and the applicability
and efficiency of the proposed control strategy developed is proved to be significant in
controlling the nonlinear vibrations of the investigated fluttering panel.
4.2 Equations of Motion
Figure 4.1 The sketch of the fluttering Euler-Bernoulli beam
The fluttering Euler-Bernoulli beam with fixed-fixed boundaries investigated in this
chapter is sketched in Fig. 4.1. The length of the beam along the x axis is given as l , the
72
thickness of the beam is h , and the density of the beam is denoted by , and the
damping coefficient of the beam is represented by c. The x axis is along the horizontal
direction of the beam, and the displacements of any point of the beam along the x- and z-
axes are designated with u and w . The elastic coefficient along the x direction is
represented by 11Q . Along the x axis, the supersonic air flows at the rate airv above the
beam, the density of the airflow is air , and the effects due to the cavity below the beam
is not considered (Oh et al., 2001). The governing equation of motion of the fluttering
beam is given in the following (Oh et al., 2001),
4
0
4
11
3
0 2
0
22
0112
0
2
2
23
2
0
2
122
1
12 x
wQ
h
x
wdx
x
whQ
ldt
wd
x
h
dt
wdh
l
0)),((1
1
2
)(
)),((2 0
2
2
00
t
ttxw
vM
M
tx
ttxwq
dt
dwc
aira
ad
, (4.1)
where the last term in Eq. (4.1) represents the aerodynamic load applied on the panel.
Since the aerodynamic load is approximated with the 1st-order piston theory
(Dowell, 1966), it can be derived as follows according to the 1st-order piston theory,
2
2
1airaird vq , 1
2 aM ,
where aM represents the Mach number.
To validate the governing equation Eq. (4.1) and facilitate the numerical simulations
in the consequent sections, the following non-dimensional variables are introduced,
73
4
2
11
12 l
hQtt
t ,
l
xx ,
h
ww 0
0 . (4.2)
Introduce the non-dimensional variables shown in Eq. (4.2) into Eq. (4.1), the non-
dimensional governing equation of the investigated nonlinear fluttering Euler-Bernoulli
beam can be expressed as,
2
0
2
td
wdB
1
0 2
0
22
0
x
wxd
x
wC
4
0
4
x
w
D
x
w
0 E
td
wd 0 F 00
td
wd. (4.3)
where,
24
2
11
2 l
hQB ,
24
11
2
12 l
QhC ,
2
2
hl
qD d ,
aira
ad
vM
M
h
qE
1
1
222
2
,
h
cF .
4.3 Series Solution
Based on the Galerkin method of discretization, the transverse displacement 0w is
expanded in a series form, in terms of a set of comparison functions as,
1
0
n
nn twxφw . (4.4)
Corresponding to the fixed-fixed boundary conditions of the fluttering beam, xφn
can be given as follows,
xxxxx nn
nn
nnnnn
sinsh
sinsh
coschcosch
. (4.5)
74
Substitute the series solution of Eq. (4.4) into Eq. (4.3), and to assist the following
presentation, replace n , nw , nw , nw , t , and l for )(xφn , twn , td
wd n , 2
2
td
wd n , t , and l
respectively, and,
1,11 ww , 1,22 ww , 1,33 ww ,
2,11,1 ww , 2,21,2 ww , 2,31,3 ww .
Therefore, with the application of the Galerkin method at 6n , the discretized
governing equations of the nonlinear fluttering beam with the fixed-fixed boundary
conditions can be obtained in the following,
6_2,6
2,61,6
5_2,5
2,51,5
4_2,4
2,41,4
3_2,3
2,31,3
2_2,2
2,21,2
1_2,1
2,11,1
vm
vm
vm
vm
vm
vm
fw
ww
fw
ww
fw
ww
fw
ww
fw
ww
fw
ww
, (4.6)
where 1_vmf , 2_vmf , 3_vmf , 4_vmf , 5_vmf , and 6_vmf are given in APPENDIX.
75
4.4 Control Design
Corresponding to the fluttering beam governed by Eq. (4.1), and the active control
strategy developed in Eqs. (2.15) ~ (2.20), the governing equation with the control input
for nonlinear fluttering beam can be given by the following expression,
0w
1
0 2
0
22
0
x
wdx
x
wB C
4
0
4
x
w
D
x
w
0 Etd
dw0 F 0w wwFU , ,(4.7)
With the application of the 6th
-order Galerkin method, Eq. (4.7) may take the
following form,
tfufw
ww
tfufw
ww
tfufw
ww
tfufw
ww
tfufw
ww
tfufw
ww
vm
vm
vm
vm
vm
vm
,
,
,
,
,
,
666_2,6
2,61,6
555_2,5
2,51,5
444_2,4
2,41,4
333_2,3
2,31,3
222_2,2
2,21,2
111_2,1
2,11,1
W
W
W
W
W
W
, (4.8)
where 1u , 2u , 3u , 4u , 5u , and 6u are derived as follows through the 6th
-order Galerkin
method,
Uu 8308686800.01 , 02 u , Uu 3637565114.03 ,
04 u , 2314425620.05 u , 06 u .
76
In the next section, it will be demonstrated in the numerical simulation that the
actual vibration of the fluttering beam at a selected point can be well synchronized to a
desired reference signal.
4.5 Numerical Simulation
To demonstrate the applicability and effectiveness of the active control strategy
developed in the Chapter 2, numerical simulations are conducted for controlling the
fluttering beam governed by Eq. (4.3). The nonlinear vibration of the beam is focused in
this section. With the numerical simulations performed, a chaotic vibration is discovered
in the six-dimensional nonlinear dynamic system of the fluttering beam. The proposed
active control strategy is then applied and found not only significantly reduces the
amplitude of the chaotic vibration, but also stabilizes the motion so that the vibration of
the fluttering beam is synchronized to a desired periodic vibration. To facilitate the
numerical simulation, the 4th
-order P-T method (Dai, 2008), is implemented.
The parameters used for the simulations are given as follows,
smvair 1250 , 21.305 msmNc , PaQ 9
11 1072 ,
ml 9.1 , mb 05.0 , mh 004.0 , 32700 mkgρ .
The non-dimensionalized initial conditions, corresponding to the displacements
described by Eqs. (4.6) after the implementation of the 6th
-order Galerkin method, are
taken as,
05.001,1 w , 1.002,1 w , 005.001,2 w , 005.002,2 w ,
77
004.001,3 w , 004.002,3 w , 003.001,4 w , 003.002,4 w ,
009.001,5 w , 009.002,5 w , 015.001,6 w , 015.002,6 w .
If the vibration of a point at 1.140m along the x-axis of the beam is selected, based
on Eq. (4.4) the non-dimensional vibration of the selected point pw can be derived as,
1,31,21,1
6
1
1, 62806.00344.14555.1 wwwwφwn
nnp
1,61,51,4 2604.122039.03935.1 www . (4.10)
A chaotic vibration is discovered when the control parameters and the unknown
external disturbance take the following values:
twr 4985.726sin75.0 , 50 , 50fsk , )sin(01.0, pwwwF . (4.12)
The vibration of the beam in the air flowing at the speed smvair 1250 is shown in
Fig. 4.2, corresponding to the non-dimensional time interval from 0t to 40t .
During this period of time, the active control strategy is applied at 20t . One may
notice in Fig. 4.2, the maximum amplitude of the vibration of the beam is around 3.
Considering that the displacement shown in the figure is non-dimensional, the actual
amplitude of the beam at the selected point is large. Thus, reduction and stabilization of
the chaotic motion may improve the operation of the fluttering beam in supersonic
airflow.
78
Starting from 20t until 40t , the active control strategy developed in Chapter 2
is continuously applied. As can be seen from Fig. 4.2, the nonlinear vibration of the beam
is indeed controlled with its amplitude significantly reduced and its chaotic vibration of
the beam is stabilized, after the application of the control strategy. That is, the chaotic
vibration of the beam is well synchronized to the reference signal applied as described in
Eq. (4.7). However, although the active control strategy is applied at 20t , as is noticed
in the numerical simulations, a short period of time about 2.5 non-dimensional time units
is needed for the vibration of the beam to be actually controlled after the application of
active the control strategy.
Figure 4.2 The wave diagram of
pw before and after the application of the active control
strategy
Fig. 4.3 shows the 2-D phase diagram of the vibration of the beam at the selected
point before the application of the active control strategy, to demonstrate the chaotic
vibration of the fluttering beam. As can be seen from Fig. 4.3, the maximum amplitude of
the vibration of the fluttering beam, is more than 2.5, or in other words 2.5 times the
thickness of the beam. Therefore, an effective active control strategy, corresponding to
the six-dimensional dynamic system, is needed for the nonlinear vibration control of the
beam.
79
Figure 4.3 The 2-D phase diagram of
pw before the application of the active control
strategy
It may also be significant to see the displacements of the beam corresponding to each
of the vibration modes via the 6th
-order Galerkin method. Before the application of the
active control strategy, the vibration of the fluttering beam represented by 1w , 2w , 3w ,
4w , 5w and 6w is shown in Fig. 4.4. As can be seen from Fig. 4.4, the vibrations of the
beam for the first six vibration modes are all chaotic before the application of the active
control strategy. In Figs. 4.4 (a-f), it is interesting to notice that the maximum amplitudes
of the vibrations are not monotonically reduced as expected: even the higher the vibration
mode significantly contributes to the vibration of the selected point. It should be noticed
the maximum amplitude of the vibration of the first vibration mode shown in Fig. 4.4 (a)
is less than 1 time the thickness of the beam, and the maximum amplitude of the sixth
vibration mode shown in Fig. 4.4 (f) is close to that of the first vibration mode. Based on
the numerical results, none of the response of the six vibration modes of the beam is
negligible. Therefore corresponding to the specified fluttering panel in this case, at least a
six-dimensional nonlinear dynamic system of the fluttering beam should be implemented
to accurately describe the vibration of a fluttering panel in practice.
80
(a)
(b)
(c)
(d)
81
(e)
(f)
Figure 4.4 The wave diagrams of the first six vibration modes before the application of
the active control strategy: (a) 1w ; (b) 2w ; (c) 3w ; (d) 4w ; (e) 5w ; (f) 6w
In the case of the application of the active control strategy, the vibration of the beam
is shown in Fig. 4.5 and Fig. 4.6. In Fig. 4.5 the vibration at the selected point, pw , is
shown, for the period of time from t=15 to t=25. It can be seen from Fig. 4.5, a short
period of time is needed for stabilizing the panel, while in the studies (Haghighi and
Markazi, 2010; Yau et al., 2011), such period is very short and the vibration of the
dynamic system in their studies is almost synchronized to the reference signal right after
the active control strategy is applied. However, in the previous studies, the FSMC
strategy, cannot be employed in the vibration control of a multi-dimensional nonlinear
dynamic system.
82
Figure 4.5 The wave diagram of
pw after the application of the active control strategy
From Figs. 4.6 (a-f), the vibration of the panel is shown in terms of 1w , 2w , 3w , 4w ,
5w and 6w . Based on these figures, through the application of the active control strategy,
each of the displacements of the fluttering beam is gradually stabilized from a chaotic
motion into a periodic one. However, it can be seen from these figures, the amplitudes of
the displacements are different and not monotonically reduced from 1w to 6w , though
they are all stabilized eventually. It should be noticed that all these stabilizations
including the variations contribute to the control and stabilization of the beam as
described by pw in Eq. (4.8). It should be also noticed the maximum amplitude of the
first vibration mode shown in Fig. 4.6 (a) is decreased to 0, while the maximum
amplitude shown in Fig. 4.6 (f) is increased from about 0.6 to 0.7. That is after the
vibration of the beam has been stabilized, the vibration of the sixth vibration mode of the
beam has become more significant. Thus, it can be confirmed that: corresponding to the
selected air flowing rate in this case, a six-dimensional nonlinear dynamic system should
be derived in the active nonlinear vibration control of a fluttering panel in practice.
83
(a)
(b)
(c)
(d)
84
(e)
(f)
Figure 4.6 The wave diagrams of the first six vibration modes after the application of the
active control strategy: (a) 1w ; (b) 2w ; (c) 3w ; (d) 4w ; (e) 5w ; (f) 6w
Fig. 4.7 shows the comparison between the actual vibration of the beam pw and that
of the reference signal rw . One may notice that the reference signal rw is periodic with
respect to time t. One may also see from Fig. 4.7, the maximum amplitude of the
vibration of the beam is different from that of the reference signal after the stabilization
of the beam with the application of the active control strategy. However, as shown in the
Fig. 4.7, the maximum amplitude of the beam is very close to that of the reference signal
and the vibration of the beam is fairly close to periodic following the pattern of the
reference signal, as desired.
85
Figure 4.7 The comparison between
pw (denoted with the continuous blue line) and rw
(denoted with the green dash line) in wave diagram
Fig. 4.8 shows the control input U . Initially, the control input displays a non-
periodic wave diagram for a short period of time corresponding to the time period of
stabilization of the vibration of the beam after the application of the active control
strategy. The maximum value of the control input in this period is around 2000, and once
the system is stabilized, the control input displays a periodic wave diagram as shown in
the figure and the maximum value of the control input is decreased to about 1500.
Figure 4.8 The control input U
86
4.6 Conclusion
In this chapter, the active control strategy developed in Chapter 2 is applied in
controlling and stabilizing the nonlinear vibration of fluttering Euler-Bernoulli beam
subject to external non-periodic aerodynamic excitation. The governing equation of the
geometrically nonlinear beam is converted into a six-dimensional dynamic system. The
numerical simulation with the application of the proposed active control strategy shows
the effectiveness of the proposed active control strategy.
87
CHAPTER 5 AXIALLY TRANSLATING EULER-
BERNOULLI BEAM OF FIXED LENGTH WITOUT
EXTERNAL EXCITATION
5.1 Introduction
In this chapter, the active vibration control strategy developed in Chapter 2 is applied
to control and stabilize the nonlinear vibrations of an axially translating Euler-Bernoulli
beam with pinned-pinned boundaries. The equations of motion of the axially translating
Euler-Bernoulli beam are established based on the von Karman-type equations. In the
development of the solutions of the beam, the equations in the forms of partial differential
equations are non-dimensionalized and transformed into six ordinary differential
equations via a 6th
-order Galerkin method. Corresponding to the derived multi-
dimensional dynamic system, the active control strategy developed in the Chapter 2 is
applied. In the numerical simulations conducted, a case of chaotic vibration of the beam
is discovered and this nonlinear vibration is suppressed and stabilized with the
application of the control strategy. The applicability and effectiveness of the control
strategy developed is also validated.
