Active Nonlinear Vibration Control of Engineering

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



3.4 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


4.3 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



5.4 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


7.3 Series Solutions ......................................................................................... 138

7.4 Control Design .......................................................................................... 139


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


8.3 Series Solutions ......................................................................................... 177

8.4 Control Design .......................................................................................... 180


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


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


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


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


................................................................................................................................. 142


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


the active control strategy in the case of 0175.0rA : (a) 1w ; (b) 2w ; (c) 3w ...... 147


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


the case of 015.0rA ............................................................................................ 150


the active control strategy in the case of 015.0rA : (a) 1w ; (b) 2w ; (c) 3w ........ 151

XVI


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


the case of 010.0rA ........................................................................................... 154


the active control strategy in the case of 010.0rA : (a) 1w ; (b) 2w ; (c) 3w ....... 155


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


the case of 0553.0r ......................................................................................... 158


the active control strategy in the case of 0553.0r : (a) 1w ; (b) 2w ; (c) 3w ...... 159


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


the case of 1107.0r ......................................................................................... 162







the case of 1660.0r ......................................................................................... 166







................................................................................................................................. 182

XVIII


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


the active control strategy: (a) 1w ; (b) 2w ; (c) 3w .................................................. 186


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)



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.


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


022 2

11 11 11 112 0

1 1( )

2 2

h l

V hE Q dV b Q dx t dz

, (3.6)


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)




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


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


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)


-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.


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



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


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 .


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 .


described by Eqs. (4.6) after the implementation of the 6th


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.


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)


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


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)


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.


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.


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.


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)


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.


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)



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)


simulations in the following sections of this chapter, the following non-dimensional


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)




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.


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 .


described in Eqs. (5.21) after the implementation of the 6th


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


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)


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)


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


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 .


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


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.


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 ,


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


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.


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

,


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)


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)



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)


simulations in the consequent sections of this chapter, the following non-dimension


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



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)



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 .


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)



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



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.


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


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


described by Eq. (6.15) after the implementation of the 3rd


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


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)


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.


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)



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.


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


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.


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)


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.


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)


The virtual work done by the force is 0 since there is external excitation applied on

the axially translating cable,

0W , (7.8)


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)


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)


simulations in the consequent sections of this chapter, the following non-dimensional


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 .




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)


-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


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.


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


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 .


described by Eq. (7.20) after the implementation of the 3rd


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.


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)



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


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)


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.



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 .


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


the case of 015.0rA


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)



152


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.





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


the control strategy. As shown in the figure, the maximum amplitude of the cable is very

close to that of the reference signal.

153



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 .


values:

001.0 , 01.0fsk , )sin(001.0, pwwwF .



speed smv 75.30 is shown in the figures in Fig. 7.12 and Fig. 7.13.

154


the case of 010.0rA




the chaotic vibration will then become a periodic one, of which the amplitude is 0.010.

155

(a)

(b)

(c)



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










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





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 .


values,

158

01.0 , 01.0fsk , )sin(001.0, pwwwF .


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.


the case of 0553.0r



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)


the active control strategy in the case of 0553.0r : (a) 1w ; (b) 2w ; (c) 3w


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.



dash line) in wave diagram in the case of 0553.0r




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





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 .


values,

009.0 , 01.0fsk , )sin(001.0, pwwwF .


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


the case of 1107.0r



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)



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










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






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 .


values:

07.0 , 001.0fsk , )sin(1660.0, pwwwF .



speed smv 75.30 is shown in Fig. 7.24 and Fig. 7.25.

166


the case of 1660.0r




the chaotic vibration will then become a periodic one, of which the angular frequency is

1660.0 .

167

(a)

(b)

(c)



168


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









the active control strategy. As shown in the figure, the frequency of the cable is very

close to that of the reference signal.

169






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


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.


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 ,


Thus, the displacement field of the cable can be derived as

kiR ttxwttxutx ,, 00 ,


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.


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)


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)



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)


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

.




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)



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 .


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)



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)



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


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


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


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.


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)



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.


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)



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



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.


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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Ew

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CwCwCw

wBwwBwwBw

wBwwBwwBw

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wBwwBwwBw

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wBwwBwwBw

wBwwBwwBw

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wwBwwwBwwwBw

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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

BwBwBw

Ew

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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.

Documents

Active Nonlinear Vibration Control of Engineering