ABSTRACT
Title of dissertation: COUPLED OSCILLATOR ARRAYS:
DYNAMICS AND INFLUENCE
OF NOISE
ABDULRAHMAN ALOFI
2021
Dissertation directed by: Professor Balakumar Balachandran
Department of Mechanical Engineering
Coupled oscillator arrays can be used to model several natural systems and
engineering systems including mechanical systems. In this dissertation work, the
influence of noise on the dynamics of coupled mono-stable oscillators arrays is inves-
tigated by using numerical and experimental methods. This work is an extension
of recent efforts, including those at the University of Maryland, on the use of noise
to alter a nonlinear system?s response. A chain of coupled oscillators is of interest
for this work. This dissertation research is guided by the following questions: i)
how can noise be used to create or quench spatial energy localization in a system
of coupled, nonlinear oscillators? and ii) how can noise be used to move the energy
localization from one oscillator to another? The coupled oscillator systems of in-
terest were harmonically excited and found experimentally and numerically to have
a multi-stability region (MR) in the respective frequency response curves. Relative
to this region, it has been found that the influence of noise depends highly on the
excitation frequency location in the MR. Near either end of the MR, the oscillator
responses were found to be sensitive to noise addition in the input and it was ob-
served that the change in system dynamics through movement amongst the stable
branches in the deterministic system could be anticipated from the corresponding
frequency response curves. The system response is found to be robust to the influ-
ence of noise as the excitation frequency is shifted toward the middle of the MR.
Also, the effects of noise on different response modes of the coupled oscillators ar-
rays were investigated. A method for predicting the behavior is based on so-called
basins of attractions of high dimensional systems. Through the findings of this work,
many unique noise influenced phenomena are found, including spatial movement of
an energy localization to a neighboring oscillator, response movement gradually up
the energy branches, and generation of energy cascades from a localized mode.
COUPLED OSCILLATOR ARRAYS:
DYNAMICS AND INFLUENCE OF NOISE
by
Abdulrahman Alofi
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2021
Advisory Committee:
Professor Balakumar Balachandran, Chair and Advisor
Professor Amr Baz, Department of Mechanical Engineering
Professor Abhijit Dasgupta, Department of Mechanical Engineering
Professor Nikhil Chopra, Department of Mechanical Engineering
Professor Sung Lee, Department of Aerospace Engineering (Dean?s Representative)
? Copyright by
Abdulrahman Alofi
2021
Dedication
To my parents, my wife and my two little kids
ii
Acknowledgments
First and foremost, I would like to thank my advisor, Professor Balakumar
Balachandran for providing me this special opportunity to pursue my Ph.D. and to
work on this exciting project. He has always been there if I needed help and guided
me throughout the whole years, and never hesitated to provide the help.
I would like to express my appreciation and deepest gratitude to Professor
Amr Baz, Professor Abhijit Dasgupta, Professor Nikhil Chopra and Professor Sung
Lee for accepting to be a part of the committee and taking the time to review my
work. I have learned a lot from my interactions with them and the courses they
have taught me.
I would like to thank my parents for all of the unconditional support provided
over the years. They made me recognize my own potential and encourged me to
always have the ambition to be the best. Also, special thanks to my beloved wife for
her patience and love. She always understood my circumstances and provided me
the support that I needed. Thanks as well for my two little kids. They are simply
creating happiness in my life. Thanks also to the rest of my family, who encouraged
me and gave me their affection.
I would like to thank all of my friends and colleagues at UMD. I always re-
ceived valuable help from them, and enjoyed the discussions on our free time. I
would like to show my appreciation to them by listing their names: Gizem Acar;
Xianbo Liu; Lautaro Cilenti;Xie Zheng; Vipin Agarwal; Guanjin Wang; Celeste Po-
ley; Thomas Breunung; Preethi Ravula; Saliou Telly; Randall Kania; Abu Bockarie
iii
Kebbie-Anthony; Nishant Nemani; and Samarpan Chakraborty. I would like to
thank as well my friend Mohammed Fayed for his kind friendship and for the nice
time spent with him.
Finally, I would like to thank King Fahd University of Petroleum and Minerals
for sponsoring my Ph.D study. Also, I am grateful for the support received from
NSF for the grant No. CMMI1760366..
iv
Table of Contents
Dedication ii
Acknowledgements iii
Table of Contents iii
List of Tables v
List of Figures vi
1 Introduction 1
1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Problem of Interest . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Overall Goal and Specific Objectives . . . . . . . . . . . . . . . . . . 7
1.3 Organization of Proposal . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Introduction and Background 9
2.1 System Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Duffing Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Frequency Response Curves for Oscillator Arrays . . . . . . . . . . . 19
2.5 Natural Frequencies and Mode Shapes . . . . . . . . . . . . . . . . . 22
2.6 Nonlinear Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 Effects of Noise in the Multi-stability Region of Coupled Oscillator Arrays 28
3.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Numerical Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3.1 Influence of Noise Near the Left Boundary of the Multi-Stability
Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3.2 Influence of Noise Near the Right Boundary of the Multi-
Stability Region . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3.3 Basins of Attraction . . . . . . . . . . . . . . . . . . . . . . . 41
iii
4 Effects of Noise on Different Response Modes of Coupled Oscillator Arrays 45
4.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1.1 Two Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1.2 Three Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Effects of Noise on Different Modes of Coupled Oscillator Arrays . . . 53
4.2.1 Near the Left End of the Multi-Stability Region . . . . . . . . 53
4.2.1.1 Two oscillators . . . . . . . . . . . . . . . . . . . . . 53
4.2.1.2 Three oscillators . . . . . . . . . . . . . . . . . . . . 58
4.2.2 Near the Right End of the Multi-Stability Region . . . . . . . 62
4.2.2.1 Two Oscillators . . . . . . . . . . . . . . . . . . . . . 62
4.2.2.2 Three Oscillators . . . . . . . . . . . . . . . . . . . . 64
4.3 Basins of Attraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4 Effects of Coupling Strength on System Response . . . . . . . . . . . 66
4.5 Effects of Increase in Number of Oscillators in an Array . . . . . . . . 69
5 Noise Based Control of System Energy of Coupled Oscillator Arrays 73
5.1 Influence of Noise Near MR Boundaries . . . . . . . . . . . . . . . . . 73
5.2 Noise Induced Response Energy Level Increase . . . . . . . . . . . . . 76
5.3 Noise Induced Spatial Movement of an Energy Localization . . . . . . 80
5.4 Influence of Noise on a Localized Mode and an Anti-localized Mode . 83
6 Summary and Recommendations for Future Work 88
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.2 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . 90
Bibliography 92
iv
List of Tables
2.1 Experimental results for system parameter values of a single oscillator. 16
3.1 Experimental results for the parameter values . . . . . . . . . . . . . 30
4.1 First oscillator parameters obtained from experimental data given in
Figure 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.1 First oscillator parameters obtained by matching the results from the
quasi-static frequency sweep shown in Figure 5.3. . . . . . . . . . . . 77
v
List of Figures
2.1 A coupled oscillator array in a free-free or in-line arrangement. . . . . 10
2.2 Experimental setup of in-line arrangement of coupled oscillators. . . . 12
2.3 Potential energy of a representative softening Duffing oscillator ob-
?2
tained by using E(x) = n0x2 + ?x4. . . . . . . . . . . . . . . . . . . 13
2 4
2.4 Potential energy of a representative hardening Duffing oscillator ob-
?2
tained by using E(x) = n0x2 + ?x4. . . . . . . . . . . . . . . . . . . 13
2 4
2.5 Frequency response curve for a representative softening Duffing oscil-
lator based on equation (2.3). . . . . . . . . . . . . . . . . . . . . . . 15
2.6 Frequency response curve for a representative hardening Duffing os-
cillator based on equation (2.3). . . . . . . . . . . . . . . . . . . . . . 16
2.7 Basin of attraction of responses of a softening oscillator given by
equation (2.3) for ?=34.86 rad/s. . . . . . . . . . . . . . . . . . . . . 17
2.8 Basin of attraction of responses of a softening oscillator given by
equation (2.3) for ?=35.39 rad/s. . . . . . . . . . . . . . . . . . . . . 18
2.9 Basin of attraction of responses of a hardening oscillator given by
equation (2.3) for ?=34.86 rad/s. . . . . . . . . . . . . . . . . . . . . 18
2.10 Basin of attraction of responses of a hardening oscillator given by
equation (2.3) for ?=36.72 rad/s. . . . . . . . . . . . . . . . . . . . . 19
2.11 All solution branches for the array of equation (2.5). . . . . . . . . . . 21
2.12 Mode shapes of the coupled oscillators array . . . . . . . . . . . . . . 23
2.13 Nonlinear modes for the coupled oscillator array. . . . . . . . . . . . . 26
2.14 Localized nonlinear modes . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1 Quasi-static frequency sweep for an array of two coupled oscillators. . 29
3.2 Time series of the responses of the array system with two coupled
oscillators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3 Experimental results for harmonic excitation frequency near the left
end of the MR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4 Numerical results for harmonic excitation frequency near the left end
of the MR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.5 Experimental results obtained when the harmonic excitation frequency
is near the right end of the MR. . . . . . . . . . . . . . . . . . . . . . 39
vi
3.6 Numerical results obtained when the harmonic excitation frequency
is near the right end of the MR. . . . . . . . . . . . . . . . . . . . . . 40
3.7 Basin of attraction for the system near the left boundary of the MR . 42
3.8 Basins of attraction for the system near the right boundary of the MR. 43
4.1 Time series of the responses of the two coupled oscillators. . . . . . . 46
4.2 Frequency responses of the two coupled oscillators. . . . . . . . . . . 47
4.3 Matching of the experimental results of the first oscillator shown in
Figure 4.2 (blue) with the analytical approximation for the parame-
ters given in Table 4.1 (black). . . . . . . . . . . . . . . . . . . . . . . 49
4.4 Time series of the responses for the array with three coupled oscillators 51
4.5 Frequency response of the system with three coupled oscillators. . . . 52
4.6 System modes for (a) first oscillator and (b) second oscillator. . . . . 54
4.7 Experimental results for the effects of noise on different modes near
the left boundary of MR of the two coupled oscillators arrays shown
in Figure 4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.8 Numerical results for the effects of noise on different modes near the
left boundary of MR for the array with two coupled oscillators shown
in Figure 4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.9 Modes of interest of the system. . . . . . . . . . . . . . . . . . . . . . 59
4.10 Experimental results for the effects of noise on different modes near
the left boundary of MR for the array with three coupled oscillators
shown in Figure 4.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.11 Numerical results for the effects of noise on different modes near the
left boundary of MR for the system with three coupled oscillators
arrays. The parameter values are based on Table 4.1. . . . . . . . . . 61
4.12 Experimental results for the effects of noise on different modes near
the right boundary of MR for the system of two coupled oscillators
shown in Figure 4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.13 Experimental results for the effects of noise on different modes near
the right boundary of MR for the system of three coupled oscillators
shown in Figure 4.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.14 Basins of attraction for the system of three coupled oscillators. . . . . 67
4.15 Effects of coupling increase on the four response branches of the set
of two coupled oscillators for the parameters provided in Table 4.1. . 69
4.16 Bifurcation sets in the (?, ?) plane for the system of two coupled
oscillators with the parameters listed in Table 4.1. . . . . . . . . . . . 70
4.17 Effects of noise on an oscillator in a state of low amplitude oscillations. 72
4.18 Phase space plots for the selected modes in Figure 4.17. . . . . . . . . 72
5.1 Numerical results for the effects of noise on 21 hardening Duffing
oscillators for the parameters listed in Table 2.1. . . . . . . . . . . . . 74
5.2 Numerical results for the effects of noise on 21 softening Duffing os-
cillators for the parameters listed in Table 2.1. . . . . . . . . . . . . . 75
vii
5.3 Experimental results for noise induced gradual energy increase in a
set of two coupled oscillators. . . . . . . . . . . . . . . . . . . . . . . 78
5.4 Numerical results for noise induced gradual energy increase in a set
of two oscillators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.5 Experimental results for noise induced spatial movement of energy in
a set of two coupled oscillators. . . . . . . . . . . . . . . . . . . . . . 81
5.6 Numerical results for noise inducing spatial movement of energy in a
set of two coupled oscillators. . . . . . . . . . . . . . . . . . . . . . . 82
5.7 Influence of noise on a LM near the left boundary of MR (?=34.90
rad/s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.8 Noise is stopped at different time instants in Figure 5.7 inducing
increase in the number of oscillators transitioning to a state of high
amplitude oscillations. . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.9 Influence of noise on an ALM near the right boundary of MR (?=36.72
rad/s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
viii
Chapter 1: Introduction
In this chapter, the author outlines the background and motivation for this
work, the problem of interest, the literature review, the overall goals, and discusses
the proposal organization.