5.2 Equations of Motion
The axially translating Euler-Bernoulli beam without external excitation considered
in this research is sketched in Fig. 5.1. The equations of motion of the beam are to be
derived based on Hamilton’s principle and von Karman-type equations. As can be seen
88
from the figure, the axially translating beam with pinned-pinned boundaries is allowed to
move axially at a constant rate 0v, and the length of the beam is given as 0l , the area of
the rectangular cross section of the beam is constant with width b and thickness h . The
x axis is along the mid-plane of the beam. The displacement of any point of the beam
along the x- and z- axes are designated as u and w .
Figure 5.1 The sketch of the axially translating Euler-Bernoulli beam
Starting from the origin at the left support of the beam, a position vector, r , of any
point ztx , of the axially translating beam without deformation is given as,
kir ztx )( , (5.1)
where i and k are the unit vectors of the fixed Cartesian coordinate.
Thus, the displacement field of the translating beam can be derived as,
kiR ttxwztx
ttxwzttxutx ,
,, 0
00
, (5.2)
where ttxu ,0 and ttxw ,0 are the displacement components along the x- and z-
directions respectively, of a point in the mid-plane ( 0z ).
Taking the total differentiation of R with respect to time t, it can be obtained,
89
ki
R
dt
ttxdw
dt
ttxdw
txz
dt
ttxdu
dt
tdx
dt
d ,,, 000
, (5.3)
where the derivative of tx with respect to time is equal to the translating rate of the
beam, and the full derivative of 0w is,
tx
ttxwv
t
ttxw
dt
ttxdw
,,, 00
00 . (5.4)
Hence, the kinetic energy of the axially translating Euler-Bernoulli beam without
external excitation over a volume V of the beam is expressible as
dxdzdt
dρbdV
dt
dρT
h
h
l
V
2
2 0
220
2
1
2
1 RR, (5.5)
where ρ denotes the density of the material of the beam.
The von Karman-type equations of strains of large deflection associated with the
displacement field, normal to the cross section of the beam along the x direction, in Eq.
(5.2), can thus be given by,
tx
ttxwz
tx
ttxw
tx
ttxu2
0
22
0011
,,
2
1,
, (5.6)
and then the total strain energy of the beam can be given by
022 2
11 11 11 112 0
1 1( )
2 2
h l
V hE Q dV b Q dx t dz
, (5.7)
where 11Q represents the elastic coefficient in the same direction with 11 .
90
The virtual work done by the force due to damping effect is defined as,
l
tdxdt
ttxdwcbW
0
0 ,, (5.8)
where c denotes the damping coefficient.
In the following analysis, the Hamilton’s principle will be employed to obtain the
nonlinear equations of motion for the beam. The mathematical statement of the
Hamilton’s principle is given by
02
1
dtWLt
t , (5.9)
where the total Lagrangian function L is given by
L T E . (5.10)
For the sake of clarity, hereafter, use x , 0u, 0w
to replace tx , ttxu ,0 , and
ttxw ,0 respectively. Substitute Eqs. (5.5), (5.7), and (5.8) into Eq. (5.10), and the
nonlinear equations of motion of the axially translating Euler-Bernoulli beam without
external excitation are derived as follows,
02
0
2
02
2
02
0
2
0112
0
2
11
dt
udI
dt
ldI
x
w
x
wA
x
uA , (5.11-a)
2
0
22
0112
0
2
0112
0
2
011
2
3
x
w
x
wA
x
w
x
uA
x
u
x
wA
02
0
2
00
4
0
4
11
dt
wdI
dt
dwc
x
wD , (5.11-b)
91
where,
hQA 1111 , hI 0 11113
Qh
D3
. (5.12)
Associated with the nonlinear dynamic equations and the boundary conditions
(Abou-Rayan et al., 1993; Younis and Nayfeh, 2003), the strain can be obtained as,
22
1
2
1 0
2
2
11
0
0
2
0
0
2
00 0 lx
dt
xd
A
Idx
x
w
lx
w
x
u l
. (5.13)
Then, substituting Eq. (5.13) into Eq. (5.11-b), the nonlinear differential governing
equation of the translating beam in z direction is derived as,
02 0
2
0
2
0
2
110
4
0
4
112
0
2
0
dx
x
w
x
w
l
A
dt
dwc
x
wD
dt
wdI
l. (5.14)
To validate the governing equation Eq. (5.14) and facilitate the numerical
simulations in the following sections of this chapter, the following non-dimensional
variables are introduced,
ttblI
IQt
4
00
11 , 0l
xx , (5.15)
and,
h
ww 0
0 , dt
dw
htd
wd 00 1
,
2
0
2
22
0
21
dt
wd
htd
wd
, 00
1l
hl ,
h
Qcc 11 . (5.16)
92
With the non-dimensional variables shown in Eqs. (5.15) and (5.16) introduced into
Eq. (5.14), the non-dimensional governing equation of the translating Euler-Bernoulli
beam without external excitation can be expressed as,
1
0
2
0
2
0
2
4
0
4
0
2
0
21
dxx
w
x
wF
x
wG
td
wdD
Atd
wd, (5.17)
where,
h
IA
0 ,
ρh
cD ,
4
0
2
11
lρh
DG
,
4
0
2
11
2
1
lρ
hAF
. (5.18)
5.3 Series Solutions
Based on the Galerkin method of discretization, the transverse displacement 0w is
expanded in a series form, in terms of a set of comparison functions as,
1
0
n
nn twxφw . (5.19)
Corresponding to the pinned-pinned boundaries of the axially translating beam,
xφn can be given as follows,
xnxφn sin . (5.20)
Substitute the series solution of Eq. (5.19) into Eq. (5.17), and to assist presentation,
replace n , nw , nw , nw , v , and t for )(xφn , twn , dt
dwn,
2
2
dt
wd n , v and t respectively. With
93
the application of the Galerkin method at 6n , the discretized governing equations of
the translating beam with pinned-pinned boundaries can be obtained in the following,
1
22
64212
1
35
24
15
16
3
8(
2wvAAvwAvwAvw
Aw
12461
4
2
1
3
4
15
8
35
12
2
1wDDvwDvwDvwGw
2
41
43
1
42
51
42
31
4 44
1
4
25
4
9wFwFwwFwwFw
)9 2
21
42
61
4 wFwwFw , (5.21-a)
2
22
1352 23
8
5
24
21
40(
2wvAAvwAvwAvw
Aw
25132
4
2
1
21
20
3
4
5
128 wDDvwDvwDvwGw
2
42
42
12
42
52
42
32
4 16259 wFwwFwwFwwFw
)436 3
2
42
62
4 FwwFw , (5.21-b)
3
22
26432
9
5
24
3
8
7
48(
2wvAAvwAvwAvw
Aw
34263
4
2
1
7
24
5
12
3
4
2
81wDDvwDvwDvwGw
2
43
42
13
42
53
43
3
4 364
9
4
225
4
81wFwwFwwFwwF
94
)981 2
23
42
63
4 wFwwFw , (5.21-c)
4
22
5134 89
80
15
16
7
48(
2wvAAvwAvwAvw
Aw
41354
4
2
1
15
8
7
24
9
40128 wDDvwDvwDvwGw
3
4
42
14
42
54
42
34
4 64410036 FwwFwwFwwFw
)16144 2
24
42
64
4 wFwwFw , (5.21-d)
5
22
64252
25
11
120
9
80
21
40(
2wvAAvwAvwAvw
Aw
56245
4
2
1
11
60
21
20
9
40
2
625wDDvwDvwDvwGw
3
45
42
15
43
5
42
35
4 1004
25
4
625
4
225wFwwFwFwwFw
)25225 2
25
42
65
4 wFwwFw , (5.21-e)
6
22
5316 1811
120
3
8
35
24(
2wvAAvwAvwAvw
Aw
63516
4
2
1
3
4
11
60
35
12648 wDDvwDvwDvwGw
2
46
42
16
42
56
42
36
4 144922581 wFwwFwwFwwFw
)36324 2
26
43
6
4 wFwwF . (5.21-f)
95
5.4 Control Design
Corresponding to the axially translating beam governed by Eq. (5.17), and the active
nonlinear control strategy developed in Eqs. (2.15) ~ (2.20), the governing equation with
the control input for the beam can be given by the following expression.
00
1
0
2
0
2
0
2
4
0
4
00 ,1
wwFUdxx
w
x
wF
x
wGwD
Aw
. (5.22)
With the application of the Galerkin discretization of 6th
-order, Eq. (5.22) may have
the following form,
tfu
wFwwFw
wFwFwwFwwFw
DwDvwDvwDvwGw
wvAAvwAvwAvw
Aw
ww
,
9
44
1
4
25
4
9
2
1
3
4
15
8
35
12
2
1
2
1
35
24
15
16
3
8
2 11
2
1,31,1
42
1,61,1
4
2
1,41,1
43
1,1
42
1,51,1
42
1,31,1
4
2,11,21,41,61,1
4
1,1
22
1,61,41,2
2,1
2,11,1
W
(5.23-a)
tfu
FwwFwwFw
wFwwFwwFw
DwDvwDvwDvwGw
wvAAvwAvwAvw
Aw
ww
,
43616
259
2
1
21
20
3
4
5
128
23
8
5
24
21
40
222
3
1,2
42
1,61,2
42
1,41,2
4
2
1,11,2
42
1,51,2
42
1,31,2
4
2,21,51,11,31,2
4
1,2
22
1,11,31,5
2,2
2,21,2
W
(5.23-b)
96
tfu
wFwwFwwFw
wFwwFwwF
DwDvwDvwDvwGw
wvAAvwAvwAvw
Aw
ww
,
98136
4
9
4
225
4
81
2
1
7
24
5
12
3
4
2
81
2
9
5
24
3
8
7
48
233
2
1,21,3
42
1,61,3
42
1,41,3
4
2
1,11,3
42
1,51,3
43
1,3
4
2,31,41,21,61,3
4
1,3
22
1,21,61,4
2,3
2,31,3
W
(5.23-c)
tfu
wFwwFw
FwwFwwFwwFw
DwDvwDvwDvwGw
wvAAvwAvwAvw
Aw
ww
,
16144
64410036
2
1
15
8
7
24
9
40128
89
80
15
16
7
48
244
2
1,21,4
42
1,61,4
4
3
1,4
42
1,11,4
42
1,51,4
42
1,31,4
4
2,41,11,31,51,4
4
1,4
22
1,51,11,3
2,4
2,41,4
W
(5.23-d)
tfu
wFwwFwwFw
wFwFwwFw
DwDvwDvwDvwGw
wvAAvwAvwAvw
Aw
ww
,
25225100
4
25
4
625
4
225
2
1
11
60
21
20
9
40
2
625
2
25
11
120
9
80
21
40
255
2
1,21,5
42
1,61,5
43
1,41,5
4
2
1,11,5
43
1,5
42
1,31,5
4
2,51,61,21,41,5
4
1,5
22
1,61,41,2
2,5
2,51,5
W
(5.23-e)
97
tfu
wFwwFwFw
wFwwFwwFw
DwDvwDvwDvwGw
wvAAvwAvwAvw
Aw
ww
,
)36324144
922581
2
1
3
4
11
60
35
12648
1811
120
3
8
35
24
266
2
1,21,6
43
1,6
42
1,41,6
4
2
1,11,6
42
1,51,6
42
1,31,6
4
2,61,31,51,11,6
4
1,6
22
1,51,31,1
2,6
2,61,6
W
(5.23-f)
where 1u , 2u , 3u , 4u , 5u and 6u are derived as follows through the 6th
-order Galerkin
method,
Uu
21 , Uu 02 , Uu
3
23 , Uu 04 , Uu
5
25 , Uu 06 .
It should be noticed that due to the pinned-pinned boundary of the translating beam,
the application of the Galerkin method based on Eq. (5.20) leaves the coefficients of 2u ,
4u , and 6u as zero. In the next section, it will be demonstrated in the numerical simulation
that the actual vibration of the axially translating Euler-Bernoulli beam at a selected point
can be well synchronized to a desired reference signal in the case that the coefficients of
2u , 4u , and 6u are zero.
5.5 Numerical Simulation
To demonstrate the applicability and effectiveness of the active control strategy
developed in the Chapter 2, numerical simulations are conducted for controlling the
axially translating beam governed by Eq. (5.17). The nonlinear vibrations of the beam are
98
focused in this section. With the numerical simulations performed, a chaotic vibration is
found when the beam is translating at a certain rate. The proposed active control strategy
is found not only effectively reduces the amplitude of the chaotic vibration, but also
stabilizes the motion so that the vibration of the axially translating beam is synchronized
to a desired periodic vibration. To facilitate the numerical simulation, the 4th
-order P-T
method (Dai, 2008), is implemented.
The parameters used for the simulations are given as follows,
smv 4.00 , 20 msmNc , PaQ 10
11 101809.1 ,
ml 5.00 , mb 02.0 , mh 002.0 , 31800 mkgρ .
The non-dimensionalized initial conditions, corresponding to the displacements
described in Eqs. (5.21) after the implementation of the 6th
-order Galerkin method, are
taken as
01.001,1 w , 002,1 w , 002.001,2 w , 002,2 w , 001,3 w , 002,3 w ,
001,4 w , 002,4 w , 001,5 w , 002,5 w , 002.001,6 w , 002,6 w .
If the vibration of a point at 0.35m along the x-axis of the beam is selected, based on
Eq. (5.19) the non-dimensional vibration of the selected point pw can be derived as,
1,31,21,1
6
1
1, 3090168873.09510564931.08090170164.0 wwwwφwn
nnp
1,61,51,4 5877853737.0000000000.15877853737.0 www . (5.24)
99
A chaotic vibration is discovered when the control parameters and the unknown
external disturbance take the following values:
twr 4966.0sin4.1 , 2750 , 375fsk , )sin(001.0, pwwwF . (5.25)
The vibration of the beam translating at the speed smv 4.00 is shown in Fig. 5.2,
corresponding to the non-dimensional time interval from 0t to 550t . During this
period of time, the active control strategy is applied at 250t . One may notice in Fig.
5.2, the maximum amplitude of the vibration of the beam can exceed 3. Considering that
the displacement shown in the figure is non-dimensional, the amplitude is large. Thus,
reduction and stabilization of the chaotic vibration may improve the operation of the
beam.
Starting from 250t until 550t , the active control strategy developed in the
Chapter 2 is continuously applied. As can be seen from Fig. 5.2, the nonlinear vibration
of the beam is indeed controlled with significantly reduced amplitude and the chaotic
vibration of the beam is stabilized, after the application of the control strategy. In other
words, the chaotic vibration of the beam is well synchronized to the reference signal
applied as described in Eq. (5.25) after 360t , and the maximum amplitude of the
vibration at the selected point has been significantly reduced from about 3.5 to 1.4.
However, although the control strategy is applied at 250t , as is noticed in the
numerical simulations, a short period of time or a few more cycles are needed for the
vibration of the beam to be actually controlled after the application of the control strategy.
In this case, it may take about 110 non-dimensional time units for the nonlinear system
of multiple dimensions to be satisfactorily stabilized.
100
Figure 5.2 The wave diagram of
pw before and after the application of the active control
strategy
Fig. 5.3 shows the phase diagram of the vibration of the beam at the selected point
before the application of the control strategy, to demonstrate the chaotic vibration of the
axially translating beam. As can be seen from Fig. 5.3, the maximum amplitude of the
vibration of the axially translating beam is more than 3.5, or in other words 3.5 times the
thickness of the beam. Therefore, an effective control strategy, corresponding to the six-
dimensional dynamic system, is needed for the vibration control of the beam.
Figure 5.3 The 2-D phase diagram of
pw before the application of the active control
strategy
It may also be significant to see the displacements of the beam corresponding to each
of the vibration modes via the 6th
-order Galerkin method. Before the application of the
active control strategy, the vibration of the beam represented by 1w , 2w , 3w , 4w , 5w and
6w is shown in Fig. 5.4. As can be seen from Fig. 5.4, the vibrations of the beam for the
101
first six vibration modes are all chaotic before the application of the active control
strategy. In Figs. 5.4 (a-f), it is interesting to notice that the amplitudes of the vibrations
are monotonically reduced as expected: the higher the vibration mode is, the lower the
amplitude of the vibration mode is. However, it should be noticed the maximum
amplitude of the vibration of the first vibration mode shown in Fig. 5.4 (a) is close to 4
times the thickness of the beam, while the maximum amplitude shown in Fig. 5.4 (f) is
close to 0.5 time the thickness of the beam. Based on the numerical results, none of the
vibration of the six vibration modes of the beam is negligible. Therefore, corresponding
to the specified axially translating speed in this case, at least a six-dimensional nonlinear
dynamic system of the beam should be implemented to accurately describe the vibration
the beam.
102
(a)
(b)
(c)
(d)
103
(e)
(f)
Figure 5.4 The wave diagrams of the first six vibration modes before the application of
the active control strategy: (a) 1w ; (b) 2w ; (c) 3w ; (d) 4w ; (e) 5w ; (f) 6w
In the case of the application of the active control strategy, the vibration of the beam
at the translating speed smv 4.00 is shown in Fig. 5.5 and Fig. 5.6. In Fig. 5.5 the
vibration at the selected point, pw , is shown, for the period of time from 250t to
550t . It can be seen from Fig. 5.5, a short period of time from 250t to about 360t
is needed for stabilizing the beam, while in the studies (Haghighi and Markazi, 2010; Yau
et al., 2011), such period is very short and the vibration of the dynamic system in their
studies is almost synchronized to the reference signal right after the control strategy is
applied. However, in the previous studies, the FSMC strategy cannot be employed in the
vibration control of a multi-dimensional nonlinear dynamic system.
104
Figure 5.5 The wave diagram of pw after the application of the active control strategy
From Figs. 5.6 (a-f), the vibration of the beam is shown in terms of 1w , 2w , 3w , 4w ,
5w and 6w . Based on these figures, through the application of the control strategy, each
of the displacements of the axially translating beam is gradually stabilized from a chaotic
vibration into a periodic one. Although it can be seen from these figures, the amplitudes
of the displacements are not monotonically reduced from 1w to 6w , they are all stabilized
eventually and the synchronization of pw to the reference signal is also completed based
on the relation given in Eq. (5.24). Also, it can be seen from Fig 5.6 (b), the stabilization
of 2w takes much longer time than the others, since it keeps slowly decreasing to the end
of the numerical simulation. It should be noticed the maximum amplitude of the first
vibration mode shown in Fig. 5.6 (a) is about 1 time the thickness of the beam, while the
maximum amplitude shown in Fig. 5.6 (f) is about 0.3 time the thickness of the beam.
That is after the vibration of the beam has been stabilized, the vibration of the sixth
vibration mode of the beam has become more significant. Thus, it can be confirmed that:
corresponding to the selected axially translating speed in this case, a six-dimensional
nonlinear dynamic system should be derived in the active nonlinear vibration control of
an axially translating beam.
105
(a)
(b)
(c)
(d)
106
(e)
(f)
Figure 5.6 The wave diagrams of the first six vibration modes after the application of the
active control strategy: (a) 1w ; (b) 2w ; (c) 3w ; (d) 4w ; (e) 5w ; (f) 6w
Fig. 5.7 shows the comparison between the actual vibration of the beam pw and that
of the reference signal rw . One may notice that the reference signal rw is perfectly
periodic with respect to time t. One may also see from Fig. 5.7, the maximum amplitude
of the vibration of the beam slightly varies after the stabilization of the beam with the
application of the control strategy. However, as shown in the Fig. 5.7, the maximum
amplitude of the beam is very close to that of the reference signal and the vibration of the
beam is fairly close to periodic following the pattern of the reference signal, as desired.
107
Figure 5.7 The comparison between
pw (denoted with the continuous blue line) and rw
(denoted with the yellow dash line) in wave diagram
Fig. 5.8 shows the control input U . Initially, the control input displays a non-
periodic wave diagram for a short period of time corresponding to the time period of
stabilization of the vibration of the beam after the application of the active control
strategy. The maximum value of the control input in this period is close to 15000, and
once the system is stabilized, the control input displays a periodic wave diagram as
shown in the figure and the maximum value of the control input is significantly decreased
to about 7500 .
Figure 5.8 The control input U
5.6 Conclusion
In this chapter, the active control strategy developed in Chapter 2 is applied in
controlling and stabilizing the nonlinear vibration of an axially translating Euler-
108
Bernoulli beam without external excitation. The governing equation of the geometrically
nonlinear beam is converted into a six-dimensional dynamic system. The numerical
simulation with the application of the proposed active control strategy shows the
effectiveness of the strategy.
109
CHAPTER 6 AXIALLY RETRACTING EULER-NOULLI
BEAM WITHOUT EXTERNAL EXCITATION
6.1 Introduction
In this chapter, the active vibration control strategy proposed in Chapter 2 is to be
applied for controlling the large-amplitude vibration of a retracting robotic arm of
multiple dimensions. The robotic arm is represented with a retracting Euler-Bernoulli
beam without external excitation. The equations of motion of the Euler-Bernoulli beam
with fixed-free boundary are to be established based on von Karman-type equations and
the consideration of the beam’s geometric nonlinearity. In developing the solutions of the
beam, the governing equation in the forms of partial differential equations are non-
dimensionalized and then converted into a multi-dimensional system through the 3rd
-
order Galerkin method. With respect to the derived multi-dimensional dynamic system,
the active control strategy previously proposed is applied, and the applicability and
efficiency of the proposed control strategy developed is significant in controlling the
nonlinear vibrations of the retracting Euler-Bernoulli beam without external excitation. A
case of large-amplitude vibration of the beam is presented to validate the effectiveness of
the proposed active control strategy in controlling such vibration of the retracting Euler-
Bernoulli beam.
6.2 Equations of Motion
The retracting Euler-Bernoulli beam investigated in this chapter is sketched in Fig.
6.l. The governing equations of motion of the beam are to be derived based on the
110
Hamilton’s principle. As can be seen from Fig. 6.1, a retracting Euler-Bernoulli beam
with fixed-free boundaries is presented, and the initial length of the beam is given as 0l .
The x axis is along the axial direction of the beam. The displacements of a point of the
retracting beam along the x- and z- axes are designated with u and w .
Figure 6.1 The sketch of the retracting Euler-Bernoulli beam
Starting from the origin of the retracting Euler-Bernoulli beam, a position vector, r ,
of any point ztx , of the beam without deformation is given as,
kir ztx ,
where i and k are the unit vectors of the fixed Cartesian coordinate shown in the Fig. 6.1.
Thus, the displacement field of the beam is,
kiΔ 00
0 wx
wzu
,
Therefore, the displacement field of the retracting beam can be derived as,
kikiΔrR ttxwzx
ttxwzttxuxwu ,
,*, 0
00
,
o
z
x
z
y
dt
tdl
o
111
where ttxu ,0 and ttxw ,0 are the displacement components along the x- and z-
directions respectively, of a point on the beam.
Taking the total differentiation of R with respect to the time t, it can be obtained
kiR
dt
ttxdw
dt
ttxdu
dt
tdx
dt
d ,, 00
.
Hence, the kinetic energy of the retracting beam arm is expressed as,
l
dxdt
d
dt
dT
0 2
1 RR , (6.1)
where ρ denotes the density of the beam per unit length, and l , the instant length of the
retracting beam, is given as,
vtll 0 ,
and v the retracting velocity of the beam is constant.
The von Karman-type equations of strains of large deflection associated with the
displacement field, normal to the cross section of the retracting beam along the x
direction, can be given by,
2
0
22
0011
,*
,
2
1,
x
ttxwz
x
ttxw
x
ttxu
,
Therefore, the total strain energy of the beam can be given by,
l
dxQU0
1111112
1 , (6.2)
112
where EbhQ 11 , and E represents the elastic coefficient in the same direction with 11 .,
b is the breadth of the beam, and h is the thickness of the beam.
The virtual work is zero since there no external excitation applied on the retracting
beam, and therefore,
0W . (6.3)
The Hamilton’s principle is employed to obtain the nonlinear equations of motion
for the retracting beam. The mathematical statement of the Hamilton’s principle is given
by,
02
1
2
1
dtWdtLt
t
t
t , (6.4)
where the total Lagrangian function L is given by,
UTL . (6.5)
For convenience, replace tx , ttxu ,0 , and ttxw ,0 with x , 0u , and 0w in the
following. Substitute Eq. (6.1), Eq. (6.2), Eq. (6.3) and Eq. (6.5) into Eq. (6.4), and then
the first term in Eq. (6.4) can be developed as,
2
1
2
1
t
t
t
tdtUTLdt
2
1
2
1111111
t
t V
t
t VdVdtQdVdt
dt
d
dt
d
RR
2
1
2
1111111
t
t VV
t
tdVdtεQdVdt
dt
d
dt
d
RR
113
2
1
2
11111112
2
0t
t VV
t
tdVdtQdVdt
dt
d R
R
2
1
2
20 2
2
2
2t
t
h
h
l
dxdzdtdt
wd
dt
udwub kiki
2
1
2
20
111111
t
t
h
h
l
dtdxdzQb
2
1
2
20 2
2
00
t
t
h
h
l
dxdzdtdt
ud
x
wzuxb
2
1
2
20 2
0
2
0
t
t
h
h
l
dxdzdtdt
wdwzb
2
1
2
20
112
0
22
0011
2
1t
t
h
h
l
dtdxdzx
wz
x
w
x
uQb . (6.6)
Eq. (6.6) is rearranged in the following,
2
1
2
20 2
2
001
t
t
h
h
l
dxdzdtdt
ud
x
wzuxbL ,
2
1
2
20 2
0
2
02
t
t
h
h
l
dxdzdtdt
wdwzbL ,
2
1
2
20
112
0
22
00113
2
1t
t
h
h
l
dtdxdzx
wz
x
w
x
uQbL ,
and,
114
2
1
2
20 2
2
001
t
t
h
h
l
dxdzdtdt
ud
x
wzuxbL
2
1
2
1
2
20
0
2
2
2
20
02
2t
t
h
h
lt
t
h
h
l
dxdzdtx
w
dt
udzbdxdzdtu
dt
udb
2
1
2
1
2
2
00 2
2
2
20
02
2
0t
t
h
h
lt
t
h
h
l
dxdzdtwx
u
dt
dzbdxdzdtu
dt
udb
2
1
2
20
00
2
2
2
0
2
2
2t
t
h
h
l
dxdzdtux
w
dt
dz
dt
ud
dt
xdb
2
1
2
2
00 2
0
2
0
2
2
10t
t
h
h
l
dxdzdtwx
wz
x
u
dt
dzb
2
1
2
20
02
0
2
2
2
0t
t
h
h
l
dxdzdtudt
ud
dt
xdb
2
1
2
2
00 2
0
2
2
2200
t
t
h
h
l
dxdzdtwdt
wd
xzb
2
1
2
20
02
0
2
2
2t
t
h
h
l
dxdzdtudt
ud
dt
xdb
2
1
2
2
00 2
0
2
2
22
t
t
h
h
l
dxdzdtwdt
wd
xzb , (6.7-a)
2
1
2
20 2
0
2
02
t
t
h
h
l
dxdzdtdt
wdwzbL
115
2
1
2
2
00 2
0
2t
t
h
h
l
dxdzdtwdt
wdb , (6.7-b)
2
1
2
20
112
0
22
00113
2
1t
t
h
h
l
dtdxdzx
wz
x
w
x
uQbL
2
1
2
20
112
0
2
00011
t
t
h
h
l
dtdxdzx
wz
x
w
x
w
x
uQb
2
1
2
20 2
0
2
111100
11110
1111
t
t
h
h
l
dtdxdzx
wzQ
x
w
x
wQ
x
uQb
2
20
011
110h
h
l
dxdzux
Qb
2
20
00
11110h
h
l
dxdzwx
w
xQb
2
20
02
11
2
1100h
h
l
dxdzwx
zQb
2
20
03
0
3
2
0
2
0
2
0
2
11
h
h
l
dxdzux
(x,t)wz
x
w
x
w
x
uQb
2
20
00
3
0
3
2
0
2
0
2
0
2
11
h
h
l
dxdzwx
w
x
wz
x
w
x
w
x
uQb
2
20
02
0
2
2
0
22
0011
2
1h
h
l
dxdzwx
w
x
wz
x
w
x
uQb
2
20
04
0
4
11
2
3
0
3
0112
0
2
2
0
2
113
0
3
11
h
h
l
dxdzwx
wQz
x
w
x
wzQ
x
w
x
wzQ
x
uzQb
116
2
20
02
0
2
0
2
0
2
11
h
h
l
dxdzux
w
x
w
x
uQb
2
20
00
2
0
2
0
2
0
2
11
h
h
l
dxdzwx
w
x
w
x
w
x
uQb
2
20
02
0
22
0011
2
1h
h
l
dxdzwx
w
x
w
x
uQb
2
20
04
0
4
11
2
h
h
l
dxdzwx
wQzb . (6.7-c)
From Eq. (6.7-a), Eq. (6.7-b), Eq. (6.7-c), the nonlinear governing equation of an
retracting Euler-Bernoulli beam without external excitation can be derived in the
following,
2
2
dt
xdh
2
0
2
dt
udh
2
0
2
11x
uhQ
0
2
0
2
011
x
w
x
whQ , (6.8-a)
2
0
2
dt
wdh
2
0
2
2
23
12 dt
wd
x
h
x
w
x
uhQ
0
2
0
2
11 2
0
2
011
x
w
x
uhQ
2
0
22
011
2
3
x
w
x
whQ
0
12 4
0
4
11
3
x
wQ
h. (6.8-b)
Associated with the nonlinear dynamic equations and the boundary conditions
(Abou-Rayan et al., 1993; Younis and Nayfeh, 2003), the strain can be obtained as,
22
1
2
12
2
110
2
0
2
00 lx
dt
xd
Qdx
x
w
lx
w
x
u l . (6.9)
117
Then, substitute Eq. (6.9) into Eq. (6.8-b), and the nonlinear differential governing
equation of the beam in z direction is derived as,
2
0
2
dt
wdh
2
0
2
2
23
12 dt
wd
x
h
l
x
wdx
x
whQ
l 0 2
0
22
011
2
10
12 4
0
4
11
3
x
wQ
h.(6.10)
To validate the governing equation Eq. (6.10) and facilitate the numerical
simulations in the consequent sections of this chapter, the following non-dimension
variables are introduced,
tblI
Qbhtt
4
00
11
3
12
1
, l
xx ,
0
00
l
ww ,
0l
ll . (6.11)
Introduce the non-dimensional variables shown in Eq. (6.11) into Eq. (6.10), the
non-dimension governing equation of the retracting beam can be expressed as,
2
0
2
td
wdA
2
0
2
2
2
td
wd
xB
1
0 2
0
22
0
x
wxd
x
wC 0
4
0
4
x
w. (6.12)
where,
2
2
12l
hA ,
24
2
11
2 l
hQB ,
24
11
2
12 l
QhC .