1.1 Background and Motivation
1.1.1 Problem of Interest
It is of interest to this work to understand the influence of noise on the dy-
namics of coupled oscillator arrays. In the past, noise has been considered as an
undesirable source of energy, and systems have been designed to overcome any pos-
sible negative outcomes due to noise. Thanks to the advances in understanding
phenomena such as stochastic resonance [1], noise nowadays might be considered
as a beneficial source of energy that helps in improving or controlling the system
dynamics. In this work, the type of noise considered in the numerical study is a
random process, white Gaussian noise, and in the experiments is a band-limited
white noise.
Coupled oscillator arrays have rich nonlinear dynamics. For example, Papan-
1
gelo et al. [2] stated that for both linear and cyclic chain of coupled oscillator arrays,
the response energy of such systems can be studied through clusters of branches in
the corresponding frequency response. Then, each branch corresponds to a certain
number of oscillators that have high amplitude oscillations. Previous efforts on in-
troducing noise into such systems revealed the possibility for creating or destroying
localization with noise [3, 4]. This work can be viewed as continuation in the spirit
of the previous efforts. Here, for the first time, to the best of the author?s knowl-
edge, the influence of noise to induce a transition between different energy branches
corresponding to high amplitude oscillations is discussed.
Here, the influence of noise on an array with two to three oscillators are first
explored to get an insight into the behavior of an array with a large number of
oscillators. As part of the study, a combination of numerical and experimental
approach is used. Different phenomena of interest include response movement up
or down in the response energy branches, and movement of the spatial localization
from one oscillator to another. The building blocks for the array are monostable
Duffing oscillators, and these oscillators are coupled through linear springs. The
aforementioned phenomena are also numerically explored with arrays with a large
number of oscillators.
This work is expected to be of interest to systems such as energy harvesting
devices [5] and rotary systems [6].
2
1.1.2 Literature Review
In small scale devices, systems vibrate at low amplitudes, wherein any small
disturbance may affect the system dynamics [7] . In these systems, one usually
encounters noise that is either derived from the surrounding environment or inherent
to the system [8]. Even though the noise levels may be low, they can alter the system
dynamics in a noticeable way. For instance, small fluctuations have been shown to
move systems with metastable states escape from a local stable state [9]. Also, the
presence of noise may induce a transition between a limit cycle and a fixed point
of a system [10]. Moreover, noise may induce a period doubling cascade, resulting
in chaos [11]. If not designed or addressed properly, not only will the performance
of these systems be affected [12] but there may be other undesirable consequences
including system malfunction [13].
In previous studies [14?17], researchers have investigated the problem of inter-
est from a different perspective. Instead of building systems that are robust to noise,
they considered noise as a beneficial source of energy that could be exploited for en-
hancing the system dynamics. In recent studies with nonlinear systems, it has been
shown that noise can be utilized to change dynamic system response. For example,
noise has been used to stabilize nonlinear systems [18?20]. Also, Gao et al. [11]
showed that noise can be used to control chaos in nonlinear systems. Moreover,
through biological studies, it has been shown that introducing noise can be bene-
ficial for human systems including heart function [21], human hearing [22, 23], and
nervous system [24?26]. In addition, enhanced response features have been obtained
3
by utilizing noise in various mechanical [27?29] and chemical systems [30?32].
In the field of mechanical systems, as an atomic force microscope (AFM)
operates in the tapping mode, a situation of interest occurs when there is zero
normal speed contact, also known as grazing [33]. It has been reported that period
doubling can exist and could be used as a new way to locate the grazing impacts
[34,35]. Building on this work, Chakraborty and Balachandran [36] showed through
experimental and numerical studies that the addition of white Gaussian noise can
help the system to move from a no contact state to a contact state, wherein the
associated response is close to that of a period-doubled orbit.
It is well known than that for an inverted pendulum, the system has an un-
stable equilibrium point at the upright position [37]. A number of researchers have
used control schemes [38, 39] or high-frequency excitations [40, 41] in their efforts
for stabilizing the upright position. By using a different approach, Perkins and
Balachandran [42] showed that noise can be used for stabilizing the upright posi-
tion when combined with a harmonic excitation. In addition, noise can force the
pendulum into rotation, which could be used for energy harvester applications [43].
With the Duffing oscillator, noise has been used to change the system dynamics
[44, 45]; depending on the application, some of these changes can be useful for the
system. Agarwal and Balachandran [46] showed that addition of noise can be used
to move the system away from an aperiodic response, including the possibility to
move the system from a chaotic motion into a fixed point of the unforced system.
Furthermore, noise has been used to reduce the hysteresis region by moving the
jump-up and the jump-down frequencies close to each other as well as for destroying
4
the hysteresis region [47].
In the 1980s, intrinsic localized modes (ILMs) were introduced as a new class
of energy localization [48?51]. In contrast to the Anderson localization [52, 53],
wherein the localization is due to the introduction of a defect or an impurity thus
creating a near-periodic system, ILMs can occur in perfectly periodic lattices and
the localization can potentially occur at any position [54]. Two important elements
for creation of ILMs are nonlinearity and discreteness [55]. These elements along
with weak coupling allow for the existence of ILMs in nonlinear systems [49]. Some
examples of systems in which ILMs occur are photonic crystals [50, 56], antiferro-
magnet lattice [57], granular media [58?61], and coupled oscillator arrays [62?64].
Coupled oscillator arrays have received considerable interest during the last
two decades. For example, Sato et al. [65] carried out a study on a micromechanical
oscillator array, and they reported a localization locking effect. By adjusting the
coupling between the oscillators, Kimura et al. [66] introduced a new means for
manipulating ILMs. In a different study, Kimura et al. [67] created an experimental
system to study ILMs.
The complexity of dynamics of coupled oscillator arrays makes these systems
quite appealing to study. The building blocks for the array can be monostable
Duffing oscillators [68, 69], wherein each oscillator has a single stable equilibrium
position. The system can be more complex, if one were to consider the building
blocks as bi-stable Duffing oscillators [70], in which case, each oscillator has two
stable equilibrium points and an unstable one.
To provide a glimpse into the dynamics of arrays with monostable and bi-
5
stable oscillators, Papangelo et al. [2,71] studied coupled oscillator arrays comprised
of monostable and bi-stable oscillators. In the case of an array with monostable
oscillators, the system response state can go into spatially localized responses. The
degree of localization ranges from being in one oscillator to the extreme case wherein
all the oscillators vibrate at a high energy level. For the array with bi-stable oscilla-
tors, the researchers were able to mimic a snaking pattern in the solution structure,
which is relevant to many continuous systems [72?74]. The response picture is com-
plex, and it is composed of isolas (closed solution branches) and similar states that
represent different energy levels.
Since coupled oscillator arrays have rich nonlinear dynamics that is sensitive
to small perturbations, attractive results have been shown by utilizing noise in those
systems. For instance, Ramakrishnan and Balachandran [75] showed that noise is
capable of strengthening and attenuating localizations in coupled oscillator arrays.
Also, Perkins et al. [3] carried out experimental and numerical investigations on
coupled cantilever beam arrays. It was shown that localization can be destroyed,
created or even shifted to a different spatial location under the influence of noise.
While the results are appealing, the subject of interest is still in its infancy. One
is still unclear on the level and/or duration of noise needed to change the system
dynamics. Also, depending on the nonlinearity and interplay with noise, there can
be new ways to positively use the noise to alter a system?s response; these aspects
remain to be explored. Moreover, even with recognition of the potential use of noise
for control, additional studies are needed to make the best use of noise.
6
1.2 Overall Goal and Specific Objectives
The overall goal of this work is to continue the recent experimental and nu-
merical efforts in the group?s prior works on exploiting the interplay between noise
and nonlinearity in mechanical and structural systems. The focus of this work
is mainly on coupled oscillator arrays arranged in a linear or in-line configuration
(free-free boundary conditions). These systems are excited by harmonic excitations
and include noise inputs in some cases. There are four specific objectives for this
dissertation work. They are as follows:
1. To understand how noise can be used to create a localization in a coupled
array of nonlinear oscillators with different numbers of oscillators
2. To examine how noise can be used to quench a spatial energy localization in
a coupled array of nonlinear oscillators with different numbers of oscillators
3. To investigating how to move a spatial localization from one oscillator to
another in an array under the influence of noise
4. To develop appropriate means to analyze responses of arrays of different sizes
1.3 Organization of Proposal
The rest of the proposal is organized as follows. In Chapter 2, the system mod-
eling is present along with the experimental setup. A background on the Duffing
oscillator is presented as it represents important information for the coupled oscil-
7
lator arrays. The frequency response curve along with the natural frequencies and
the nonlinear modes of the system are presented. In Chapter 3, when the harmonic
excitation frequency is close to the boundaries of the multi-stability region of the
coupled oscillators arrays, the influence of noise on the system response is discussed.
A method for exploring how the oscillators respond to disturbances is presented
using basins of attractions. Following that, in the next chapter, the effects of noise
on different modes are investigated. In addition, the effects of coupling strength and
the number of oscillators on how an oscillator responds to noise are explored. In
Chapter 5, different phenomena are presented for the coupled oscillator arrays. To
close, the work is summarized and recommendations for future work are outlined,
in the last chapter.
8
Chapter 2: Introduction and Background
In this chapter, the author presents the model that is used for the study. In
addition, a brief background on the Duffing oscillator is provided. The results are
presented in the form of the frequency response curves, natural frequencies and
mode shapes, and nonlinear modes.
2.1 System Modeling
In the considered model, there are N identical oscillators that are intercon-
nected through linear springs. Each oscillator has a linear stiffness as well as a
cubic stiffness. The oscillator array has free-free boundary conditions (i.e., the two
oscillators at the boundary are only connected from one side and free from the other
side). A schematic for the model is shown in Figure 2.1. The governing equations
of motion for the system can be written as shown next:
2
md x1 + cdx12 + k x + k x
3
dt dt l 1 nl 1
+ kc(x1 ? x2) = f1(t)
md
2xn + cdxn2 + k
3
dt dt l
xn + knlxn + kc(2xn ? xn+1?
(2.1)
xn?1) = fn(t), for 2 ? n ? N ? 1
d2m xN + cdxN2 + klxN + k
3
nlxN + kc(xN ? xN?1) = f (t)dt dt N
9
Figure 2.1: A coupled oscillator array in a free-free or in-line arrangement.