6.3 Series Solution
Based on the Galerkin method of discretization, the transverse displacement 0w is
expanded in a series form, in terms of a set of comparison functions as,
118
1
0
n
nn twxφw . (6.13)
Corresponding to the fixed-free boundary conditions of the retracting beam, xφn
can be given as follows,
xxxxx nn
nn
nnnnn
sinsh
sinsh
coschcosch
. (6.14)
Substitute the series solution of Eq. (6.14) into Eq. (6.13), and to assist the following
presentation, replace n , nw , nw , nw , t , and l for )(xφn , twn , td
wd n , 2
2
td
wd n , t , and l
respectively, and,
1,11 ww , 1,22 ww , 1,33 ww ,
2,11,1 ww , 2,21,2 ww , 2,31,3 ww .
Therefore, with the application of the Galerkin method at 3n , the discretized
governing equations of the retracting Euler-Bernoulli beam with the fixed-free boundary
conditions can be obtained in the following,
213132123312231321
2133121323212311232,3
2,31,3
213132123312231321
3121321323123213212,2
2,21,2
213132123312231321
2313213212313213212,1
2,11,1
w
ww
w
ww
w
ww
, (6.15)
119
where,
003836562.18581959666.01 A ,
240039336562.0873752475.12 A ,
680041373701.0564284688.13 A ,
A74232364.11240039336562.01 ,
A29402727.13003953771.12 ,
A230548650.3100041702276.03 ,
A45045556.27680041373701.01 ,
A0324787618.9100041702276.02 ,
A90565282.45004115395.13 ,
1,3
2
1,2
1,31,21,1
2
1,31,2
2
1,31,11,3
2
1,1
2
1,21,11,2
2
1,1
3
1,3
3
1,2
3
1,1
1
00.448
40.039.012.140606.359
52.56604.026.96205.035.151
ww
wwwwwwwww
wwwwwww
,
1,3
2
1,2
1,31,21,1
2
1,31,2
2
1,31,11,3
2
1,1
2
1,21,11,2
2
1,1
3
1,3
3
1,2
3
1,1
2
04.2
00.89663.455439.020.0
15.052.56646.150.212001.0
ww
wwwwwwwww
wwwwwww
,
120
1,3
2
1,2
1,31,21,1
2
1,31,2
2
1,31,11,3
2
1,1
2
1,21,11,2
2
1,1
3
1,3
3
1,2
3
1,1
3
63.4554
78.037.478.288612.1406
00.44820.088.978268.069.119
ww
wwwwwwwww
wwwwwww
,
and, , given as the non-dimensional axial translating velocity of the retracting beam, is
expressed as,
dt
dl
l
x .
6.4 Control Design
Corresponding to the retracting beam governed by Eq. (6.12), and the active control
strategy developed in Eqs. (2.15) ~ (2.20), the governing equation with the control input
for the beam can be given by the following expression,
0w A
2
0
2
2
2
dt
wd
xB
1
0 2
0
22
0
x
wdx
x
wC
4
0
4
x
w
wwFU , , (6.16)
With the application of the 3rd
-order Galerkin method, Eq. (6.16) may take the
following form,
tfuw
ww
tfuw
ww
tfuw
ww
,
,
,
3332,3
2,31,3
2222,2
2,21,2
11112,1
2,11,1
W
W
W
, (6.17)
121
where 1u , 2u , and 3u are derived as follows through the 3rd
-order Galerkin method,
Uu 7849249756.01 , Uu 4319801434.02 , Uu 256487792.03 .
In the next section, it will be demonstrated with the numerical simulation that the
actual vibration of the retracting Euler-Bernoulli beam without external excitation at a
selected point can be well synchronized to a desired reference signal.
6.5 Numerical Simulation
The vibration of the retracting beam is investigated numerically utilizing the multi-
dimensional system developed, with concentration on a randomly selected point on the
beam. With the numerical simulations performed, a large-amplitude vibration of the
selected point is discovered. Then, the proposed active control strategy in Chapter 2 is
found not only effective in reducing the amplitude of the discovered, but also
synchronizing the motion to the given frequency of the desired reference signal. To
facilitate the numerical simulation with higher accuracy and reliability, the 4th
-order P-T
method (Dai, 2008), is implemented.
The primary parameters of the retracting beam are given as below,
PaQ 10
11 1023.0 , ml 20 , mb 03.0 , mh 02.0 , 3
1000 mkgρ ,
and the constant velocity the Euler-Bernoulli beam retracting at is,
smv 02.0 .
122
The non-dimensionalized initial conditions, corresponding to the displacements
described by Eq. (6.15) after the implementation of the 3rd
-order Galerkin method, are
taken as,
2.001,1 w , 5.002,1 w , 11.001,2 w , 45.002,2 w ,
1.001,3 w , 4.002,3 w .
The length of the beam retracts from the initial length 2 meters to 1 meter with
respect to time, and pw given as the transverse displacement at a selected point, which is
0.05 meter from the moving end of the beam, is expressed as below,
3
1
1,
n
npnp wxw .
where 1,nw , 2,nw , and 3,nw respectively denote the contributions of the first three
vibration modes to the actual vibration of the selected point pw .
A large-amplitude vibration of the retracting beam occurs as shown in Fig. 6.2, while
the developed control strategy is not applied.
123
Figure 6.2 The wave diagram of pw without the application of the active control strategy
The vibration of the retracting beam is shown in Figure 6.2, corresponding to the
non-dimensional time from 0t to 45.109t . During this period of time, one may
notice in Figure 6.2, the maximum amplitude of the vibration of the retracting beam can
exceed 5. Considering that the displacement shown in Fig. 6.2 is non-dimensional, and
the dimensional amplitude is actually 5 times the initial thickness of the beam, the
maximum amplitude observed from Fig. 6.2 is large. Besides, the period of the beam
seems to increase with respect to the non-dimensional time and the amplitude of the
selected point gradually decreases once it reaches a certain value around 5.5, and
therefore the discovery of the large-amplitude vibration requires to be suppressed.
Besides, from Fig. 6.3 (a), Fig. 6.3 (b) and Fig. 6.3 (c), it can be learned that
although the contribution of the first vibration mode in Fig. 6.3 (a) is larger than those of
the other two vibration modes as shown in Fig. 6.3 (b) and Fig. 6.3 (c), the contributions
of the second vibration mode is obviously not negligible. Actually it can be learned that
the second vibration mode also significantly contributes to the actual vibration of the
selected point. Thus, the development of a multi-dimensional dynamic system is
necessary for the accurate prediction of the dynamics of the retracting beam.
124
(a)
(b)
(c)
Figure 6.3 The wave diagrams of the first three vibration modes without the application
of the active control strategy: (a) 1w ; (b) 2w ; (c) 3w
125
As shown in Fig. 6.4, the proposed active control strategy is applied at the non-
dimensional time 25t , and the control parameters and the unknown external
disturbance take the following values,
twr 3518.14sin35.0 , 220 , 1fsk
, )sin(2.0, pwwwF
.
As can be seen from Fig. 6.4, the maximum amplitude of the vibration of the
retracting beam is reduced significantly from about 5.5 to the value 0.35. The
synchronization of the vibration of the beam also results in a periodic vibration of the
actual vibration of the selected point.
Figure 6.4 The wave diagram of pw with the application of the active control strategy
From Fig. 6.5 (a), Fig. 6.5 (b) and Fig. 6.5 (c), following should be noticed: although
the contributions of the first three vibration modes of the beam are indeed affected with
the application of the proposed control strategy, the suppression of the vibration of the
first vibration mode is more significant than that of the other two vibration modes. That is:
despite the continuous increase in the amplitude of the second vibration mode, the actual
response of the beam at the selected point can be finally synchronized to the reference
signal.
126
(a)
(b)
(c)
Figure 6.5 The wave diagrams of the first three vibration modes with the application of
the active control strategy: (a) 1w ; (b) 2w ; (c) 3w
Fig. 6.6 is presented to fully demonstrate the effectiveness of the proposed active
control strategy. In Fig. 6.6, the difference is very small between the actual vibration of
127
the retracting Euler-Bernoulli beam and that of the reference signal, and the
synchronization between the actual vibration of the beam at the selected point and the
reference signal shows the significant effectiveness of the proposed control strategy.
Figure 6.6 The comparison between
pw (the continuous blue line) and rw (the green
dash line) in wave diagram
In Fig. 6.7, the control input required for the vibration control of the selected point
on the retracting Euler-Bernoulli beam is given. As can be found in Fig. 6.7, the value of
the control input required goes to a high value at the beginning of the control application,
and then it quickly decreases once the actual vibration of the selected point is
synchronized to the desired reference signal.
128
Figure 6.7 The control input U
6.6 Conclusions
The active control strategy developed in Chapter 2 has been applied to control the
large-amplitude vibration of a retracting Euler-Bernoulli beam without external excitation.
The active control strategy shows effectiveness in controlling the vibration of the
retracting Euler-Bernoulli beam governed by a nonlinear multi-dimensional system and
therefore is suitable for controlling multi-dimensional dynamic systems of retracting
beam. The application of such active control strategy is not seen in the current literature
concerning the vibration of an axially retracting beam with a decreasing length. In
concluding the findings of the research in this chapter, the following needs to be
emphasized.
Firstly, the model of the retracting beam established in the research is nonlinear in
comparing with the existing models which are mainly linear.
Secondly, in controlling the retracting Euler-Bernoulli beam, a multi-dimensional
control strategy shows great advantages as it better represents the dynamics of the
129
retracting beam and better controls the vibration especially the large vibrations of the
beam. With the results of the research in this chapter, the first two modes of vibration
make contributions to the actual vibration of the beam, and therefore the development of
a multi-dimensional dynamic system based on the vibration modes in the Galerkin
discretization is evidently necessary for controlling the retracting beam.
Lastly, with the active control strategy developed, the small difference between the
controlled vibration of the retracting beam and that of the reference signal demonstrates
the high efficiency in the synchronization and low consumption of the control energy.
The research results in this chapter show the significant effectiveness in controlling
the axially retracting Euler-Bernoulli beam without external excitation and may provide
potential guidance for controlling the robotic arms in industrial applications.
130
CHAPTER 7 AXIALLY TRANSLATING CABLE
WITHOUT EXTERNAL EXCITATION
7.1 Introduction
In this chapter, the control strategy proposed in Chapter 2 is to be applied to control
the nonlinear vibrations of an axially translating cable with fixed-fixed boundary
conditions. A cable system model consisting of the equations of motion is to be
established based on the von Karman-type equations. In developing for the solutions of
the cable system and for the sake of applying the active control strategy in the nonlinear
vibration control, the governing equations in the forms of partial differential equations
will be non-dimensionalized and then transformed into three ordinary differential
equations via a 3rd
-order Galerkin method. Corresponding to the derived multi-
dimensional dynamic system, the proposed active control strategy is applied. The
applicability of the control strategy developed will be demonstrated in some numerical
simulations based on the model established. The chaotic vibrations of the cable system
are considered for applying the control strategy. The suppression and stabilization of the
chaotic vibrations are to be demonstrated graphically to show the application and
efficiency of the active control strategy in controlling the nonlinear axially translating
cable system of multiple-dimensions.
7.2 Equations of Motion
The axially translating cable considered in this chapter is sketched in Fig. 7.l. The
equations of motion of the cable are to be derived based on Hamilton’s principle and von
131
Karman-type equations. As can be seen from Fig. 7.1, the axially translating cable with
fixed-fixed boundaries is allowed to move axially at a constant rate 0v , and the length of
the cable is given as l . The displacement of any point of the axially translating cable
along the x- and z- axes are designated as u and w .
Figure 7.1 The sketch of the axially translating cable with fixed-fixed ends
Starting from the origin at the left fixed end of the axially translating cable, a
position vector, r , of any point tx of the translating cable without deformation is given
as,
kir 0)( tx , (7.1)
where i and k are the unit vectors of the fixed Cartesian coordinate shown in the figure.
Thus, the displacement field of the axially translating cable can be derived as,
kiR ttxwttxutx ,, 00 , (7.2)
where ttxu ,0 and ttxw ,0 are the displacement components along the x- and z-
directions respectively, of a point of the axially translating cable.
132
Taking the total differentiation of R with respect to time t, one may obtain,
ki
R
dt
ttxdw
dt
ttxdu
dt
tdx
dt
d ,, 00
, (7.3)
where the derivative of tx with respect to time is equal to the translating rate of the
cable, and the full derivative of ttxw ,0 is,
tx
ttxwv
t
ttxw
dt
ttxdw
,,, 00
00 , (7.4-a)
tx
ttxwv
tx
ttxwv
t
ttxw
dt
ttxwd2
0
22
00
02
0
2
2
0
2 ,,2
,,
. (7.4-b)
Hence, the kinetic energy of the translating cable over a volume V of the cable is
expressible as,
l
dxdt
d
dt
dT
0 2
1 RR , (7.5)
where ρ denotes the mass per unit length of the axially moving cable.
The von Karman-type equations of strains of large deflection associated with the
displacement field, normal to the cross section of the cable along the x direction, in Eq.