Here, xn is the displacement of the n
th oscillator, m is the oscillator mass, c is
the damping coefficient, kl and knl are the linear and the nonlinear stiffness coeffi-
cients, respectively, kc is the linear coupling coefficients, fn is the force acting on the
nth oscillator. This forcing is composed of a deterministic component in the form of
a harmonic excitation and a stochastic input. In the stochastic input, which is in
the form of white Gaussian noise (?? W? (t)), ?? is the noise intensity, and W (t) is the
Wiener process. With the harmonic excitation given by a? cos(?t), a? is the forcing
amplitude and ? is the forcing frequency. Introducing the following parameters
? = ?c , ?2 = kl , ? = knl , ? = kc , a = a? , and ? = ?? ,
2 mk n0l m m m m m
and assuming that the forcing on all the oscillators to have similar phase and am-
plitude values (fn = f), equation (2.1) can be rewritten as follows:
10
x?1 + 2??n0x?
2
1 + ?n0x1 + ?x
3
1 + ?(x1 ? x2) = a cos(?t) + ? W? (t)
x?n + 2??n0x?n + ?
2 x + ?x3n0 n n + ?(2xn ? xn?1?
(2.2)
xn+1) = a cos(?t) + ? W? (t), for 2 ? n ? N ? 1
x?N + 2??
2
n0x?N + ?n0xN + ?x
3
N + ?(xN ? xN?1) = a cos(?t) + ? W? (t)
Here, an overdot indicates the derivative with respect to time t. The param-
eters of the system are determined from the experimentally obtained frequency-
response curves for the system shown in Figure 2.2, as done in the group?s prior
studies (e.g., [47]).
2.2 Experimental Setup
In Figure 2.2, the experimental setup that has been used to perform the study
is shown. A spring coupled set of metallic cantilever oscillators is secured to a base,
which is connected to an electrodynamic shaker used to generate the excitation.
Permanent magnets are used to realize the nonlinear spring characteristics of each
oscillator. For each cantilever oscillator, one of the magnets is located on top of
the cantilever, while the other is fixed on a plate above the free end. The spacing
between this pair of magnets can be adjusted to realize an oscillator with a harden-
ing or softening characteristic. LabVIEW software is used to generate a combined
excitation consisting of harmonic and noise components. The excitation input is
filtered by using a low pass filter so that only frequencies that are below a chosen
cutoff frequency are allowed. The transverse deflections of the cantilevers in the
11
Figure 2.2: Experimental setup of in-line arrangement of coupled oscillators.
excitation direction are measured by using strain gauges that are attached close to
each cantilever?s base. LabVIEW software is also used for data acquisition of the
strain gauge signals.
2.3 Duffing Oscillator
Since the Duffing oscillator represents the main oscillator component in equa-
tions (2.2), exploring the effects of parameter changes on the response of a single
oscillator can give one a glimpse into the behavior of the full system. Considering
only a single beam oscillator, the equation of motion reduces to the following Duffing
equation:
12
Figure 2.3: Potential energy of a representative softening Duffing oscillator obtained
?2
by using E(x) = n0x2 + ?x4. The parameter values are similar to those given in
2 4
Section 5.3 and listed in Table 2.1.
Figure 2.4: Potential energy of a representative hardening Duffing oscillator ob-
?2
tained by using E(x) = n0x2 + ?x4. The parameter values are similar to those
2 4
given in Section 5.3 and listed in Table 2.1.
13
x?+ 2??n0x?+ ?
2
n0x+ ?x
3 = f(t), (2.3)
Here, the forcing f(t) = F cos(?t).
In this work, the author considers mainly systems with a monostable hardening
Duffing oscillator with a few cases of a softening monostable Duffing oscillator. For
both configurations, the coefficients of x is positive; that is, > 0. For monostable
hardening case, the coefficients of x3 is > 0, while for a monostable softening case,
this coefficient is < 0. The energy of the unforced and undamped system for both
configurations has a single potential well at the origin resulting corresponding to a
stable equilibrium position, as shown in Figure 2.3 and 2.4 for the softening and
hardening cases, respectively.
For the forced and damped system, the frequency response curves are shown
in Figures 2.5 and 2.6, for the softening and hardening cases, respectively. The
parameters are similar to those obtained from the experimental results given in
Section 5.3 and summarized in Table 2.1. For certain forcing frequencies, the system
has a region with three solutions, two stable periodic solutions (solid branches) and
one unstable periodic solution (dashed branch). Due to the dissipation, each stable
solution has a basin of attraction, in which trajectories are drawn towards this stable
solution.
In order to gain an insight into the coupled oscillator array dynamics, the
effects of system parameters on the basins of attraction of the periodic orbits of the
single oscillator are studied. To obtain the basins of attraction, equation (2.3) is
14
Figure 2.5: Frequency response curve for a representative softening Duffing oscillator
based on equation (2.3).The parameters are similar to the obtained experimental
values given in Section 5.3 and listed in Table 2.1. A normalized response amplitude
is shown on the vertical axis and the excitation frequency is shown on the horizontal
axis.
solved by using Matlab built-in function ode45 for a set of initial conditions. For
each set of initial conditions, the steady state response ends up in either the high
amplitude attractor or the low amplitude attractor. The corresponding results are
plotted with a unique color specified for each corresponding attractor region. In
Figures 2.7 and 2.8, the author has shown the effects of increasing the excitation
frequency in a quasi-static manner on a softening Duffing oscillator, while keeping
all of the other parameters fixed. In the figures, there are two attractors, one
corresponding to the high amplitude response (small green point) and the other
corresponding to the low amplitude response (small black point). Each attractor
has an associated basin of attraction (shown in red for the high amplitude one and
blue for the low amplitude one) wherein for trajectories initiated inside a basin,
15
Table 2.1: Experimental results for system parameter values of a single oscillator.
Oscillator 1 (Softening) Oscillator 2 (Hardening)
Parameter Value Parameter Value
?n0 36.22 rad/s ?n0 33.76 rad/s
? -150 ? 270
F 18 F 18
? 0.00745 ? 0.00717
they are attracted to the corresponding attractor. At ? = 34.86 rad/s, the basin of
attraction of the high amplitude response is relatively small compared to the basin of
the low amplitude response, as illustrated in Figure 2.7. As the excitation frequency
is increased, the basin of attraction of the high amplitude response increases in size
while the basin of attraction of the low amplitude response decreases in size. This
Figure 2.6: Frequency response curve for a representative hardening Duffing oscilla-
tor based on equation (2.3). The parameters are similar to the obtained experimental
values given in Section 5.3 and listed in Table 2.1. A normalized response amplitude
is shown on the vertical axis and the excitation frequency is shown on the horizontal
axis.
16
trend continues until the basin of the high amplitude response reaches its largest
size at the end of the hysteresis region, as shown in Figure 2.8 for ? = 35.39 rad/s.
However for the hardening Duffing oscillator, an opposite behavior is observed.
For ?=34.86 rad/s , the basin of attraction of the high amplitude response is larger
than the size of the basin of attraction of the low amplitude response. As one
increases the forcing frequency, the basin of the high amplitude response shrinks in
size until it reaches a minimum just before the jump down frequency, as shown in
Figure 2.10 for ?=36.72 rad/s.
The previous results are important for understanding the dynamics of the
coupled oscillator array, as will be shown in the coming sections.
Figure 2.7: Basin of attraction of responses of a softening oscillator given by equation
(2.3) for ?=34.86 rad/s. The high amplitude stable response is labelled as the HA
attractor and the low amplitude stable response is labelled as the LA attractor in
this figure and other similar figures that follow.
17
Figure 2.8: Basin of attraction of responses of a softening oscillator given by equation
(2.3) for ?=35.39 rad/s.
Figure 2.9: Basin of attraction of responses of a hardening oscillator given by equa-
tion (2.3) for ?=34.86 rad/s.
18
Figure 2.10: Basin of attraction of responses of a hardening oscillator given by
equation (2.3) for ?=36.72 rad/s.
2.4 Frequency Response Curves for Oscillator Arrays
To get a clear picture for all available solutions for equation (2.2), the author
followed a similar procedure as in earlier work [2]. A harmonic balance method is
first used to express the periodic solution of equation (2.2) as follows:
?M
xn = [An,m cos(m?t) +Bn,m sin(m?t)] (2.4)
m=0
Here, ? is the excitation frequency, M is the chosen number of harmonics, An,m
and Bn,m are the amplitudes of cosine and sine terms, respectively. After substitut-
ing equation (2.4) with only the first harmonic in equation (2.2), the result is the
following:
19
(??2 + ?2n0 + 3?|B1|2 + ?)A1 + 2?? 3 2n0B1 + ?|A1| A1 ? ?A4 4 2 = 0
(??2 + ?2n0 + 3?|A |2 + ?)B ? 2?? A 3 24 1 1 n0 1 + ?|B1| B1 ? ?B2 = a4
(??2 + ?2 3n0 + ?|Bn|2 + 2?)An + 2??n0B + 3n ?|An|2A4 4 n?
?(An?1 + An+1) = 0, for 2 ? n ? N ? 1
(2.5)
(??2 + ?2 + 3?|A |2 + 2?)B ? 2?? A + 3?|B |2n0 n n n0 n n B4 4 n?
?(Bn?1 +Bn+1) = a, for 2 ? n ? N ? 1
(??2 + ?2 + 3?|B |2n0 N + ?)AN + 2?? 3n0BN + ?|A 2N | AN ? ?A = 04 4 N?1
(??2 + ?2 3n0 + ?|AN |2 + ?)BN ? 2?? 3n0AN + ?|B 2N | BN ? ?B4 4 N?1 = a
The obtained system of equation is in the form of N algebraic equations with
N + 1 unknowns. To find all the frequency response branches of equation (2.2),
Auto software [76] is used by considering the solution of equation (2.5) as an initial
conditions and the excitation frequency ? as the continuation parameter. Equation
(2.5) is solved for 5 coupled oscillators array by considering highly localized modes as
initial conditions; for example, (0, 1, 1, 1, 0) and (1, 0, 1, 1, 1), where 1 corresponds to
a high amplitude oscillator response and 0 corresponds to a low amplitude oscillator
response. The system parameters are similar to the obtained experimental results
given in Section 4.1.1 and summarized in Table 4.1. The solution b?ra?nches are plot-
ted in terms of the norm of all x solutions, which is defined by L2 = N A2 +B2n=0 n n.
This norm is used as a measure of system energy. The frequency response curve
for the oscillator array is shown in Figure 2.11. The energy of the system differs
from one branch to another. On the upper branch (P5 Mode), the system has the
20
Figure 2.11: All solution branches for the array of equation (2.5): (a) all solution
branches and (b) corresponding distribution of responses for branches P0-P5.
highest possible energy at the corresponding excitation frequency, and the energy
is equally distributed between the oscillators. Moving to a lower solution branch
(P4), one of the oscillators is in a low energy state, and the rest of them are in
a high energy state. This pattern continues until one reaches the (P0) mode, in
which the system energy has its minimum value, and each of the oscillators has a
low amplitude response.
21
2.5 Natural Frequencies and Mode Shapes
Determining the linear natural frequencies of a system is helpful for character-
izing the system responses. Toward this, the conservative, unforced linear system
about the zero equilibrium position is first obtained from system (2.2) as follows
x?1 + ?
2
n0x1 + ?(x1 ? x2) = 0
x? + ?2n n0xi + ?(2xn ? xn?1 ? xn+1) = 0 for n = 2, 3, ..., N ? 1 (2.6)
x? + ?2N n0xN + ?(xN ? xN?1) = 0
The solution of this system has an oscillatory behaviour. Thus, the solution is
assumed to have the following trial functions
xn(t) = X e
i?t
n (2.7)
where Xn is the amplitude of the n
th oscillator and ? is the linear natural frequency.