(7.2) can thus be given by,
2
0011
,
2
1,
tx
ttxw
tx
ttxu . (7.6)
Therefore, the total strain energy of the cable can be given by,
133
l
dxQU0
1111112
1 , (7.7)
where 11Q represents the elastic coefficient in the same direction with 11 .
The virtual work done by the force is 0 since there is external excitation applied on
the axially translating cable,
0W , (7.8)
In the following analysis, the Hamilton’s principle will be employed to obtain the
nonlinear equations of motion for the axially translating cable. The mathematical
statement of the Hamilton’s principle is given by,
02
1
dtWLt
t , (7.9)
where the total Lagrangian function L is given by,
L T E . (7.10)
For the sake of clarity, hereafter, use x , 0u , 0w to replace tx , ttxu ,0 , and
ttxw ,0 respectively. Substitute Eqs. (7.5), (7.7), and (7.8) into Eq. (7.9), and the first
term in Eq. (7.9) can be developed as below,
2
1
2
1
t
t
t
tdtUTLdt
2
1 0111111
t
t
l
dtQdt
d
dt
d
RR
134
2
1
2
1 0111111
0
t
t
lt
t
l
dtQdtdt
d
dt
d
RR
2
1
2
11111112
2
0t
t VV
t
tdVdtQdVdt
dt
d R
R
2
1 0 2
0
2
2
2
0
t
t
l
dxdtdt
ud
dt
xdux
2
1 0 2
0
2
0
t
t
l
dxdtdt
wdw
2
1 011
2
0011
2
1t
t
l
dxdtx
w
x
uQ
321 LLL ,
where,
2
1 0 2
0
2
2
2
01
t
t
l
dxdtdt
ud
dt
xduxL , (7.11-a)
2
1 0 2
0
2
02
t
t
l
dxdtdt
wdwL , (7.11-b)
2
1 011
2
00113
2
1t
t
l
dxdtx
w
x
uQL . (7.11-c)
Since the cable is moving axially at a constant velocity ( 02
0
2
dt
vd), from Eq. (7.11-a),
it can be derived,
135
2
1 0 2
0
2
2
2
01
t
t
l
dxdtdt
ud
dt
xduxL
dtduQuQt
t
ll
2
1 011011001111
2
1 002
0
2t
t
l
dxdtudt
ud . (7.12-a)
From Eq. (7.11-c), it can be derived in as follows
2
1 011
2
00113
2
1t
t
l
dxdtx
w
x
uQL
2
1 011
2
0011
2
1t
t
l
dxdtx
w
x
uQ
2
1 0
01111
t
t
l
dxdtx
uQ
2
1 0
001111
t
t
l
dxdtx
w
x
wQ
dtduQuQt
t
ll
2
1 011011001111
dtx
wdwQw
x
wQ
t
t
ll
2
1 0
001111
0
00
1111
dtduQt
t
l
2
1 0110110
dtx
wdwQ
t
t
l
2
1 0
0110110
136
dtxdux
Qt
t
l
2
1 00
1111
dtdxw
x
wQ
t
t
l
2
1 002
0
2
1111
dtdxwx
w
xQ
t
t
l
2
1 00
01111
dtxdux
w
x
u
xQ
t
t
l
2
1 00
2
0011
2
1
dtdxwx
w
x
w
x
uQ
t
t
l
2
1 002
0
22
0011
2
1
dtdxwx
w
x
w
x
u
xQ
t
t
l
2
1 00
0
2
0011
2
1
dtxdux
w
x
w
x
uQ
t
t
l
2
1 002
0
2
0
2
0
2
11
dtdxwx
w
x
w
x
w
x
uQ
t
t
l
2
1 002
0
22
0
2
0
2
011
2
1
dtdxwx
w
x
w
x
w
x
uQ
t
t
l
2
1 00
2
0
2
0
2
0
2
0
2
11 . (7.12-b)
From Eqs. (7.12-a), (7.12-b), (7.11-b), the nonlinear governing equation of an axially
translating cable can be derived in the following,
2
0
2
dt
ud 0
2
0
2
0
2
0
2
11
x
w
x
w
x
uQ , (7.13-a)
137
2
0
2
dt
wd
x
w
x
uQ
0
2
0
2
11 2
0
2
011
x
w
x
uQ
0
2
32
0
22
011
x
w
x
wQ , (7.13-b)
Associated with the nonlinear dynamic equations and boundary conditions (Abou-
Rayan et al., 1993; Younis and Nayfeh, 2003), the strain can be obtained as,
l
dxx
w
lx
w
x
u
0
2
0
2
00
2
1
2
1. (7.14)
Then, substitute Eq. (7.14) into Eq. (7.13-b), and the nonlinear differential governing
equation of the axially translating cable in z direction is derived as,
02
12
0
2
0
2
0112
0
2
x
wdx
x
wQ
ldt
wd l
. (7.15)
To validate the governing equation Eq. (7.15) and facilitate the numerical
simulations in the consequent sections of this chapter, the following non-dimensional
variables are introduced,
ttQ
t
11 , l
xx ,
l
ww 0
0 , dt
dw
ltd
wd 00 1
,
2
0
2
22
0
21
dt
wd
ltd
wd
, (7.16)
With the non-dimensional variables shown in Eq. (7.16) introduced into Eq. (7.15),
the non-dimensional governing equation of the axially translating cable with fixed-fixed
ends can be expressed as,
2
0
2
td
wd 0
2
0
21
0
2
0
x
wxd
x
w, (7.17)
138
where,
211
11
2
1
lQ
l .
7.3 Series Solutions
Based on the Galerkin method of discretization, the transverse displacement 0w is
expanded in a series form, in terms of a set of comparison functions as,
1
0
n
nn twxφw . (7.18)
Corresponding to the fixed-fixed ends of the axially translating cable, xφn can be
given as follows,
xxxxx nn
nn
nnnnn
sinsh
sinsh
coschcosch
. (7.19)
Substitute the series solution of Eq. (7.19) into Eq. (7.17), and to assist presentation,
replace n , nw , nw , nw , v , and t for )(xφn , twn , dt
dwn , 2
2
dt
wd n , 0v and t respectively. With
the application of the Galerkin method at 3n , and,
1,11 ww , 1,22 ww , 1,33 ww ,
2,11,1 ww , 2,21,2 ww , 2,31,3 ww .
The discretized governing equations of the axially translating cable with the fixed-
fixed supports can be obtained as the following.
139
332,3
2,31,3
222,2
2,21,2
112,1
2,11,1
2
2
2
w
ww
w
ww
w
ww
(7.20)
where,
1,1
22
1,212
1
3
8wvvw ,
1,2
22
1,11,32 23
8
5
24wvvwvw ,
1,3
22
1,332
9
5
24wvvw
,
2
1,21,1
42
1,31,1
43
1,1
4
14
9
4
1wwwww ,
3
1,2
42
1,31,2
42
1,11,2
4
2 49 wwwww ,
2
1,21,3
43
1,3
42
1,11,3
4
3 94
81
4
9wwwww .
7.4 Control Design
Corresponding to the axially translating cable governed by Eq. (7.17), and the active
control strategy developed in Eqs. (2.15) ~ (2.20), the governing equation with the
control input for the cable can be given by the following expression,
140
wwFUx
wdx
x
ww ,
2
0
21
0
2
00
, (7.21)
With the application of the 3rd
-order Galerkin discretization, Eq. (7.22) may have the
following form:
tfuw
ww
tfuw
ww
tfuw
ww
,2
,2
,2
33332,3
2,31,3
22222,2
2,21,2
11112,1
2,11,1
W
W
W
, (7.22)
where 1u , 2u , and 3u , are derived as follows through the 3rd
-order Galerkin method,
Uu 8309.01 , Uu 02 , Uu 3638.03 .
It should be noticed that due to the fixed-fixed boundaries of the axially-translating
cable, the application of the Galerkin method based on Eq. (7.19) makes the coefficients
of 2u equal to zero. In the next section, it will be demonstrated in the numerical
simulation that the actual response of the axially-translating cable at a selected point can
be well synchronized to a desired reference signal in the case that the coefficients of 2u
equals to zero.
7.5 Numerical Simulation
To demonstrate the applicability and effectiveness of the control strategy developed
in Chapter 2, numerical simulations are conducted for controlling the nonlinear vibration
141
of the axially-translating cable governed with Eq. (7.17). The nonlinear vibrations of the
cable are emphasized in this research. With the numerical simulations performed, a
chaotic motion is found when the cable is translating at certain rates. The proposed active
control strategy is found not only effectively reduces the amplitude of the chaotic motion,
but also stabilizes the motion so that the response of the translating cable is controlled to
a desired periodic motion. To facilitate the numerical simulation, the 4th
-order P-T
method (Dai, 2008), is implemented.
The parameters used for the simulations of the responses of the axially-translating
cable are given as follows,
NQ 4
11 109.2 , ml 5.0 , mkgρ 00.1 ,
and the constant axially-translating rate is given as below,
smv 75.30 .
The non-dimensionalized initial conditions, corresponding to the displacements
described by Eq. (7.20) after the implementation of the 3rd
-order Galerkin method, are
taken as,
001.001,1 w , 005.002,1 w ,
0001.001,2 w , 0025.002,2 w ,
00005.001,3 w , 00125.002,3 w .
142
If the vibration of a point at m375.0 along the x-axis of the cable is selected, based
on Eqs. (7.18) and (7.20) the non-dimensional response of the selected point pw can be
derived as,
1,31,21,1
3
1
1, 3710.14449.18632.0 wwwwφwn
nnp
. (7.23)
7.5.1 Chaotic Vibration
The response of the cable translating at the speed smv 75.30 is shown in Fig. 7.2,
corresponding to the non-dimensional time from 0t to 20000t . During this period
of time, one may notice: in Fig. 7.2, it is a chaotic vibration discovered; and the
maximum amplitude of the vibration of the cable can exceed 0.03. In considering that the
displacement shown in the figure is non-dimensional, the amplitude is very large. Thus,
the reduction and stabilization of the chaotic vibration may improve the operation of the
cable.
Figure 7.2 The wave diagram of pw without the application of the active control strategy
Besides, from Fig. 7.3 (a), Fig. 7.3 (b) and Fig. 7.3 (c), it can be learned: although
the contribution of the first vibration mode in Fig. 7.3 (a) is larger than those of the other
143
two vibration modes as shown in Fig. 7.3 (b) and Fig. 7.3 (c), the contributions of the
other two vibration modes are obviously not negligible. Actually it can be found that the
other two vibration modes also significantly contribute to the actual response of the
selected point. Thus, the development of a multi-dimensional dynamic system is
necessary for the accurate prediction of the nonlinear dynamics of the axially translating
cable.
144
(a)
(b)
(c)
Figure 7.3 The wave diagrams of the first three vibration modes without the application
of the active control strategy: (a) 1w ; (b) 2w ; (c) 3w
145
7.5.2 Amplitude Synchronization
In this section, the active control strategy proposed in Chapter 2 will be applied to
show its effectiveness and efficiency in the amplitude synchronization. Three different
desired amplitudes will be specified as the amplitude of the reference signal.
7.5.2.1 0175.0rA
In the application of active the control strategy, the desired amplitude of the
reference is set as,
twr 0646.0sin0175.0 .
and the other control parameters and the unknown external disturbance take the following
values,
01.0 , 01.0fsk , )sin(001.0, pwwwF .
As shown in Fig. 7.4, the proposed active control strategy is applied at 4000t .
After the application of the control strategy, the vibration of the cable at the translating
speed smv 75.30 is shown in Fig. 7.4 and Fig. 7.5.
146
Figure 7.4 The wave diagram of pw with the application of the active control strategy in
the case of 0175.0rA
In Fig. 7.4 the vibration of the selected point, pw , is shown, for the period of time
from 4000t to 6500t . It can be seen from Fig. 7.4, a time period from the non-
dimensional time to about 4800t is needed for stabilizing the cable after the
application of the active control strategy. After the short period, the chaotic vibration will
then become a periodic one, of which the amplitude is 0.0175.
4000t
147
(a)
(b)
(c)
Figure 7.5 The wave diagrams of the first three vibration modes with the application of
the active control strategy in the case of 0175.0rA : (a) 1w ; (b) 2w ; (c) 3w
From Fig. 7.5 (a-c), the vibration of the selected point is shown in terms of 1w , 2w ,
and 3w . Based on these figures, through the application of the active control strategy,
148
each of the vibration modes of the axially-translating cable is gradually stabilized from a
chaotic vibration into a periodic one.
Figure 7.6 The comparison between
pw (the continuous blue line) and rw (the green
dash line) in wave diagram in the case of 0175.0rA
Fig. 7.6 shows the comparison between the actual vibration of the cable pw and the
reference signal rw . One may notice that the reference signal rw is well periodic with
respect to time t. One may also see from the figure, the maximum amplitude of the
cable’s vibration slightly varies after the stabilization of the cable with the application of
the active control strategy. However, as shown in the figure, the maximum amplitude of
the cable is very close to that of the reference signal on the whole.
149
Figure 7.7 The control input U in the case of 0175.0rA
Fig. 7.7 shows the control input U . Initially, the control input reaches a peak very
quickly after the application of the active control strategy. Once the system is stabilized,
the control input displays a periodic wave diagram as shown in the figure and the
maximum value of the control input is significantly decreased to very small value.
7.5.2.2 015.0rA
In the application of the control strategy, the desired amplitude of the reference
signal is set as,
twr 0646.0sin015.0 .
and the other control parameters and the unknown external disturbance take the following
values,
003.0 , 01.0fsk , )sin(001.0, pwwwF .
As shown in Fig. 7.8, the proposed control strategy is applied at 4000t . After the
application of the control strategy, the vibration of the cable at the translating speed
smv 75.30 is shown in Fig. 7.8 and Fig. 7.9.
150
Figure 7.8 The wave diagram of pw with the application of the active control strategy in
the case of 015.0rA
In Fig. 7.8 the vibration of the selected point, pw , is shown, for the period of time
from 4000t to 6500t . It can be seen from Fig. 7.8, a short period of time is needed
for stabilizing the cable after the application of the control strategy. After the short period,
the chaotic vibration will then become a periodic one, of which the amplitude is 0.015.
151
(a)
(b)
(c)
Figure 7.9 The wave diagrams of the first three vibration modes with the application of
the active control strategy in the case of 015.0rA : (a) 1w ; (b) 2w ; (c) 3w
152
From Fig. 7.9 (a-c), the vibration of the selected point is shown in terms of 1w , 2w ,
and 3w . Based on these figures, through the application of the control strategy, each of
the vibration modes of the axially-translating cable is gradually stabilized from a chaotic
vibration into a periodic one.
Figure 7.10 The comparison between
pw (the continuous blue line) and rw (the green
dash line) in wave diagram in the case of 015.0rA
Fig. 7.10 shows the comparison between the actual vibration of the cable pw and the
reference signal rw applied. One may notice that the reference signal rw is well periodic
with respect to time t. One may also see from the figure, the maximum amplitude of the
cable’s vibration slightly varies after the stabilization of the cable with the application of
the control strategy. As shown in the figure, the maximum amplitude of the cable is very
close to that of the reference signal.
153
Figure 7.11 The control input U in the case of 015.0rA
Fig. 7.11 shows the control input U . Initially, the control input reaches a peak very
quickly after the application of the control strategy. Once the system is stabilized, the
control input displays a periodic wave diagram as shown in the figure and the maximum
value of the control input is significantly decreased to a very small value.
7.5.2.3 010.0rA
In the application of the control strategy, the desired amplitude of the reference
signal is set as,
twr 0646.0sin010.0 .
and the other control parameters and the unknown external disturbance take the following
values:
001.0 , 01.0fsk , )sin(001.0, pwwwF .
As shown in Fig. 7.12, the proposed active control strategy is applied at 4000t .
After the application of the control strategy, the vibration of the cable at the translating
speed smv 75.30 is shown in the figures in Fig. 7.12 and Fig. 7.13.
154
Figure 7.12 The wave diagram of pw with the application of the active control strategy in
the case of 010.0rA
In Fig. 7.12 the vibration of the selected point, pw , is shown, for the period of time
from 4000t to 6500t . It can be seen from Fig. 7.12, a short period of time is needed
for stabilizing the cable after the application of the control strategy. After the short period,
the chaotic vibration will then become a periodic one, of which the amplitude is 0.010.
155
(a)
(b)
(c)
Figure 7.13 The wave diagrams of the first three vibration modes with the application of
the active control strategy in the case of 010.0rA : (a) 1w ; (b) 2w ; (c) 3w
From Fig. 7.13 (a-c), the displacement of the selected point is shown in terms of 1w ,
2w , and 3w . Based on these figures, through the application of the active control strategy,
156
each of the vibration modes of the axially-translating cable is gradually stabilized from a
chaotic vibration into a periodic one.
Figure 7.14 The comparison between
pw (the continuous blue line) and rw (the green
dash line) in wave diagram in the case of 010.0rA
Fig. 7.14 shows the comparison between the actual vibration of the cable pw and the
reference signal rw applied. One may notice that the reference signal rw is well periodic
with respect to time t. One may also see from the figure, the maximum amplitude of the
cable’s vibration slightly varies after the stabilization of the cable with the application of
the active control strategy. As shown in the figure, the maximum amplitude of the cable
is very close to that of the reference signal.
157
Figure 7.15 The control input U in the case of 010.0rA
Fig. 7.15 shows the control input U . Initially, the control input reaches a peak very
quickly after the application of the active control strategy. Once the system is stabilized,
the control input displays a periodic wave diagram as shown in the figure and the
maximum value of the control input is significantly decreased to a very small value.
7.5.3 Frequency Synchronization
In this section, the active control strategy proposed in Chapter 2 will be applied to
show its effectiveness and efficiency in the frequency synchronization. Three different
desired frequencies will be specified as the frequency of the reference signal.
7.5.3.1 0553.0r
In the application of the control strategy, the desired frequency of the reference is set
as,
twr 0553.0sin018.0 .
and the other control parameters and the unknown external disturbance take the following
values,
158
01.0 , 01.0fsk , )sin(001.0, pwwwF .
As shown in Fig. 7.16, the proposed active control strategy is applied at 4000t .
After the application of the active control strategy, the vibration of the cable at the
translating speed smv 75.30 is shown in Fig. 7.16 and Fig. 7.17.
Figure 7.16 The wave diagram of pw with the application of the active control strategy in
the case of 0553.0r
In Fig. 7.16 the vibration of the selected point, pw , is shown, for the period of time
from 4000t to 6500t . It can be seen from Fig. 6.16, a short period of time is needed
for stabilizing the cable after the application of the active control strategy. After the short
period, the chaotic motion will then become a periodic one, of which the angular
frequency is 0.0553.
159
(a)
(b)
(c)
Figure 7.17 The wave diagrams of the first three vibration modes with the application of
the active control strategy in the case of 0553.0r : (a) 1w ; (b) 2w ; (c) 3w
From Fig. 7.17 (a-c), the vibration of the selected point is shown in terms of 1w , 2w ,
and 3w . Based on these figures, through the application of the control strategy, each of
160
the vibration modes of the axially-translating cable is gradually stabilized from a chaotic
motion into a periodic one.
Figure 7.18 The comparison between
pw (the continuous blue line) and rw (the green
dash line) in wave diagram in the case of 0553.0r
Fig. 7.18 shows the comparison between the actual vibration of the cable pw and the
reference signal rw applied. One may notice that the reference signal rw is well periodic
with respect to time t. One may also see from the figure, the maximum amplitude of the
cable’s response slightly varies after the stabilization of the cable with the application of
the control strategy. As shown in the figure, the frequency of the cable is very close to
that of the reference signal.
161
Figure 7.19 The control input U in the case of 0553.0r
Fig. 7.19 shows the control input U . Initially, the control input reaches a peak very
quickly after the application of the active control strategy. Once the system is stabilized,
the control input displays a periodic wave diagram as shown in the figure and the
maximum value of the control input is significantly decreased to a very small value.
7.5.3.2 1107.0r
In the application of the active control strategy, the desired frequency of the
reference is set as,
twr 1107.0sin018.0 .
and the other control parameters and the unknown external disturbance take the following
values,
009.0 , 01.0fsk , )sin(001.0, pwwwF .
As shown in Fig. 7.20, the proposed active control strategy is applied at 4000t .
After the application of the active control strategy, the vibration of the cable at the
translating speed smv 75.30 is shown in in Fig. 7.20 and Fig. 7.21.
162
Figure 7.20 The wave diagram of pw with the application of the active control strategy in
the case of 1107.0r
In Fig. 7.20 the vibration of the selected point, pw , is shown, for the period of time
from 4000t to 6500t . It can be seen from Fig. 7.20, a short period of time is needed
for stabilizing the cable after the application of the active control strategy. After the short
period, the chaotic motion will then become a periodic one, of which the angular
frequency is 0.1107.
163
(a)
(b)
(c)
Figure 7.21 The wave diagrams of the first three vibration modes with the application of
the active control strategy in the case of 1107.0r : (a) 1w ; (b) 2w ; (c) 3w
From Fig. 7.21 (a-c), the vibrations of the selected point are shown in terms of 1w ,
2w , and 3w . Based on these figures, through the application of the active control strategy,
164
each of the vibration modes of the axially-translating cable is gradually stabilized from a
chaotic vibration into a periodic one.
Figure 7.22 The comparison between
pw (the continuous blue line) and rw (the green
dash line) in wave diagram in the case of 1107.0r
Fig. 7.22 shows the comparison between the actual vibration of the cable pw and the
reference signal rw applied. One may notice that the reference signal rw is well periodic
with respect to time t. One may also see from the figure, the maximum amplitude of the
cable’s vibration slightly varies after the stabilization of the cable with the application of
the active control strategy. As shown in the figure, the angular frequency of the cable is
very close to that of the reference signal.
165
Figure 7.23 The control input U in the case of 1107.0r
Fig. 7.23 shows the control input U . Initially, the control input reaches a peak very
quickly after the application of the active control strategy. Once the system is stabilized,
the control input displays a periodic wave diagram as shown in the figure and the
maximum value of the control input is significantly decreased to a very small value.
7.5.3.3 1660.0r
In the application of the control strategy, the desired frequency of the reference is set
as
twr 1660.0sin018.0 .
and the other control parameters and the unknown external disturbance take the following
values:
07.0 , 001.0fsk , )sin(1660.0, pwwwF .
As shown in Fig. 7.24, the proposed active control strategy is applied at 4000t .
After the application of the control strategy, the vibration of the cable at the translating
speed smv 75.30 is shown in Fig. 7.24 and Fig. 7.25.
166
Figure 7.24 The wave diagram of pw with the application of the active control strategy in
the case of 1660.0r
In Fig. 7.24 the vibration of the selected point, pw , is shown, for the period of time
from 4000t to 6500t . It can be seen from Fig. 6.24, a short period of time is needed
for stabilizing the cable after the application of the control strategy. After the short period,
the chaotic vibration will then become a periodic one, of which the angular frequency is
1660.0 .
167
(a)
(b)
(c)
Figure 7.25 The wave diagrams of the first three vibration modes with the application of
the active control strategy in the case of 1660.0r : (a) 1w ; (b) 2w ; (c) 3w
168
From Fig. 7.25 (a-c), the vibration of the selected point is shown in terms of 1w , 2w ,
and 3w . Based on these figures, through the application of the active control strategy,
each of the vibration modes of the axially translating cable is gradually stabilized from a
chaotic vibration into a periodic one.
Figure 7.26 The comparison between
pw (the continuous blue line) and rw (the green
dash line) in wave diagram in the case of 1660.0r
Fig. 7.26 shows the comparison between the actual vibration of the cable pw and the
reference signal rw applied. One may notice that the reference signal rw is well periodic
with respect to time t. One may also see from the figure, the maximum amplitude of the
cable’s vibration slightly varies after the stabilization of the cable with the application of
the active control strategy. As shown in the figure, the frequency of the cable is very
close to that of the reference signal.
169
Figure 7.27 The control input U in the case of 1660.0r
Fig. 7.27 shows the control input U . Initially, the control input reaches a peak very
quickly after the application of the active control strategy. Once the system is stabilized,
the control input displays a periodic wave diagram as shown in the figure and the
maximum value of the control input is significantly decreased to a very small value.
7.6 Conclusions
The active control strategy developed in Chapter 2 is applied to control the chaotic
vibration of an axially moving cable. The active control strategy shows its effectiveness
in controlling the vibration of the cable governed by a nonlinear multi-dimensional
system and is suitable for controlling multi-dimensional dynamic systems of nonlinear
elastic cables.
170
CHAPTER 8 AXIALLY EXTENDING CABLE
WITHOUT EXTERNAL EXCITATION
8.1 Introduction
In this chapter, the active vibration control strategy proposed in Chapter 2 is to be
applied for controlling the large-amplitude vibration, which is discovered from a multi-
dimensional dynamic system of an extending nonlinear elastic cable without external
excitation. The equations of motion of the cable with fixed-fixed boundary are to be
established based on von Karman-type equations and the consideration of the cable’s
geometric nonlinearity. In the development of the solutions of the extending cable, the
equations in the forms of partial differential equations are non-dimensionalized and then
converted into a multi-dimensional system through the 3rd
-order Galerkin method. With
respect to the derived multi-dimensional dynamic system, the active control strategy
previously proposed is to be applied and the applicability and efficiency of this control
strategy will be demonstrated in controlling the nonlinear vibrations of the elastic cable.
A case of large-amplitude vibration of the extending nonlinear elastic cable is presented
to show the effectiveness of the proposed active control strategy in controlling such
vibration of the cable.
8.2 Equations of Motion
The extending nonlinear elastic cable without external excitation investigated in this
chapter is sketched in Fig. 8.l. The equations of motion of the cable are to be derived
based on the Hamilton’s principle. As can be seen from Fig. 8.1, the cable is placed
171
between two fixed-fixed ends, and the initial length of the beam is given as 0l , the area of
the cross section of the cable is A . The x axis is along the axial direction of the cable.
The displacements of a point of the elastic cable along the x- and z- axes are designated
with u and w .
Figure 8.1 The sketch of the extending nonlinear elastic cable
Starting from the origin at the upper support of the cable, a position vector, r , of
any point ztx , of the cable without deformation is given as,
kir 0 tx ,
where i and k are the unit vectors of the fixed Cartesian coordinate shown in the Fig. 8.1.
Thus, the displacement field of the cable can be derived as
kiR ttxwttxutx ,, 00 ,
where ttxu ,0 and ttxw ,0 are the displacement components along the x- and z-
directions respectively, of a point on the cable.
Taking the total differentiation of R with respect to the time t, it can be obtained,
172
kiR
dt
ttxdw
dt
ttxdu
dt
tdx
dt
d ,, 00
.
Therefore, the kinetic energy of the cable is expressed as,
l
dxdt
d
dt
dT
0 2
1 RR , (8.1)
where ρ denotes the mass of the cable per unit length, and l , the instant length of the
cable, is given as,
vtll 0 ,
and v the extending velocity of the cable is constant.
The von Karman-type equations of strains of large deflection associated with the
displacement field, normal to the cross section of the cable along the x direction, can be
given by,
2
0011
,
2
1,
x
ttxw
x
ttxu ,
Therefore, the total strain energy of the cable can be given by,
l
dxQU0
1111112
1 , (8.2)
where EAQ 11 , and E represents the elastic coefficient in the same direction with 11 .