After substituting equation 2.7 into equation 2.6, the following eigenvalue problem
can be obtained:
(K ? ?2M)X = 0 (2.8)
where, M is the N ?N identity matrix, X is N ? 1 vector of the unknowns Xn, K
is the stiffness matrix, which is defined by
22
Figure 2.12: Mode shapes of the coupled oscillators array.
? ?
????2n0 + ? ?? 0 0 0 ????
??
? ?? ?2n0 + 2? ?? 0 0 ?
K= ???
??
???
?
0 ?? ?2 + 2? ?? 0 ????,n0
?? 0 0 ?? ?2 ?n0 + 2? ?? ???
0 0 0 ?? ?2n0 + ?
The experimental parameters that are shown in Table 4.1 are chosen for finding
the solutions for a coupled array of five oscillators and parameters as in the previous
subsection. The linear natural frequencies for the system are obtained as
?=37.0000, 37.0052, 37.0187, 37.0354, and 37.0489 rad/s
The corresponding mode shapes are shown in Figure 2.12.
23
2.6 Nonlinear Modes
To find the nonlinear modes of the system, the concept of nonlinear modes
is considered [77, 78] following a procedure that is similar to that used for a cyclic
coupled oscillator array [2]. To start the analysis, the unforced conservative system
of equation (2.2) is first considered
x?1 + ?
2
n0x1 + ?(x1 ? x2) + ?x31 = 0
x? + ?2 x + ?(2x ? x ? x ) + ?x3n n0 i n n?1 n+1 n = 0 for n = 2, 3, ..., N ? 1 (2.9)
x?N + ?
2
n0xN + ?(xN ? x 3N?1) + ?xN = 0
The solution of equation (2.9) is expressed using only one harmonic as
xn = Xnexp(i?t) + c.c., (2.10)
whereXn is a complex valued amplitude and ? is the the nonlinear natural frequency.
After substituting equation (2.10) into equations (2.9) and neglecting the higher
order terms, the following set of algebraic equations is obtained
(??2 + ?2n0 + ?)X1 + 3?X31 ? ?X2 = 0
(??2 + ?2n0 + 2?)Xn + 3?X3n ? ?(Xn?1?
(2.11)
Xn+1) = 0 for n = 2, 3, ..., N ? 1
(??2 + ?2 3n0 + ?)XN + 3?XN ? ?XN?1 = 0
The nonlinear modes of the system can be obtained by solving equation (2.11).
24
The solution of equation (2.11) is found using MANLAB package [79]. The details
of which are summarized in [80]. As it can be seen from equations (2.11), the
nonlinear modes of the system show the dependence between the amplitude and
frequency of the system. For frequency values close to the linear frequencies of the
system, the linear modes are considered good initial conditions for equations (2.11).
The obtained solution branches are shown in Figure 2.13 with solid blue lines. No
bifurcations are detected in all of the five branches. For low amplitudes, the shapes
of the nonlinear modes resembles the shapes of the corresponding linear modes.
However, as the amplitude increases, the energy gets focused in only a subgroup of
the oscillators, while the other oscillators loose the energy gradually. For sufficiently
high amplitudes, the energy of the system is completely localized in only a subgroup
of the oscillators. The only exception for this scenario happens for the in-phase
mode, wherein all the oscillators keep the homogeneity of the mode shape as the
amplitude increases. The final shape for the nonlinear modes corresponding to the
five linear modes are shown in Figure 2.14 .
To obtain the other nonlinear modes of the system, highly localized solution at
high frequency values are considered. Since the oscillators should vibrate at the same
frequency at the highly localized modes, the oscillators will either have similar non-
zero amplitudes or zero amplitudes. Thus, the three possibilities for each oscillator
are Xn = (?1, 0, 1). Toward this, there are 35 = 243 possible mode outcomes. Many
of these modes belong to the same family due to the axial symmetry or a change
in sign. After reducing the similar modes and the five corresponding modes of the
linear modes, only 65 different modes are left. These remaining modes are plotted
25
Figure 2.13: Nonlinear modes for the coupled oscillator array. The solid blue lines
are the nonlinear modes that are obtained from the linear modes. The dashed
red lines are the remaining modes obtained from the localized modes. The starting
points of different branches are shown in the top inset. The majority of the branches
are shown in the bottom inset.
in Figure 2.13 with dashed red lines. As it can be seen from the top zoomed portion,
there is no presence of those modes at low amplitudes. As the amplitude increases
more branches are born. Also, the majority of these branches bifurcate into one or
more than one branches. The bottom zoomed portion shows the majority of these
branches. Seven samples of the obtained localized solutions are shown in Figure
2.14.
26
Figure 2.14: The localized nonlinear modes obtained from solving equation (2.11).
The first five modes correspond to the linear modes of the system. The other seven
modes are representative examples of the 65 modes.
27
Chapter 3: Effects of Noise in the Multi-stability Region of Coupled
Oscillator Arrays
In this chapter, the effects of noise on the system response in the multi-stability
region of coupled oscillator arrays are discussed. The considered systems are sub-
jected to harmonic excitations with and without noise additions. First, the influ-
ence of noise is experimentally studied with an array of two coupled oscillators for
harmonic excitation values in the hysteresis region. The experimental results are
compared qualitatively to the numerical results obtained by using Euler Maruyama
simulations. The findings help understand the level of noise and the duration of
noise required to induce a change in the system dynamics as well as whether one
can create or destroy a response for frequency values in the multi-stability region.
3.1 Experimental Setup
An array consisting of two coupled oscillators is considered here. The experi-
mental setup is similar to the one shown in Figure 2.2, with only two oscillators in
the present case. The beam oscillator on the left is called the first oscillator and
the other one is called the second oscillator. The oscillators are tuned to realize a
multi-stability region in the same frequency range.
28
Figure 3.1: Quasi-static frequency sweep for an array of two coupled oscillators:
(a) the quasi-static frequency sweep of the first (blue) and second (red) oscillators
and (b) matching of the experimental results of the first oscillator (blue) with the
numerical solution (black) obtained for the parameters given in Table 3.1. The
unstable branch is depicted with dashed lines.
29
Table 3.1: Experimental results for the parameter values
Parameter Value Parameter Value
?n0 32.78 rad/s ? 225.5
F 17.95 ? 0.00787
? 1
Each oscillator?s response is normalized with the maximum response amplitude
value. A quasi-static frequency sweep is conducted with the system shows that the
responses of both of the oscillators to have multi-stability region for the range of
? =33.86 rad/s to ? = 35.22 rad/s as illustrated in Figure 3.1 (a). Localized modes
(LMs), in which the system response is localized in which the responses of one of
the two beam oscillators is on the high amplitude branch while the response of
the other beam oscillator is on the low amplitude branch exist within this range;
however, they are only obtainable when a certain disturbance is applied to the beam
oscillators. For simplicity, both oscillators are assumed to have similar properties,
as the first oscillator, when carrying out the numerical studies. The matching of
the first oscillator?s response from the experiments to the numerical solution of the
Duffing oscillator [81] with the parameters provided in Table 3.1 is shown in Figure
3.1 (b). In Figure 3.2, the time series of the beam oscillator responses for a quasi-
static frequency sweep up (i.e., increasing excitation frequency) and a quasi-static
frequency sweep down (i.e., decreasing excitation frequency) are shown. No LMs
are found during these sweeps.
30
Figure 3.2: Time series of the responses of the array system with two coupled
oscillators: (a) quasi-static frequency sweep up for the first oscillator, (b) quasi-
static frequency sweep up for the second oscillator, (c) quasi-static frequency sweep
down for the first oscillator, and (d) quasi-static frequency sweep down for the second
oscillator. The window where the system has multiple stable responses is marked in
this figure.
3.2 Numerical Scheme
Here, the scheme used for numerical simulations is described. In equations
(2.2), the derivative of Wiener process does not exist and to carry out the simula-
tions, these equations are converted to a Langevin form, which has an incremental
?
noise dW with zero mean and a standard deviation dt. The result is
31
dz1 = z2dt
dz2 = (?2??n0z2 ? ?2n0z1 ? ?z31 ? ?(z1 ? z3) + a cos(?t))dt+ ? dW (t)
dzn?2?3 = zn?2?2dt, for 2 ? n ? N ? 1
dzn?2?2 = (?2?? 2 3n0zn?2?2 ? ?n0zn?2?3 ? ?zn?2?3 ? ?(2zn?2?3 ? zn?2?1 ? zn?2?5)+
a cos(?t))dt+ ? dW (t), for 2 ? n ? N ? 1
dzN?2?1 = zN?2dtdzN?2 = (?2?? 2n0zN?2 ? ?n0zN?2?1 ? ?z3N?2?1 ? ?(zN?2?1 ? zN?2?3)+
a cos(?t))dt+ ? dW (t)
(3.1)
For two oscillators, equations (3.1) are reduced to
dz1 = z2dt
dz = (?2?? z ? ?2 z ? ?z32 n0 2 n0 1 1 ? ?(z1 ? z3) + a cos(?t))dt+ ? dW (t)
(3.2)
dz3 = z4dt
dz4 = (?2?? 2 3n0z4 ? ?n0z3 ? ?z3 ? ?(z3 ? z1) + a cos(?t))dt+ ? dW (t)
The solution is obtained by numerically integrating system (3.2) by using the Euler-
Maruyama scheme [82]. To obtain the localized modes, the authors have used the
anti-continuous limit based method(e.g., [83]). With this method, the uncoupled
system is considered first and the initial conditions are determined for the high
amplitude and the low amplitude responses of an individual oscillator. Then, after
introducing coupling and increasing the coupling strength gradually, and using the
shooting method with an initial guess determined from the uncoupled system, the
32
periodic solutions for the coupled system are found.
3.3 Results and Discussion
In this section, the influence of noise on system responses are explored for
different excitation frequency values, for which jumps between different solution
branches may occur. Two regions of interest, which are at either end of the multi-
stability region are explored.
3.3.1 Influence of Noise Near the Left Boundary of the Multi-Stability
Region
When the harmonic excitation frequency is close to the left side or the low
frequency end of the multi-stability region (MR), the experimental results obtained
are shown in Figure 3.3. At each frequency, the experiment has been run for twelve
times for 100 s time window for different noise intensities, starting from the low-low
mode, a mode in which both beam oscillators have a low response amplitude when
noise is not present. The root mean square (RMS) value over each period in each run
is calculated for the whole interval. The RMS value for the 12 runs is then averaged
and plotted for the chosen noise input and excitation frequency. The averaged RMS
value for each oscillator?s response is represented in color over the corresponding
time window with the blue color corresponding to low amplitude oscillations and
the red color corresponding to high amplitude oscillations. The horizontal black and
white bar on the top is used to show the time windows over which noise is applied.
33
In the present case, noise has been applied over the whole interval shown.
The two oscillators are set to vibrate at the low-low mode in the beginning of
the experiments. Then, noise has been applied and the two oscillators tend to move
toward the high-high mode, which is a mode in which each of the beam oscillators
have a high amplitude response in the absence of noise. Close to the left boundary
of the MR (?=33.82 rad/s), for ? = 0.5units, the oscillator responses jumped to
the high-high mode eight times with the average time being 77.5 s. In Figure 3.3 (a),
the author shows the experimental results for this case. As it can be seen, the color
gradient for both oscillators tend to the red color at the end of the time interval.