The virtual work due to the weight of the cable is zero considering the constant
extending velocity and no external excitation applied (Tang et al., 2011),
173
0W . (8.3)
The Hamilton’s principle is employed to obtain the nonlinear equations of motion
for the extending elastic cable. The mathematical statement of the Hamilton’s principle is
given by,
02
1
2
1
dtWdtLt
t
t
t , (8.4)
where the total Lagrangian function L is given by,
UTL . (8.5)
For convenience, replace tx , ttxu ,0 , and ttxw ,0 with x , 0u , and 0w in the
following. Substitute Eqs. (8.1), (8.2), (8.3) and (8.5) into Eq. (8.4), and then the first
term in Eq. (8.4) can be developed as,
2
1
2
1
t
t
t
tdtUTLdt
2
1 0111111
t
t
l
dtQdt
d
dt
d
RR
2
1
2
1 0111111
0
t
t
lt
t
l
dtQdtdt
d
dt
d
RR
2
1
2
11111112
2
0t
t VV
t
tdVdtQdVdt
dt
d R
R
2
1 0 2
0
2
2
2
0
t
t
l
dxdtdt
ud
dt
xdux
174
2
1 0 2
0
2
0
t
t
l
dxdtdt
wdw
2
1 011
2
0011
2
1t
t
l
dxdtx
w
x
uQ
321 LLL
where,
2
1 0 2
0
2
2
2
01
t
t
l
dxdtdt
ud
dt
xduxL , (8.6-a)
2
1 0 2
0
2
02
t
t
l
dxdtdt
wdwL , (8.6-b)
2
1 011
2
00113
2
1t
t
l
dxdtx
w
x
uQL . (8.6-c)
Since the cable is moving axially at a constant velocity ( 02
2
dt
xd), from Eq. (8.6-a),
it can be derived,
2
1 0 2
0
2
2
2
01
t
t
l
dxdtdt
ud
dt
xduxL
2
1 0 2
0
2
0 00t
t
l
dxdtdt
udu
2
1 002
0
2t
t
l
dxdtudt
ud . (8.7-a)
175
From Eq. (8.6-c), it can be derived as follows,
2
1 011
2
00113
2
1t
t
l
dxdtx
w
x
uQL
2
1 011
2
0011
2
1t
t
l
dxdtx
w
x
uQ
2
1 0
01111
t
t
l
dxdtx
uQ
2
1 0
001111
t
t
l
dxdtx
w
x
wQ
dtduQuQt
t
ll
2
1 011011001111
dtx
wdwQw
x
wQ
t
t
ll
2
1 0
001111
0
00
1111
dtduQt
t
l
2
1 0110110
dtx
wdwQ
t
t
l
2
1 0
0110110
dtxdux
Qt
t
l
2
1 00
1111
dtdxw
x
wQ
t
t
l
2
1 002
0
2
1111
dtdxwx
w
xQ
t
t
l
2
1 00
01111
dtxdux
w
x
u
xQ
t
t
l
2
1 00
2
0011
2
1
176
dtdxwx
w
x
w
x
uQ
t
t
l
2
1 002
0
22
0011
2
1
dtdxwx
w
x
w
x
u
xQ
t
t
l
2
1 00
0
2
0011
2
1
dtxdux
w
x
w
x
uQ
t
t
l
2
1 002
0
2
0
2
0
2
11
dtdxwx
w
x
w
x
w
x
uQ
t
t
l
2
1 002
0
22
0
2
0
2
011
2
1
dtdxwx
w
x
w
x
w
x
uQ
t
t
l
2
1 00
2
0
2
0
2
0
2
0
2
11 , (8.7-b)
From Eqs. (8.7-a), (8.7-b), (8.6-b), the nonlinear governing equation of an extending
elastic cable can be derived in the following,
2
0
2
dt
ud 0
2
0
2
0
2
0
2
11
x
w
x
w
x
uQ , (8.8-a)
2
0
2
dt
wd
x
w
x
uQ
0
2
0
2
11 2
0
2
011
x
w
x
uQ
0
2
32
0
22
011
x
w
x
wQ , (8.8-b)
Associated with the nonlinear dynamic equations and the boundary conditions
(Abou-Rayan et al., 1993; Younis and Nayfeh, 2003), the strain can be obtained as,
l
dxx
w
lx
w
x
u
0
2
0
2
00
2
1
2
1. (8.9)
177
Then, substitute Eq. (8.9) into Eq. (8.10), and the nonlinear differential governing
equation of the cable in z direction is derived as,
02
12
0
2
0
2
0112
0
2
x
wdx
x
wQ
ldt
wd l
. (8.10)
To validate the governing equation Eq. (8.10) and facilitate the numerical
simulations in the consequent sections, the following non-dimension variables are
introduced,
tQ
lt
11
0
1 t ,
l
xx ,
0
00
l
ww ,
0l
ll . (8.11)
Introduce the non-dimensional variables shown in Eq. (8.11) into Eq. (8.10), the
non-dimension governing equation of the investigated nonlinear extending elastic cable
can be expressed as,
02
0
21
0
2
0
2
0
2
x
wxd
x
w
td
wd . (8.12)
where,
2
0
3
011
11
ll
lQ
l
.
8.3 Series Solutions
Based on the Galerkin method of discretization, the transverse displacement 0w is
expanded in a series form, in terms of a set of comparison functions as,
178
1
0
n
nn twxφw . (8.13)
Corresponding to the fixed-fixed boundaries of the cable, xφn can be given as
follows,
xxxxx nn
nn
nnnnn
sinsh
sinsh
coschcosch
. (8.14)
Substitute the series solution of Eq. (8.14) into Eq. (8.13), and to assist the following
presentation, replace n , nw , nw , nw , t , and l for )(xφn , twn , td
wd n , 2
2
td
wd n , t , and l
respectively, and,
1,11 ww , 1,22 ww , 1,33 ww ,
2,11,1 ww 2,21,2 ww , 2,31,3 ww .
Therefore, with the application of the Galerkin method at 3n , the discretized
governing equations of the nonlinear extending elastic cable with the fixed-fixed
boundary can be obtained in the following,
332,3
2,31,3
222,2
2,21,2
112,1
2,11,1
w
ww
w
ww
w
ww
, (8.15)
where,
179
2
2
1,32,32
2
1,22,22
2
1,12,1
1
58.153.134.334.305.400.1
l
w
l
w
l
w
l
w
l
w
l
w ,
2
2
1,32,32
2
1,22,22
2
1,12,1
2
15.1051.523.1400.1
34.334.3
l
w
l
w
l
w
l
w
l
w
l
w ,
2
2
1,32,32
2
1,22,22
2
1,12,1
3
80.3000.115.1051.5
58.153.1
l
w
l
w
l
w
l
w
l
w
l
w ,
1,3
2
1,2
1,31,21,1
2
1,31,2
2
1,31,11,3
2
1,1
2
1,21,11,2
2
1,1
3
1,3
3
1,2
3
1,1
1
00.448
40.039.012.140606.359
52.56604.026.96205.035.151
ww
wwwwwwwww
wwwwwww
,
1,3
2
1,2
1,31,21,1
2
1,31,2
2
1,31,11,3
2
1,1
2
1,21,11,2
2
1,1
3
1,3
3
1,2
3
1,1
2
04.2
00.89663.455439.020.0
15.052.56646.150.212001.0
ww
wwwwwwwww
wwwwwww
,
1,3
2
1,2
1,31,21,1
2
1,31,2
2
1,31,11,3
2
1,1
2
1,21,11,2
2
1,1
3
1,3
3
1,2
3
1,1
2
63.4554
78.037.478.288612.1406
00.44820.088.978268.069.119
ww
wwwwwwwww
wwwwwww
,
where,
, given as the non-dimensional axially translating velocity of the elastic cable, is
expressible as,
dt
dl
l
x .
180
8.4 Control Design
Corresponding to the axially extending cable governed by Eq. (8.12), and the active
control strategy developed in Eqs. (2.15) ~ (2.20), the governing equation with the
control input for the cable can be given by the following expression,
wwFUx
wxd
x
ww ,
2
0
21
0
2
00
, (8.16)
With the application of the 3rd
-order Galerkin method, Eq. (8.16) may take the
following form,
tfuw
ww
tfuw
ww
tfuw
ww
,
,
,
3332,3
2,31,3
2222,2
2,21,2
11112,1
2,11,1
W
W
W
, (8.17)
where 1u , 2u , and 3u are derived as follows through the 3rd
-order Galerkin method,
Uu 83.01 , Uu 02 , Uu 36.03 .
In the next section, through the numerical simulation, it will be demonstrated that the
actual vibration of the cable at a selected point can be well synchronized to a desired
reference signal.
181
8.5 Numerical Simulation
The vibration of the extending nonlinear elastic cable is investigated numerically
utilizing the multi-dimensional system developed, with concentration on a randomly
selected point in the cable. With the numerical simulations performed, a large-amplitude
vibration of the selected point is discovered. Then, the proposed active control strategy is
found not only effective in reducing the amplitude of the large-amplitude vibration, but
also synchronizing the motion to the given frequency of the desired reference signal. To
facilitate the numerical simulation with higher accuracy and reliability, the 4th
-order P-T
method (Dai, 2008), is implemented.
The three parameters of the extending nonlinear elastic cable are given as those from
the work (Tang et al., 2011),
NQ 7
11 109.2 , ml 300 , mkgρ 00.1 ,
and the constant velocity the cable moves at is,
smv 75.3 .
The non-dimensionalized initial conditions, corresponding to the vibrations
described in Eqs. (8.15) after the implementation of the 3rd
-order Galerkin method, are
given as,
0001.001,1 w , 0005.002,1 w , 0002.001,2 w , 0001.002,2 w ,
001,3 w , 002,3 w .
182
The length of the cable increases from the initial length 30 meters to 180.41 meters
with time, and pw given as the transverse displacement at a selected point, which is 25
meters from the moving end of the cable, is expressed as below,
321
3
1
1, WWWwxwn
npnp
,
where 1W , 2W , and 3W respectively denote the contributions of the first three vibration
modes to the actual vibration of the selected point pw , and they are given as below,
1,111 wxW p , 1,222 wxW p , 1,333 wxW p .
A large-amplitude vibration of the cable occurs as shown in Fig. 8.2, while the
developed control strategy is not applied.
Figure 8.2 The wave diagram of pw without the application of the active control strategy
The vibration of the cable is shown in Figure 8.2, corresponding to the non-
dimensional time from 0t to 7200t . During this period of time, one may notice: in
Figure 8.2, the maximum amplitude of the vibration of the cable can exceed 0.02.
Considering that the displacement shown in Fig. 8.2 is non-dimensional, and the
dimensional amplitude is actually 0.02 time the initial length of the elastic cable, the
183
maximum amplitude observed from Fig. 8.2 is large. Although the period of the cable
seems to increase with respect to the non-dimensional time and the amplitude of the
selected point gradually decreases once it reaches a certain value around 0.02, the
discovery of the large-amplitude vibration of the selected point requires to be suppressed.
Besides, from Fig. 8.3 (a), Fig. 8.3 (b) and Fig. 8.3 (c), it can be learned that
although the contribution of the first vibration mode in Fig. 8.3 (a) is larger than those of
the other two vibration modes as shown in Fig. 8.3 (b) and Fig. 8.3 (c), the contributions
of the other two vibration modes are obviously not negligible, since their maximum
contributions are about both one third of that of the first vibration mode. Actually it can
be learned that the other two vibration modes also significantly contributes to the actual
vibration of the selected point. Thus, the development of a multi-dimensional dynamic
system is necessary for the accurate prediction of the dynamics of the extending nonlinear
elastic cable.
184
(a)
(b)
(c)
Figure 8.3 The wave diagrams of the first three vibration modes without the application
of the active control strategy: (a) 1w ; (b) 2w ; (c) 3w
As shown in Fig. 8.4, the proposed active control strategy is applied at 1800t , and
the control parameters and the unknown external disturbance take the following values,
185
twr 0350.0sin001.0 , 6.0 , 1fsk , )sin(0001.0, pwwwF .
As can be seen from Fig. 8.4, the maximum amplitude of the vibration of the cable is
reduced significantly from about 0.02 to 0.001. The synchronization of the vibration of
the cable also responds periodically.
Figure 8.4 The wave diagram of pw with the application of the active control strategy
From Fig. 8.5 (a), Fig. 8.5 (b) and Fig. 8.5 (c), following should be noticed: although
the contributions of the first three vibration modes of the cable are indeed affected with
the application of the proposed control strategy, only the contribution of the first
vibration mode are obviously affected; as a result the second and the third vibration
modes play more important roles in the case of the application of the control strategy.
186
(a)
(b)
(c)
Figure 8.5 The wave diagrams of the first three vibration modes with the application of
the active control strategy: (a) 1w ; (b) 2w ; (c) 3w
Fig. 8.6 is presented to fully demonstrate the effectiveness of the proposed active
control strategy. In Fig. 8.6, the difference is very small between the actual vibration of
187
the cable and that of the reference signal, and the synchronization between the actual
vibration of the cable at the selected point and the reference signal shows the significant
effectiveness of the proposed active control strategy
Figure 8.6 The comparison between
pw (the continuous blue line) and rw (the green
dash line) in wave diagram
In Fig. 8.7, the control input required for the vibration control of the selected point
on the elastic cable is given. As can be found in Fig. 8.7, the value of the control input
required goes to a high value at the beginning of the control application, but it quickly
decrease to a very small value after the actual vibration of the selected point is
synchronized to the desired reference signal.
Figure 8.7 The control input U
188
8.6 Conclusions
The active control strategy proposed in Chapter 2 is applied in this chapter to control
the large-amplitude vibration of an extending nonlinear elastic cable without external
excitation. The active control strategy shows effectiveness in controlling the motion of
the cable governed by a nonlinear multi-dimensional system and is suitable for
controlling multi-dimensional dynamic systems of nonlinear elastic cables. The
application of such active control strategy is not seen in the current literature concerning
the vibration of axially extending cable with an increasing length. In concluding the
findings of the research in this chapter, the following needs to be emphasized.
Firstly, the cable model established in the research is nonlinear in comparing with
the existing models that are mainly linear.
Secondly, in controlling the nonlinear elastic cables, a multi-dimensional model
shows its great advantages as it better represents the dynamics of the cable and better
controls the vibration especially the large-amplitude vibrations of the cable. With the
results in this chapter, all the three vibration modes of dynamic system make significant
contributions to the vibration of the cable, and the development of a multi-dimensional
dynamic system is evidently necessary for controlling the nonlinear elastic cables.
Thirdly, with the active control strategy developed, the small difference between the
controlled vibration of the cable and that of the reference signal demonstrates the high
efficiency of the synchronization and minute consumption of the control energy.
189
Lastly, the results presented show significant effectiveness in controlling the axially
translating structures with varying lengths and may provide guidance for controlling the
elevator cables in industrial applications.
190
CHAPTER 9 CONCLUSIONS AND FUTURE
WORKS
9.1 Conclusion
With the research of this PhD dissertation, the following can be concluded.
1. A control strategy for actively controlling the nonlinear vibrations of structures
of multiple dimensions is developed, and the control strategy shows
effectiveness in controlling the nonlinear vibrations of various typical
engineering structures.
2. Conditions and characteristics of applying the control strategy in controlling the
nonlinear vibrations of each of the seven typical engineering structures
considered are expressed in details. This provides a guidance for applying the
control strategy in the real world, to control the nonlinear vibrations of typical
engineering structures which are commonly seen in mechanical, civil, aeronautic
and aerospace engineering fields.
3. The vibrations of all the typical multi-dimensional engineering structures
subjected to periodic and non-periodic excitations are complex, showing periodic,
non-periodic, chaotic and the other nonlinear behaviors. The nonlinear vibrations
of the structures can all be controlled and stabilized to periodic motions by
utilizing the control strategy developed.
191
4. As proven in the research, the control strategy with multi-dimensional approach
is necessary for many engineering structures. The availability of a control
strategy with multi-dimensional approach is therefore significant. The multi-
dimensional approach provides more accurate and reliable results in comparison
with that of the conventional single dimensional approaches which may even
lead to incorrect results.
5. The conventional FSMC was originally designed for controlling chaotic
vibrations. The active nonlinear vibration control strategy developed, however,
can be used to control chaotic vitiation as well as the linear and nonlinear
vibrations with large-amplitudes in terms of stabilizing frequency and reducing
amplitudes.
6. The applications of the active control strategy in the engineering structures
featuring no bending moment can be synchronized to that with almost exact
frequency and amplitude as that of the reference signal. The corresponding
control input to maintain the vibration control may be decreased to a very small
value after the vibration synchronizations. This is significant in control
application.