However, the responses still do not clearly reach a red color, indicating that jumps
did not occur in all runs. At the same frequency value, when the noise intensity
is increased to ? = 0.8units, the responses in all of the cases were found to have
moved to the high-high mode except in one occasion with the average time being
50.5 s. The results are plotted in Figure 3.3 (b), and it is clearly evident that the
response gradient color for both oscillators becomes red indicating almost all runs
induced the jump. As the noise intensity is increased, an observation is that the
time required to induce jumps is reduced. While the change in the response color
gradient for ? = 0.5units is limited in the first 50 s, there is a significant change
for ? = 0.8units.
Another experiment was conducted at ?=33.84 rad/s for two different noise
intensities, namely ? = 0.8units and ? = 1.2units, and the results are shown in
Figures 3.3 (c) and (d), respectively. While the noise intensity ? = 0.8units is found
to have a pronounced impact on the response for ?=33.82 rad/s, there is almost
34
Figure 3.3: Experimental results for harmonic excitation frequency near the left end
of the MR.
no impact on the response for ?=33.84 rad/s. Only in two out of the 12 runs, the
introduction of noise is found to induce jumps. However, for ? = 1.2units, the
responses of the oscillators jumped into the high-high mode for 8 cases, indicating
a noticeable effect on the low-low mode. The test are run further on ?=33.86 rad/s
for two noise intensities ? = 1.2units and ? = 1.5units, and these results are shown
in Figures 3.3 (e) and (f), respectively. As it can be seen from the response color
gradient, for ? = 1.2units only a small number of responses cases (4) have moved
into the high-high mode. On the other hand, for ? = 1.5units more number of runs
reveal a change in the system response dynamics (7 in this case).
35
Figure 3.4: Numerical results for harmonic excitation frequency near the left end of
the MR.
The numerical results are obtained by solving equation (3.2) using Euler-
Maruyam simulations. The average RMS value for the numerical simulation is ob-
tained by considering the RMS average over 50 runs by using a different noise vector
for each run. Similar ? values to the experimental tests are considered (?=33.82,
33.84, and 3.86 rad/s) for three different noise intensities (? = 0.5, 1.0, and 1.5
units ) for each ? value. For the first frequency ?=33.82 rad/s, successful jumps
are observed to be induced for all noise intensities with the differences in the time
required to induce that jump decreasing as the noise intensity was increased. For
?=33.84 rad/s, a low noise intensity almost has no impact on the system dynam-
ics, but the intermediate and high noise levels being sufficient to induce a mode
change with a transition time for the high noise level case being less than that for
the medium level. For the last ? value (?=33.86 rad/s),again no effect on the re-
36
sponse was observed when a low noise level was used. For the medium noise level,
the change in responses occurred in less trials than the observed for the previous
? value. Only the high noise level was found to have a discernible impact on the
low-low mode for this excitation frequency.
From the previous experimental and numerical results, one can note that the
level of noise and the required time to move the system from the low-low mode to
the high-high mode depend on the excitation frequency. Close to the left bound-
ary or low frequency end of the MR, the low-low mode is sensitive to disturbances.
Small disturbances move the system response from the low-low mode to the high-
high mode in a small amount of time. When the harmonic excitation frequency was
moved towards the center of the MR, the low-low mode was found to become robust
to disturbances. High noise intensities are found to be needed to induce a response
jump. The average time to induce a jump was found to increase as one moves toward
the center of the MR and this time needed was found to decrease when the noise
intensity was increased. It is clear that the experimental and numerical results agree
qualitatively. It is worth mentioning that when starting from the high-high mode
and applying comparable noise intensities no change was observed in the system
response dynamics.
37
3.3.2 Influence of Noise Near the Right Boundary of the Multi-Stability
Region
In Figure 3.5, the author has shown the experimental results obtained when
the harmonic excitation frequency is near to right end of the MR. Three different ?
values are chosen for the study, which are 35.2, 35.12, and 35.0 rad/s. Similar to the
previous experimental studies, for each parameter set, the test was run for twelve
times in 100 s time windows and the average of the RMS value over the 12 runs is
plotted. All runs are set to start from the high-high mode to explore the possibility
for changing the system dynamics to the low-low mode.
At ?=35.2 rad/s, the high-high mode was found to be quite sensitive for a low
level noise disturbance. For a low noise intensity (? = 0.25units), in eleven of the
twelve cases, the system response collapsed to the low-low mode with the average
time being 44.17 s. The associated dynamics is illustrated in Figure 3.5 (a). When
the noise intensity is increased to ? = 0.5units, in all of the twelve cases, the system
response collapsed within an average time of only 5.17 s, as illustrated in Figure 3.5
(b) . Moving toward the center of the MR, the effects of two noise intensities are
explored at ?=35.12 rad/s. These intensities are ? = 0.5units and ? = 0.8units,
and the corresponding results are depicted in Figures 3.5 (c) and (d), respectively.
For ? = 0.5units, only in two of the twelve cases, the chosen noise input induced
a transition from the high-high mode to the low-low mode, but by increasing the
intensity further to ? = 0.8units, in all the twelve cases, the response collapsed to
the low-low mode within an average time of 14.67 s. At the last chosen ? value
38
Figure 3.5: Experimental results obtained when the harmonic excitation frequency
is near the right end of the MR.
(?=35.0 rad/s), the high-high mode was found to become more robust to noise.
When applying noise with ? = 0.8units, almost no effect on the response mode was
observed. In none of the twelve cases, there was a change in the system dynamics,
as shown in Figure 3.5 (e). In order to have response jump downs occur, higher
noise intensity needed to be applied, as shown in Figure 3.5 (f) for ? = 1.2units.
It is clear that at the right side of the MR of identical oscillators, the oscillator
responses tend to move toward the low amplitude responses. Applying sufficient
noise can alter the system dynamics and move each oscillator?s response to the low-
amplitude response. On the other hand, at the left side of the MR, the oscillator
39
Figure 3.6: Numerical results obtained when the harmonic excitation frequency is
near the right end of the MR.
responses tend to move toward the high amplitude oscillations. Applying noise can
change the system dynamics and induce more high amplitude oscillations. Another
observation based on the results from Figures 3.3-3.6 is that the sensitivity of the
oscillators to disturbances for the right region of the MR is higher than the left
region. As a result, when moving from one frequency value to another in the right
of the hysteresis region, less noise intensity was found to be required than when is at
the left region and considering similar increment for the frequency. The reason for
this might related to the basin of attraction of an uncoupled oscillator; that is the
single Duffing oscillator. It is well known that for the Duffing oscillator the ratio of
the high amplitude attractor basin to the low amplitude attractor basin close to the
jump down frequency is much smaller than the ratio of the low amplitude attractor
basin to the high amplitude attractor basin close to the jump up frequency.
40
3.3.3 Basins of Attraction
To better understand the influence of disturbances on the system, the basin
of attractions near the left and right boundaries of MR are obtained and shown in
Figure 3.7. While the basins of attraction are usually illustrated for two-dimensional
systems, they can be found for a four-dimensional system. The procedure is similar
to the one used in the work of Ikeda et. al. [68]. To find the basins, the initial
conditions for a mode of interest (L-L, H-L, L-H and H-H) is selected as the starting
point for the numerical integration of equation by using the ode45 solver in Matlab
[84]. Next, a grid of 250*250 with (0,0) as the center point is defined as a disturbance
for the system. For each point, the first value is added for each displacement (x1
and x2) and the second point is added for each velocity (x?1 and x?2) to represent
deviations for the system. Equation (2.2) is then solved for two oscillators for each
point and the solution is traced after reaching the steady state solutions. Depending
on the final state, each point is colored as red (H-H), green (H-L), magenta (L-H) or
blue (L-L). In Figures 3.7 (a)-(c), the basins of attraction corresponding to Section
3.3.1 are shown for the three previously considered ? values,; that is, ?=33.82,
33,84 and 33.86 rad/s. The initial state for the system is on the L-L mode, and it is
represented by the black point and this is the state at which no deviations have been
applied. For ?=33.82 rad/s, when small deviations are applied, the system response
is found to return to the original state of the L-L mode, and this is represented by
the small blue region. However, if the deviation is relatively higher, the response no
longer return to the L-L mode but to the H-H mode. This is illustrated by the red
41
Figure 3.7: Basin of attraction for the system near the left boundary of the MR
starting from the L-L mode: (a) ?=33.82 rad/s, (b) ?=33.84 rad/s, and (c) ?=33.86
rad/s.
region. When one moves towards the center of the MR, the blue region is found to
expand further and the black dot is found to move more inside of the L-L mode, as
shown in Figures 3.7 (b) and (c). The previous results might give an indication for
what to expect in the response change after applying noise to the coupled oscillator
arrays. For ? values close to the left boundary, the L-L mode region is small in
size indicating how sensitive this mode is to a noise disturbance. As a result, small
disturbances might be enough to move the response mode into the H-H response
42
Figure 3.8: Basins of attraction for the system near the right boundary of the MR
starting from the H-H mode: (a) ?=35.2 rad/s, (b) ?=35.12 rad/s, and (c) ?=35.0
rad/s.
mode. As one moves toward the center, the blue region expands in size and higher
noise intensities would be needed to induce a change in the system dynamics.
For the right side of the MR, three ? values are considered as in Section 3.3.2,
which are: 35.2, 35.12 and 35.0 rad/s. The basins of attraction for this region are
shown in Figures 3.8 (a)-(c). For ?=35.2 rad/s, the H-H mode has a small basin
that is shown as a small red stripe in Figure 3.8 (a). When one moves towards the
center of the MR region, the red area expands in size as shown in Figures 3.8 (b)
and (c). These plots might give an indication for how robust the H-H mode is to a
43
disturbance in this region. When the red area is small in size, it is expected that
the H-H mode is vulnerable to a disturbance and the system response can change
with a low intensity disturbance. However, as one moves toward the center, the red
region is found to expand in size, and high noise intensities are needed to alter the
system dynamics. In the considered basin sets, there is no presence of basins of the
LMs responses. As a result, it is considered less likely to obtain LMs by introducing
noise to the system for the chosen initial conditions.
44
Chapter 4: Effects of Noise on Different Response Modes of Coupled
Oscillator Arrays
In this chapter, the influence of noise on different response modes of coupled
oscillator arrays is investigated. In the experimental part, arrays of two and three
coupled oscillators are investigated. The results are then compared with results
from numerical simulations obtained by using the Euler-Maruyama scheme. Fur-
thermore, the effects of coupling and changing the number of oscillators on the
system responses are discussed, in the presence of noise.
4.1 Experimental Setup
4.1.1 Two Oscillators
An array composed of two coupled-oscillators is set to have similar properties.
The time series for the system responses obtained by performing a quasi-static fre-
quency sweep is shown in Figure 4.1. For the quasi-static frequency sweep up, the
system response is on the H-H branch as shown for the first and second oscillators
in Figures 4.1 (a) and (b), respectively. When the critical excitation frequency of
?=38.97 rad/s is reached, the responses of both oscillators collapse in amplitude val-
45
Figure 4.1: Time series of the responses of the two coupled oscillators: responses
to quasi-static frequency sweep up for (a) first oscillator and (b) second oscillator,
and responses to quasi-static frequency sweep down for (c) first oscillator and (d)
second oscillator.
ues. During the quasi-static frequency sweep down, the system response is initially
on the L-L branch. As the frequency is swept down, the system response continues
on the L-L branch until one reaches the critical frequency value of ?=37.97 rad/s,
at which location, the response of both oscillators jump to higher amplitude values,
as shown in Figures 4.1 (c) and (d) for the first and second oscillators, respectively.
Localized modes within the considered frequency region exist, but they are
not attainable using the quasi-static frequency sweep for the system. In Figure
4.2, all the possible solution branches for the system are shown. To obtain the
46
Figure 4.2: Frequency responses of the two coupled oscillators: (a) all the four
branches of the four modes H-H, L-H, H-L and L-L arranged from top, (b)-(e) time
series of the first (blue) and second (red) oscillators at the marked red location (X)
in (a) for the different: (b) H-H, (c) L-H, (d) H-L, and (e) L-L.
47
Table 4.1: First oscillator parameters obtained from experimental data given in
Figure 4.2
Parameter Value Parameter Value
?n0 37.0 rad/s ? 270
F 18.5 ? 0.0063
? 1
L-H mode, the experiment is first set to run in the middle of the MR on the H-H
branch. Next, a disturbance is applied to the first oscillator to change its dynamics
from high amplitude oscillations to low amplitude oscillations. Then, a quasi-static
frequency sweep up is performed to obtain the right side of the branch. Next,
the same procedure is performed but with a quasi-static frequency sweep down to
obtain the left side of the branch. A similar procedure can be applied to obtain the
H-L mode, but with a disturbance applied to the second oscillator. Four possible
branches exist, which are H-H, L-H, H-L and L-L modes as arranged from the top
to bottom in Figures 4.2 (a). Here, the localized modes (L-H and H-L modes) have
different branches due to the imperfections in matching the two oscillators, which is
the case in real-life situations. These imperfections between the two beam oscillators
might appear due to the difference in the beam properties, magnets characteristics
or the boundary conditions. The time series obtained for the four possible modes
at ?=38.35 rad/s are plotted in Figures 4.2 (b) to (e).
From the experiment results, it is noted that the H-H branch is destroyed when
one reaches to the end of the high amplitude branch at ?= 38.97 rad/s and jumps
to the L-L branch, even though the LMs exist beyond this frequency, as shown in
48
Figure 4.1. For the numerical study, both oscillators are assumed to have the same
properties as the first oscillator. The matching of the experimental results with the
analytical approximation based on the parameters given in Table 4.1 is shown in
Figure 4.3.
4.1.2 Three Oscillators
For the experimental setup for three oscillators, an additional oscillator is
added next to the second oscillator. The first and third oscillators are each free at
one end and coupled with the second oscillator from the other end. The oscillators
are set to have comparable properties, since this helps to better understand the
Figure 4.3: Matching of the experimental results of the first oscillator shown in
Figure 4.2 (blue) with the analytical approximation for the parameters given in
Table 4.1 (black). Dashed lines are used for the unstable branch.
49
transition of energy in the MR. The time series for the responses obtained during the
quasi-static frequency sweep up and down are shown in Figures 4.4 (a)-(c) and (d)-
(e), respectively. When performing the quasi-static frequency sweep up, the system
response follows the H-H-H branch. At ?=38.40 rad/s, the first oscillator collapses to
a state of low amplitude oscillations while the other two oscillators responses remain
in a state of high amplitude oscillations. As the frequency sweep up is continued, the
second oscillator response falls to a state of low amplitude oscillations at ?=38.87
rad/s. The remaining (third) oscillator response moves to a state of low amplitude
oscillations at ?=39.83 rad/s. During the quasi-static frequency sweep down, the
system response starts on the L-L-L branch, and continues without change until one
reaches ?=37.58 rad/s, wherein all of the responses of the oscillators jump to a state
of high amplitude oscillations simultaneously.
All the solution branches for the system are shown in Figure 4.5 (a). As
it can be seen, responses on all LMs branches are not observed during the quasi-
static frequency sweep. Applying different disturbances, as explained in the two
oscillators case, and running the experiments with frequency sweep in the forward
and backward directions lead to obtaining of all the system response branches. Based
on the system energy, the response branches can be classified into four different
levels. The highest energy level is the one wherein all of the oscillators vibrate
in a state of high amplitude oscillations. The next level is when two oscillators
vibrate in high amplitude states, containing three different energy branches, which
are H-L-H, L-H-H, and H-H-L as arranged from top to bottom in Figure 4.5 (a).
The next level is when only one oscillator vibrates in a high amplitude state. This
50
Figure 4.4: Time series of the responses for the array with three coupled oscillators:
responses to quasi-static frequency sweep up for (a) first oscillator, (b) second oscil-
lator, and (c) third oscillator, and responses to quasi-static frequency sweep down
for (d) first oscillator, (e) second oscillator, and (f) third oscillator.
51
Figure 4.5: Frequency response of the system with three coupled oscillators: (a)
Eight branches of the eight modes H-H-H, H-L-H, L-H-H, H-H-L, L-H-L, H-L-L,
L-L-H, and L-L-L as arranged from top, (b)-(e) Time series of the responses of the
first (blue), second (red) and third (green) oscillators at the marked red location
(X) in (a) for: (b) H-H-H, (c) H-L-H, (d) L-H-H, (e) H-H-L, (f) L-H-L, (g) H-L-L,
(h) L-L-H, and (i) L-L-L. In plot (a), the L2 norm based on the response amplitudes
is plotted on the y-axis.
52
corresponds to the three branches L-H-L, H-L-L, and L-L-H as arranged from the
top. The last branch is for the lowest energy level, which is the L-L-L branch. Due
to the experiment imperfections, as explained in the two oscillators case, a difference
between the L-H-H and H-H-L modes branches is observed as well as between the
H-L-L and L-L-H modes branches. The time series for each response branch at
?=38.35 rad/s is shown in Figures 4.5 (b)-(i).
4.2 Effects of Noise on Different Modes of Coupled Oscillator Arrays
4.2.1 Near the Left End of the Multi-Stability Region
4.2.1.1 Two oscillators
Here, noise has been applied on the system of two coupled oscillators system
of Figure 4.2 on different modes. The study is first run near the left end of the
MR to investigate the increase of system energy, and then near the right end to
investigate the decrease of energy. Black arrows are included in Figures 4.6 (a) and
(b), to show the transition locations on the branches, for the responses of the first
and second oscillators, respectively.
To start the investigation, ?=38.2 rad/s is chosen, since it is close enough to
the left boundary of the MR and all response branches are present at this frequency;
that is, L-H, H-L, L-L, and H-H modes. For the L-H mode, applying noise at
? = 0.75units is enough to alter the system dynamics from the L-H mode to the
H-H mode, as can be seen from Figure 4.7 (a). However, applying a similar noise
53
Figure 4.6: System modes for (a) first oscillator and (b) second oscillator. The black
arrows indicate the expected transition in the response of an oscillator under the
influence of noise.
54
intensity to the H-L mode has almost no effects. To induce a change on this mode,
higher noise intensity needs to be considered such as ? = 1.5units, as can be seen
from Figure 4.7 (b). For the L-L mode, the dynamics is robust to noise, and no
change is found to occur for ? = 1.5units, see Figure 4.7 (c). It is expected that
noise can induce a change for high noise intensities, but this could not pursued due
to the experimental limitations in applying noise intensities (?) > 1.5units, which
could damage the experimental setup. However, this mode becomes more sensitive
to perturbations for ? values closer to the MR boundary, and moving to the high
amplitude oscillations is possible, as can be seen in Figure 4.7 (d) for ?=38.1 rad/s
and ? = 1.2units.
Numerical studies were also conducted. To obtain the numerical results, equa-
tion (3.2) is solved by using the Euler-Maruyama simulations for 50 different noise
vectors and the resulting RMS displacement average is plotted in Figure 4.8. Due
to the homogeneity of the two oscillators, the H-L mode is similar to the L-H mode.
As a result, only two modes are considered for the test, which are the H-L and
the L-L modes. When considering similar noise intensities (? = 2.5units) at the
same frequency value (?=38.2 rad/s), the H-L mode is found to be more sensitive
to the noise perturbation. With the H-L mode, in 49 tests, there was a successful
transition to the H-H mode compared to transitions observed in 34 tests with the
L-L mode. The summary of the results is plotted in Figures 4.8 (a) and (b) for the
H-L and L-L modes, respectively.
Through The results, it is shown that the influence of noise on the different
modes near the left boundary of the MR can be anticipated from the end of each
55
Figure 4.7: Experimental results for the effects of noise on different modes near the
left boundary of MR of the two coupled oscillators arrays shown in Figure 4.2.
response branch. The branch that has an end longer than the others is expected
to be more robust to noise. As can be seen from Figure 4.2 (a), the L-L mode has
an end at ?=37.97 rad/s compared to the frequency values of ?=38.18 rad/s and
?=38.19 rad/s for the H-L and the L-H modes, respectively. As a result, the modes
that are more vulnerable to change in system dynamics under the influence of noise
can be arranged in order as L-H mode, H-L mode, followed by the L-L mode.
56
Figure 4.8: Numerical results for the effects of noise on different modes near the left
boundary of MR for the array with two coupled oscillators shown in Figure 4.2: (a)
starting point is H-L mode and (b) starting point is L-L mode. The horizontal bar
on the top is indicative of the duration over which noise is applied. Here, noise is
applied throughout the window shown.
57
4.2.1.2 Three oscillators
A similar type of study has been made with a system consisting of three
coupled oscillators. Four different modes near the left boundary of the MR are
considered. These modes are H-L-H, L-L-H, H-L-L, and L-L-L, and they are depicted
in Figure 4.9 (a).
At ?=38.2 rad/s, the H-L-H and the H-L-L modes are found to be influenced
by noise, with the H-L-H mode requiring lesser noise intensity than the H-L-L mode,
as shown in Figures 4.10 (a) and (b). The other two modes are found to remain
robust to noise, even when high noise intensity is applied; that is, ? = 1.5units. As
with the two oscillators array, the mode that has a longer end in the MR is found
to be more robust to noise, as shown in the zoomed in portion of Figure 4.9 (a). As
noise intensities (?) > 1.5units might damage the experiment, the other two modes
are tested for another frequency value (?=37.8 rad/s) that is closer to the left end
of the MR. While the L-L-H mode responds to noise and a change is observed in the
resulting system dynamics, the L-L-L mode is found to be not responsive to noise;
the results are illustrated in Figures 4.10 (c) and (d). The L-L-L mode is found to
become sensitive to noise very close to the end of the L-L-L branch, as shown in
Figure 4.10 (g) at ?=37.61 rad/s .
In the numerical simulations, the system is studied at ?=38.2 rad/s with a
noise input of ? = 2.5units. Due to the homogeneity of the chosen oscillators, only
three modes are considered, namely, the H-L-H, H,L,L, and the L-L-L modes. As
observed in the experiments, the mode that has a shorter branch end in the MR is
58
Figure 4.9: Modes of interest of the system: (a) creation of response in the second
oscillator and (b) attenuation of response in the first oscillator. The black arrows
indicate the expected transition for each oscillator to follow under the influence of
noise.
59
Figure 4.10: Experimental results for the effects of noise on different modes near the
left boundary of MR for the array with three coupled oscillators shown in Figure
4.5.
found to respond quicker than the other modes. In most of the runs starting with
the H-L-H mode, a change is noted in the system dynamics within the first 100 s,
as shown in Figure 4.11 (a). With the H-L-L mode, the second oscillator response
jumps to a state of high amplitude oscillations in the most runs within about 300
s. With regard to the results of Figure 4.11 (c), when noise is applied to the L-L-L
mode over a similar time window, the system dynamics is found to be altered in
only 34 runs, which is the lowest among all the mode cases. This can be explain
60
Figure 4.11: Numerical results for the effects of noise on different modes near the left
boundary of MR for the system with three coupled oscillators arrays. The parameter
values are based on Table 4.1.