7. In implementing the active control strategy in the vibration control of a MEMS
beam, the selection of the control parameters can be difficult to specify. A new
control method named two-phase control method is developed for conveniently
determining for the control parameters.
192
In applying the active control strategy in the vibration controls of engineering
structures, the following conditions and characteristics of control need to be taken into
considerations.
1. The frequency of the vibration of the selected point on the Euler-Bernoulli beam
type structures should be synchronized to that of the same frequency as the
external excitation.
2. The applications of the active control strategy in the engineering structures
featuring bending moment would not have exact frequency synchronization, and
the control input should remain stable after the vibration synchronization.
9.2 Future Works
In the future, the following works may further improve the study presented in this
thesis:
The internal vibrations, which may come up with large-amplitude vibration
and can only be described in a multi-dimensional system, should be applied
to validate the proposed active control strategy.
The principle, which may facilitate the selection of control parameters,
should be developed in order to increase both the efficiency and applicability
of the proposed active control strategy.
The influence of the external disturbance should be evaluated to guarantee
the reliance of the proposed active control strategy.
193
Experiments should be conducted to corroborate the application of the
control strategy developed.
In the current research, focus is on the axially translating structures with
constant axial moving velocity. However, structures moving at an accelerated
velocity are also commonly seen in engineering field, and therefore can be
targeted as a potential research topic in the future.
In comparing to the one-cable system studied in the current research, a two-
cable system or more complicated ones would be a promising direction in the
future.
194
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203
APPENDIX
This appendix lists the detailed expressions of nvmf _ (n=1…6) shown in Eq. (4.6),
3
1,4
3
1,3
3
1,5
2,1
1,61,51,4
1,31,21,1
2
1,41,11,5
2
1,2
2
1,31,1
1,6
2
1,21,3
2
1,2
2
1,61,2
1,5
2
1,4
2
1,51,41,6
2
1,1
2
1,51,21,4
2
1,2
2
1,21,1
2
1,31,21,5
2
1,1
2
1,41,3
2
1,51,3
2
1,61,31,5
2
1,3
1,6
2
1,31,4
2
1,3
2
1,41,2
1,6
2
1,5
2
1,61,51,6
2
1,4
2
1,61,4
3
1,6
3
1,1
3
1,21,3
2
1,11,4
2
1,1
2
1,61,11,2
2
1,1
2
1,51,1
2,1
1,61,51,3
1,61,51,4
1,31,21,1
10
1,51,41,1
1,41,21,11,41,31,21,61,31,2
1,51,31,21,61,51,21,61,41,2
1,51,41,21,51,31,11,61,51,1
1,61,41,11,61,41,31,51,41,3
1,61,51,41,51,21,11,61,21,1
1,31,21,11,41,31,11,61,31,11_
05534226.02555090.962436105.2010
000008613.1
3083676.440815484.000555718.0
48439795.0110825921.05509646.500
886239.21106179220.350120087.1406
87019741.49983634.447905175039.2
428511.130681312980.0862127136.4
50277859.002162959.05174152.566
39251344.09182285.280275268.1669
102709.2198204598.36434200708.279
46820535.052949616.0171689986.0
0950169.16207090.285124925156.18
67960208.68752968.393518176.151
0490439489.00646447.359013460385.0
193520.4607039304074.0286692.3364
000008613.1
41880502.28
4375818703.0330001816814.09069393239.0
120002127373.0342074372.31046.2
187001371.1
4064158.4212455405.3338788691.301
455994249.02602844.23672532877.3
8085173.2607594919.30260264702.15
7116859.7853423719.621475071371.0
2677687.4863904193380.07437281.381
404587272.0696137655.0844944021.7
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wBwBwBw
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wwBwwwBwwwBw
wwBwwwBwwwBwfvm
204
3
1,4
3
1,3
3
1,5
2,2
1,61,51,4
1,31,21,1
2
1,41,11,5
2
1,2
2
1,31,1
1,6
2
1,21,3
2
1,2
2
1,61,2
1,5
2
1,4
2
1,51,41,6
2
1,1
2
1,51,21,4
2
1,2
2
1,21,1
2
1,31,21,5
2
1,1
2
1,41,3
2
1,51,3
2
1,61,31,5
2
1,3
1,6
2
1,31,4
2
1,3
2
1,41,2
1,6
2
1,5
2
1,61,51,6
2
1,4
2
1,61,4
3
1,6
3
1,1
3
1,21,3
2
1,11,4
2
1,1
2
1,61,11,2
2
1,1
2
1,51,1
2,2
1,61,51,3
1,61,51,4
1,31,2
8
1,1
1,51,41,1
1,41,21,11,41,31,21,61,31,2
1,51,31,21,61,51,21,61,41,2
1,51,41,21,51,31,11,61,51,1
1,61,41,11,61,41,31,51,41,3
1,61,51,41,51,21,11,61,21,1
1,31,21,11,41,31,11,61,31,12_
631468.293846341521.1621005609.1
000026073.1
2571692.1019573689.1904867762.0
46760866.2239836.3803014588769.0
169493224.0622686683.1393208356.0
426376.2139040590038.263228.17723
5840721.0199365.45228284566.189
92257.12158012066.2366147109409.0
628221.455409173427.052606012.3
60779294.305178770.211131490.0
180872.1526978076.1693805788.8487
223789.407465819160.26705456.1553
319903.5428404395.57780131001898.0
502974.21202027428576.0346260306.0
223179.3643201602710.0115410.2198
000026073.1
2280479.751
10160685750.0725829398.1760001641568.0
515469709.51002164.1342074392.3
6716905.260
037423479.0669611942.354785522.48
023127.224233595646.73983293.1880
693974280.3292800689.08485932.234
024994419.566143834.178644011.833
17026590.258731120.70023003161.13
9986237.8952437793.3332282026.300
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BwwBwww
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wwBwwwBwwwBwfvm
205
3
1,4
3
1,3
3
1,5
2,3
1,61,51,4
1,31,21,1
2
1,41,11,5
2
1,2
2
1,31,1
1,6
2
1,21,3
2
1,2
2
1,61,2
1,5
2
1,4
2
1,51,41,6
2
1,1
2
1,51,21,4
2
1,2
2
1,21,1
2
1,31,21,5
2
1,1
2
1,41,3
2
1,51,3
2
1,61,31,5
2
1,3
1,6
2
1,31,4
2
1,3
2
1,41,2
1,6
2
1,5
2
1,61,51,6
2
1,4
2
1,61,4
3
1,6
3
1,1
3
1,21,3
2
1,11,4
2
1,1
2
1,61,11,2
2
1,1
2
1,51,1
2,3
1,61,51,3
1,61,51,4
1,31,21,1
1,51,41,1
1,41,21,11,41,31,21,61,31,2
1,51,31,21,61,51,21,61,41,2
1,51,41,21,51,31,11,61,51,1
1,61,41,11,61,41,31,51,41,3
1,61,51,41,51,21,11,61,21,1
1,31,21,11,41,31,11,61,31,13_
9339442.4881329.9782203666.6431
9999644052.0
8535950.1464273982.5048828720.7
16811.14619647477583.00169938735.0
284152.1669594565.1121775212.2886
15117979.18627050.455442970860.17
149553.417992142140.9918369773.1
17344372.382779341.10007499.448
373804020.45732134.15188093.16970
01044.2730257610.37040714570.7224
3184810.140551711433.8507731838.3
01127856.63685240.912009621228.64
77271896.139029584.1436886824.119
677051622.0120780.1406346260306.0
223179.3643201602710.0115410.2198
9999644047.0
2410279.112
557108889.2570011198659.0633842917.7
20000063757.0515474735.560002155551.0
473689853.0
2473138.333980308.3387133586.3096
443352389.17454684.75511573519.13
3038316.8348107682.55723982668.28
3477965.621830902.631626196853.12
566264.1555455006438.08790934.301
783910225.004460276.101796733.11
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wBwwBwwBw
wBwBwBw
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wwBwwwBwwwBwfvm
206
3
1,4
3
1,3
3
1,5
2,4
1,61,51,4
1,31,21,1
2
1,41,11,5
2
1,2
2
1,31,1
1,6
2
1,21,3
2
1,2
2
1,61,2
1,5
2
1,4
2
1,51,41,6
2
1,1
2
1,51,21,4
2
1,2
2
1,21,1
2
1,31,21,5
2
1,1
2
1,41,3
2
1,51,3
2
1,61,31,5
2
1,3
1,6
2
1,31,4
2
1,3
2
1,41,2
1,6
2
1,5
2
1,61,51,6
2
1,4
2
1,61,4
3
1,6
3
1,1
3
1,21,3
2
1,11,4
2
1,1
2
1,61,11,2
2
1,1
2
1,51,1
2,4
1,61,51,3
1,61,51,4
1,31,21,1
1,51,41,1
1,41,21,11,41,31,21,61,31,2
1,51,31,21,61,51,21,61,41,2
1,51,41,21,51,31,11,61,51,1
1,61,41,11,61,41,31,51,41,3
1,61,51,41,51,21,11,61,21,1
1,31,21,11,41,31,11,61,31,14_
39666.29440811503198.25826387.8
000016701.1
8012791.2082209351.969839.39942
64143504.2080251646.00003228710.0
1747679078.099843178.05286544510.0
9985876.930814494579.1270729.5428
98181774.2122826.453057059801.390
160031.4522830588.8487025100673.0
963462.169340364808.077847040.14
088100802.92059301.2259894615.4
128480.314194936.16970934731.8815
609211.838530865980.6240534.16407
07734.6628412698.118930030389427.0
6600024.788344896374.0894211.2110
077209590.96998347.21055995013.0
000016701.1
251884.1546
20328068849.0701931920.970000329384.0
633821984.7260001442159.09069293027.0
51536.2611
343754809.0026710027.766042409.17
856288.83310458705.25521603.3142
242074060.21877944293.03571698.483
50156688.498378231.173984272.8353
9855698.2756694744.260020343678.5
2409108.333570205.33389454445.617
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207
3
1,4
3
1,3
3
1,5
2,5
1,61,51,4
1,31,21,1
2
1,41,11,5
2
1,2
2
1,31,1
1,6
2
1,21,3
2
1,2
2
1,61,2
1,5
2
1,4
2
1,51,41,6
2
1,1
2
1,51,21,4
2
1,2
2
1,21,1
2
1,31,21,5
2
1,1
2
1,41,3
2
1,51,3
2
1,61,31,5
2
1,3
1,6
2
1,31,4
2
1,3
2
1,41,2
1,6
2
1,5
2
1,61,51,6
2
1,4
2
1,61,4
3
1,6
3
1,1
3
1,21,3
2
1,11,4
2
1,1
2
1,61,11,2
2
1,1
2
1,51,1
2,5
1,61,51,3
1,61,51,4
1,31,21,1
1,51,41,1
1,41,21,11,41,31,21,61,31,2
1,51,31,21,61,51,21,61,41,2
1,51,41,21,51,31,11,61,51,1
1,61,41,11,61,41,31,51,41,3
1,61,51,41,51,21,11,61,21,1
1,31,21,11,41,31,11,61,31,15_
29384372.6718418.240893843.69710
9999347969.0
7375895.22146644.8913427717606.4
90989821.008529825.00005660884.0
248304.130651283.121571386689.279
62951953.27429384.112159026909.21
80454.4529992718202.34113730478.5
90218862.9401191064.15695438.350
112600575.0894033.3363532791.4178
48495.19284332315.911981415.27298
59427225.3303338932.518588969.0
7231429.57304886.988713204203.104
69480876.515738176.22265878919.93
387099031.04733170.151457145942.0
810268.28501196321858.0485226.6028
9999347971.0
7139783.205
79185211.1160007680836.0702575634.9
00005707872.0726191766.1720000120169.0
489847278.1
7725675.2601808567.8346421920.755
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PEER REVIEWED PUBLICATIONS OF THE AUTHOR
Journal publications:
Dai, L., Sun, L., and Chen, C. (2014), Control of an extending nonlinear elastic cable
with an active vibration control strategy, Communication in Nonlinear Science
and Numerical Simulation, 19, 3901-3912.
Dai, L., Sun, L., and Chen, C. (2014), A control approach for vibrations of a nonlinear
microbeam system in multi-dimensional form, Nonlinear Dynamics, 77, 1677-
1692.
Dai, L. and Sun, L. (2014), Controlling chaotic vibrations of an Euler-Bernoulli beam
with an active control strategy, International Journal of Dynamics and Control, 1-
12.
Dai, L., Chen, C., and Sun, L. (2013), An active control strategy for vibration control of
an axially translating beam, Journal of Vibration and Control, accepted in press,
available on line: Doi: 10.1177/1077546313493312.
Dai, L. and Sun, L. (2012), On the Fuzzy Sliding Mode Control of nonlinear motions in a
laminated beam, Journal of Applied Nonlinear Dynamics,1, 287-307.
210
Conference publications
Dai, L. and Sun, L. (2014), Nonlinear vibration control of an axially translating string,
5th International Conference on Nonlinear Science & Complexity (NSC 2014),
Xi’an, China.
Sun, L. and Dai, L. (2013), Nonlinear vibration control of a translating beam with an
active control strategy, 24th Canadian Congress of Applied Mechanics
(CANCAM 2013), Saskatoon, Canada.
Dai, L. and Sun, L. (2013), Vibration control of a translating beam with an active control
strategy on the basis of the Fuzzy Sliding Mode Control, ASME 2013
International Mechanical Engineering Congress & Exposition (IMECE 2013), San
Diego, United States.
Dai, L. and Sun, L. (2012), Control of chaotic responses of a laminated composite beam
subjected to external excitation, 4th IEEE International Conference on Nonlinear
Science & Complexity (NSC 2012), Budapest, Hungary.
Dai, L. and Sun, L. (2012), Analysis and control of chaotic responses of a cantilever
beam subjected to sinusoidal excitation, ASME 2012 International Mechanical
Engineering Congress & Exposition (IMECE 2012), Houston, United States.
211
Book chapter contributions:
Dai, L., and Sun, L. (2015), “Active Control of Nonlinear Axially Translating Cable
Systems of Multi-Dimensions,” chapter contribution in the book: Nonlinear
Approaches in Engineering Applications – Dynamic Systems and Control, edited
by L. Dai and R. Jazar, ISBN: : 978-3-319-09461-8, Springer, New York.
Dai, L., Han, L., Sun, L. and Wang, X. (2013), “Diagnosis and Control of Nonlinear
Oscillations of a Fluttering Plate,” chapter contribution in the book: Nonlinear
Approaches in Engineering Applications 2, edited by R. Jazar and L. Dai, ISBN:
978-1-4614-6876-9, Springer, New York, 2013.