61
by noting that with this mode, one has the longest end in the frequency response
curve.
4.2.2 Near the Right End of the Multi-Stability Region
4.2.2.1 Two Oscillators
For the system in Figure 4.2, the results obtained by applying noise when the
harmonic excitation frequency is near the right side of the MR in the frequency
response curve are shown in Figure 4.12. Three modes are considered, namely,
the H-H, L-H, and H-L modes. From Figure 4.12 (a), the H-H mode at ?=38.96
rad/s is found to be vulnerable even when noise with a low intensity is applied,
such as, ? = 0.25units. The L-H mode is found to be responsive to a moderate
noise intensity, as shown in Figure 4.12 for ? = 1.2units. With regard to the H-L
mode, no change in the system dynamics is observed for all noise intensities less
than ? = 1.5units. This mode can collapse when noise with a strong noise intensity
is applied. However, this is not safe for the experiment. Alternatively, this can also
happen when one moves closer to the branch end, such as the excitation frequency
of 39.15 rad/s. The results obtained for this mode are shown in Figures 4.12 (c) and
(d). As with the previous study, a quasi-static frequency sweep can be used to get
a clue to how robust this system mode is. A response branch, which has a longer
end, is supposed to be more robust to disturbances.
Through the numerical studies, the author is not able to capture similar qual-
itative behavior as that observed in the experiments. This can be reasoned by
62
Figure 4.12: Experimental results for the effects of noise on different modes near
the right boundary of MR for the system of two coupled oscillators shown in Figure
4.2.
comparing the frequency response curves obtained from the experiments and the
numerical work. The LMs observed in the experiments have longer ends compared
to the response state in which all oscillators are in a high mode. However, in the
numerical results, they have shorter ends than that of the all high mode. As a result,
in the numerical study, the LMs are more vulnerable to a change under the influence
of noise, which is not in agreement with the experimental results. A better system
63
model is needed to predict the qualitative behavior within this region.
4.2.2.2 Three Oscillators
For the array with three coupled oscillators, four different modes are chosen
near to the right end of the MR, which are H-H-H, H-H-L, H-L-H, and H-L-L
modes, as shown in Figure 4.9 (b). The influence of noise on these modes is shown
in Figure 4.13. The results show that the H-H-H mode is the the most vulnerable
mode to perturbations. In Figure 4.13 (a), applying ? = 0.5units at ?=38.29 rad/s
is sufficient to induce a change in the system dynamics. As one moves down in the
energy branches, the modes become more robust to noise. A change in the H-L-H
mode is possible for higher noise intensities, as shown in Figure 4.13 (b). For the
last two modes, no change is noticed when applying noise at ? = 1.5units. One
needs to either apply higher noise intensities or move closer to the two modes ends
for a change to happen, as shown in Figures 4.13 (c)-(g).
The results agree well with the previous findings obtained with two oscilla-
tors. The mode that has the longest branch is found to be the most robust one to
disturbances. In the numerical part, the the author is not able to capture the mode
response change for the same reason as described in the previous section for the case
with two coupled oscillators.
64
Figure 4.13: Experimental results for the effects of noise on different modes near the
right boundary of MR for the system of three coupled oscillators shown in Figure
4.5.
4.3 Basins of Attraction
The basins of attraction for the three previous oscillators starting from H-H-
H, H-L-H, H-L-L, and L-L-L are generated as described in Chapter 2 and shown
in Figures 4.14 (a), (b), (c), and (d), respectively. For an initial condition in the
H-H-H mode, a large basin of the H-H-H mode is found surrounding the original
initial condition with only a relatively small sized basin for the L-L-L mode. This
65
result indicates the robustness of the H-H-H mode to noise at this frequency. For a
start from the H-L-H mode, it is found that the basin of this mode is surrounded
by the basin of the H-H-H mode in mostly all directions. Thus, if noise is applied to
response in the H-L-H mode, the middle oscillator response is most likely to jump
from a low amplitude state to a high amplitude state. For dynamics initiated from
the H-L-L mode, a small sized basin is found for the H-H-L mode surrounding the
basin of the H-L-L mode, and a large sized basin for the H-H-H mode covering
mostly the whole remaining region. This would mean that when noise is applied,
a transition from the H-L-L mode into the H-H-L mode, followed by a transition
to H-H-H is possible. For the last case, when the dynamics is initiated from the
L-L-L mode, the H-H-H mode basin is found to cover the whole region, which is
expected as one jumps from the L-L-L mode immediately to the H-H-H mode after
application of noise.
4.4 Effects of Coupling Strength on System Response
For low coupling strength to zero coupling, the system dynamics is similar
to that of individual Duffing oscillators discussed in Chapter 2. As the coupling
strength increases, the coupling between the oscillators is expected to influence the
system dynamics. Understanding how the coupling influences the dynamics could
be helpful in better utilizing noise to change the system dynamics. To that end,
the Van der Pol?s method is used to obtain the steady state solutions for different
coupling values [68]. To begin with, the solution of equation (2.2) for a set of two
66
Figure 4.14: Basins of attraction for the system of three coupled oscillators, when
starting from the following response modes: (a) H-H-H, (b) H-L-H, (c) H-L-L, and
(d) L-L-L.
coupled oscillators is assumed to have the following form:
xi = ui cos?t? vi sin?t for i = 1 and 2 (4.1)
where, ui and vi are unknown variab?les of time. The amplitude and phase of each
oscillator are represented by Ai = u2i + v
2
i and ? = vi/ui , respectively. The
system parameters are scaled using a book keeping parameter  as follows:
ui, vi = O(1), ?, ?, a = O() u?
2 4 (4.2)
i, v?i = O( ), u?i, v?i = O( )
67
After substituting equation (4.1) into equation (2.2) with consideration of the scal-
ing provided in equation (4.2), the governing equation for the approximate steady
solutions can be found, up to the accuracy of O(2), as follows:
2?u?1 = ?2??? 2 2 3 2 2n0u1 + ?v2 + (? ? ?n0 ? ?)v1 ? ?v (u + v )4 1 1 1
2?v?1 = ?2??? v ? ?u + (??2 + ?2 + ?)u + 3n0 1 2 n0 1 ?u (u21 1 + v24 1)? a
(4.3)
2?u?2 = ?2??? 2 2n0u2 + ?v1 + (? ? ?n0 ? ?)v2 ? 3?v (u2 + v24 2 2 2)
2?v?2 = ?2??? 2 2n0v2 ? ?u1 + (?? + ?n0 + ?)u 3 2 22 + ?u2(u2 + v2)? a4
To obtain the solutions for the different response branches from equation (4.3), the
initial condition of (1,0) is first assigned for each oscillator based on the H or L
amplitude assumptions. The solutions are then used in Auto software to continue
the solution along each branch. In Figures 4.15 (a)-(d), the solution of equation
(4.3) for the parameters shown in Table 4.1 for the first oscillator with the coupling
parameters ? = 1, ? = 3, ? = 5 , and ? = 20 are presented, respectively. Since the
numerical results do not show accurate results for the right boundary of the MR, the
focus will be only on the left boundary. For small coupling (? = 1), the ends of the
two LMs are just before the ends for the unison modes, wherein all oscillators have
the same response amplitudes. As the coupling strength is increased, the amplitude
gap between the LMs and the unison modes is found to increase. Increasing the
coupling strength further will make this gap larger with the inducement of a Hopf
bifurcation as shown for ? = 20.
In Figure 4.16, the bifurcation sets are shown in the ? - ? space. The solid
68
Figure 4.15: Effects of coupling increase on the four response branches of the set
of two coupled oscillators for the parameters provided in Table 4.1: (a) ? = 1, (b)
? = 3, (c) ? = 5, and (d) ? = 20.
black lines are the boundary of the unison mode with the region for the H-H and
the L-L modes illustrated by the black arrow. These modes are not affected by
the increase in the coupling strength. The LMs are bounded by the two green
curves, which is illustrated by the red arrow. For a high coupling strength, a Hopf
bifurcation is induced, and the loci of the Hopf instability points is shown by using
the red curve.
4.5 Effects of Increase in Number of Oscillators in an Array
In this section, numerical simulations are used to study how the increase the
number of oscillators affects the noise influenced responses. Arrays of one to five
69
Figure 4.16: Bifurcation sets in the (?, ?) plane for the system of two coupled
oscillators with the parameters listed in Table 4.1. The black lines are the boundaries
of the MR, the green curves are the boundaries of the LMs, the red curve is the loci
of Hopf bifurcation points, the blue curve is a stable branch near the left boundary
and the black dashed curves are unstable saddle points. The black and red arrows
represent the range for the MR and LMs, respectively.
oscillators are considered near to the left boundary (?=38.3 rad/s) of the MR of
the system with parameters as listed in Table 4.1. The corresponding modes are
are L, H(L), H(L)H, HH(L)HH and HHH(L)HHH. The goal is to track how the
responses of the low amplitude oscillators induce a transition to a state of the high
amplitude oscillations. The average RMS displacements, which were obtained with
50 noise vectors for all the selected modes at ? = 2.5 units, are shown in Figures
4.17 (a)-(e). For a single oscillator, it is clear that over a time window of 500 s,
in only a few trials (8 out of 50), the addition of noise is successful in inducing a
70
jump. When another oscillator in a state of high amplitude oscillations is coupled to
a low amplitude state oscillator, the number of trials in which a noise induced jump
occurs increases significantly. For the same time window, in only 8 of the 50 trials,
a jump did not occur. Next, two neighboring oscillators with a high amplitude state
are coupled to an oscillator in a low amplitude state. Within almost no time, the
responses of all of the oscillators jumped to a state of high amplitude oscillators.
When increasing the number of oscillators further, no significant change is observed
in the responses of the three oscillators. Through the numerical results, it is shown
that the two neighboring oscillators have a high impact on the middle oscillator?s
response sensitivity to noise. When they have different amplitude oscillations they
facilitate the transition of the middle oscillator?s response to another state. The
plots for all of the low amplitude oscillations in the previous cases as well as the
final state of the high amplitude oscillations are shown in Figure 4.18. From these
results, it can be noted that when an oscillator has neighboring oscillators with
different amplitude responses, the addition of noise gets the response closer to the
final H state, wherein all oscillators are in a high amplitude state. Beyond three
oscillators, the phase space plots are found to not change, and this is reflected in
the previous results.
71
Figure 4.17: Effects of noise on an oscillator in a state of low amplitude oscillations:
(a) L, (b) H-L, (c) H-L-H, (d) H-H-L-H-H, and (e) H-H-H-L-H-H-H
Figure 4.18: Phase space plots for the selected modes in Figure 4.17.
72
Chapter 5: Noise Based Control of System Energy of Coupled Oscil-
lator Arrays
In this chapter, possible noise based methods for controlling the system energy
of coupled oscillator arrays are discussed. Different responses are generated under
the influence of noise. Both experimental and numerical efforts are used here.
5.1 Influence of Noise Near MR Boundaries
The results of Chapter 2 are used as a clue to test a large array size. Hardening
and softening oscillators in arrays, each composed of 21 oscillators, are considered
for the study. All of the oscillators are assumed to be identical with the properties
of an oscillator as listed in Table 2.1. For the hardening case, the qualitative results
obtained for the influence of noise on a large oscillator array agree well with those
provided in Chapter 2 for a set of two coupled oscillators. Near the left boundary
or the low frequency end of the MR, all of the oscillators tend to move towards a
state of high amplitude oscillations under the influence of noise, while near the right
boundary or the high frequency end of the MR, the oscillators tend to move toward
towards a state of low amplitude oscillations, as can be seen from Figure 5.1. For
the case with softening oscillators, the opposite is true. Near the left boundary of
73
Figure 5.1: Numerical results for the effects of noise on 21 hardening Duffing oscilla-
tors for the parameters listed in Table 2.1: (a) near left boundary of MR (?=34.86
rad/s) and (b) near right boundary of MR (?= 36.72 rad/s). The dark horizon-
tal bar on the top is used to indicate that noise is applied throughout the chosen
duration.
74
Figure 5.2: Numerical results for the effects of noise on 21 softening Duffing oscilla-
tors for the parameters listed in Table 2.1: (a) near left boundary of MR (?=34.86
rad/s) and (b) near right boundary of MR (?= 35.39 rad/s). The dark horizon-
tal bar on the top is used to indicate that noise is applied throughout the chosen
duration.
75
the MR, the oscillators tend to move towards a state of low amplitude oscillations,
whereas near the right boundary, the oscillators tend to move towards a state of
high amplitude oscillations, as can be seen from Figure 5.2.
These findings can be understood by studying the basins of attractions for the
single Duffing oscillators, since they represent the building blocks for the oscillator
arrays. For the hardening type of oscillators, near the left boundary of the MR, the
basins of attraction of the high amplitude oscillations is larger in size than that of
the basin of the low amplitude oscillations. As perturbations are applied, the system
in all likelihood transitions to a state of high amplitude oscillations. Near the right
boundary of the MR, the basin of attraction for the high amplitude oscillations is
small in size compared to the basin of the low amplitude oscillations. Thus, it is
expected that the coupled oscillator arrays move from a state of high amplitude
oscillations to a state of low amplitude oscillations under the influence of noise. For
a system composed of softening type oscillators, the opposite behavior is expected.
5.2 Noise Induced Response Energy Level Increase
In all of the previous findings, applying noise for unison modes (all high or
all low) of homogeneous oscillators results in a system response wherein all of the
oscillators either jump to high amplitude oscillations or drop to low amplitude os-
cillations simultaneously. The oscillators could have different transition times from
one response to another, if there are imperfections/non-uniformity in the coupling
between the oscillators. To that end, two coupled oscillators are tuned experimen-
76
Table 5.1: First oscillator parameters obtained by matching the results from the
quasi-static frequency sweep shown in Figure 5.3.
Parameter Value Parameter Value
?n01 33.25 rad/s ?n02 33.15 rad/s
? 205 F 18.0
? 0.0075 ? 1
tally such that the oscillators have different jump up frequencies in the coupled
system. From the corresponding results shown in Figure 5.3 (a), it is clear that
during a quasi-static frequency sweep down the first oscillator jumps to a state of
high amplitude oscillations at ?=34.24 rad/s, whereas the second oscillator jumps
at ?=34.19 rad/s. The system is first set to the L-L mode at ?=34.26 rad/s, as can
be seen from Figure 5.3 (b). Noise is applied at t=113 s for about 13 s. Since this
frequency is close to the jump up frequency of the first oscillator, a localized mode
is created when the first oscillator response is induced a jump to a state of high am-
plitude oscillations. Stopping noise input at this time shows that the localized mode
persists as long as no further disturbances are applied. Noise is reapplied at t=210 s
for about 190 s. Due to the influence of noise, the second oscillator response jumps
as well to a state of high amplitude oscillations. As a result, noise can be used to
jump between the energy branches of the oscillator arrays when nonhomogeneities
exist.
Similar qualitative results are obtained by using numerical simulations. The
oscillator in the arrays are assumed to have properties close to that of first oscillator
in Figure 5.3 with the natural frequencies being the only difference between the two
77
Figure 5.3: Experimental results for noise induced gradual energy increase in a set
of two coupled oscillators: (a) a quasi-static frequency change for the system and the
responses for the first (blue) oscillator and second (red) oscillator and (b) increase
of energy in the first oscillator followed by that in the second oscillator. The dark
regions in the horizontal bar on the top in (b) are regions where noise is applied.
78
Figure 5.4: Numerical results for noise induced gradual energy increase in a set of
two oscillators: (a) quasi-static frequency sweep for the system and the responses
for the first (blue, OS1) oscillator and second (red, OS2) oscillator and (b) increase
of energy in the first oscillator followed by that in the second oscillator. The dark
regions in the horizontal bar on the top in (b) are regions where noise is applied.
79
oscillators, as shown in Table 5.1. The analytical approximations for the responses
of the uncoupled oscillators are shown in Figure 5.4 (a). Similar to the experiments,
noise is used to induce energy transition from the L-L response mode to the H-L
response mode and then to the H-H response mode.
5.3 Noise Induced Spatial Movement of an Energy Localization
As discussed previously, the hardening and softening oscillators arrays have
opposite properties. For this section, two oscillators are coupled together, with
the first oscillator having a softening character and the second oscillator having a
hardening character. The two oscillators are tuned so that the jump down in the
first oscillator?s response is near the jump up in the second oscillator?s response.
Results from the experimental quasi-static frequency sweep for the array is shown
in Figure 5.5 (a). At ?=34.86 rad/s, the oscillators are set to have a localized mode
in the first oscillator, as shown in Figure 5.5 (b). Noise is then applied for a time
duration of about 70 s. Due to the application of noise, the first oscillator response
is collapsed to a state of low amplitude oscillations, and the energy of the system is
decreased. Since the considered harmonic excitation frequency is close to the jump
up frequency for the second oscillator, a continued application of noise induces the
second oscillator to move to a state of high amplitude oscillations. Thus, noise
has been used to induce a spatial movement of energy in the considered coupled
oscillator array.
Similar results were obtained during numerical simulations. The properties of
80
Figure 5.5: Experimental results for noise induced spatial movement of energy in a
set of two coupled oscillators: (a) quasi-static frequency sweep for the system and
the responses for the first (blue) oscillator and second (red) oscillator, and (b) spatial
energy movement from the first oscillator to second oscillator.The dark regions in
the horizontal bar on the top in (b) are regions where noise is applied.
81
Figure 5.6: Numerical results for noise inducing spatial movement of energy in a set
of two coupled oscillators: (a) quasi-static frequency sweep for the system and the
responses for the first (blue, OS1) oscillator and second (red, OS2) oscillator, and
(b) spatial energy movement from the first oscillator to second oscillator. The dark
regions in the horizontal bar on the top in (b) are regions where noise is applied.
82
the first oscillator and the second oscillator are first obtained from the frequency
response curve, which is shown in Figure 5.6 (a), and the corresponding parameter
values are listed in Table 2.1. The oscillators are first set to have a LM in the first
oscillator. The application of noise results in a change in the spatial location of the
LM to the second oscillator, as shown in Figure 5.6 (b).
5.4 Influence of Noise on a Localized Mode and an Anti-localized
Mode
For a 21 oscillator array, the influence of noise on a LM and and anti-localized
mode (ALM) is numerically investigated. All of the oscillators are assumed to be
homogeneous with the parameter values listed in Table 2.1. For the LM case, a state
of high amplitude oscillations is created at ?=34.9 rad/s near the jump up frequency
for the middle oscillator, while all the other oscillators are vibrating in a state of low
amplitude oscillations, as shown in Figure 5.7 (a). When noise is introduced, the
two neighboring oscillators adjacent to the localized oscillator start to respond and
move to a state of high amplitude oscillations. As the noise input is continued, these
immediate neighboring oscillators respond and the result is a movement to a state
of high amplitude oscillations. This pattern continues resulting in energy cascades
until all of the oscillators vibrate in a state of high amplitude oscillations, as shown
in Figure 5.7 (b). In Figure 5.8, it is shown that different number of high amplitude
oscillations can be obtained by stopping the noise input at different times.
For the ALM, the oscillators are first set to have a low amplitude oscillation
83
Figure 5.7: Influence of noise on a LM near the left boundary of MR (?=34.90
rad/s): (a) no noise is applied and (b) ? = 1 units. In case (b), noise is applied
throughout the duration of interest.
84
Figure 5.8: Noise is stopped at different time instants in Figure 5.7 inducing increase
in the number of oscillators transitioning to a state of high amplitude oscillations
from one to three in (a) and eleven in (b). The length of the black horizontal bar
corresponds to the duration of the noise input.
85
only in the middle oscillator at ?=36.72 rad/s near the jump down frequency lo-
cation, as shown in Figure 5.9. The application of noise shows that there is no
transition resulting in drop for the oscillator responses to low amplitude oscilla-
tions. Instead, all of the oscillators drop simultaneously to a state of low amplitude
oscillations. As such, there is no noise induced transition between different energy
branches. Rather, there is only a drop to all low amplitude mode across the array.
86
Figure 5.9: Influence of noise on an ALM near the right boundary of MR (?=36.72
rad/s): (a) no noise is applied and (b) ? = 2 units. In case (b), noise is applied
throughout the considered duration as indicated by the black horizontal bar.
87
Chapter 6: Summary and Recommendations for Future Work
6.1 Summary
Here, a summary for the dissertation work and the key findings are discussed.
In this work, experimental and numerical studies have been carried out to
study the influence of noise on the responses of coupled oscillators arrays. In Chap-
ter 3, as an initial step, responses of an array of two coupled oscillators of the
hardening Duffing type are investigated through experiments and numerical stud-
ies. Responses to harmonic excitations in the presence and absence of noise in
the system input are studied. It has been shown that near the multi-stability re-
gion boundaries, the responses of the coupled oscillators arrays are highly sensitive
to noise. Close to each boundary, a low level noise intensity is found to induce a
change in the system dynamics. As one moves into the center of the frequency range
of the considered multistability region, a higher level noise intensity is found to be
needed to induce a response change. For the hardening case, near the left boundary
of the multi-stability region, the responses of the oscillators tend to move toward
high amplitude oscillations, whereas near the right boundary of the multi-stability
region, the responses of the oscillators tend to move toward low amplitude oscilla-
tions. A method for recognizing how much noise is required to induce a change in
88
the system dynamics is developed by using the basins of attraction. Starting from
a certain response mode, disturbances are applied to all of the oscillators and the
resulting steady state responses are tracked and plotted. The size of the initial mode
region is found to change with respect to the excitation frequency.
In Chapter 4, the effects of noise on different responses modes of two oscilla-
tor and three oscillator systems are discussed. For an array with two homogeneous
oscillators, near the left boundary of the considered multi-stability region, the ap-
plication of noise to a localized mode response is expected to induce a quick change
to a state of all high amplitude response modes. Near the right boundary of the
considered multi-stability region, the experimental results reveal that the applica-
tion of noise to a response state with all high amplitudes has discernible influence
on the response. The frequency response curves can be used to predict how sensitive
each mode is compared to the other modes. While the numerical simulations are
found to predict accurately the results close to the left boundary of a multi-stability
region, the same is not true near the right boundary of a multi-stability region.
Through different studies in Chapter 4, the effects of coupling strength as
well as the number of oscillators are investigated. With regard to the coupling, it
has been shown that increasing the coupling facilitates a change in system dynamics
under the influence of noise. From the study on the number of oscillators, it is noted
that the influence of noise on the response of an oscillator depends highly on the
character (i.e., hardening or softening characteristic) of the neighboring oscillators.
The oscillators far from the location almost have no effect except for the neighbors.
In Chapter 5, noise input based control of the energy of coupled oscillators
89
arrays have been discussed. Noise is added to a harmonic excitation to move a
high number of oscillators in an array from a high amplitude response state to a
low amplitude response state and vice versa. It is shown that noise can be used
to gradually move the system response up in the energy branches for an array of
non-homogeneous oscillators. Also, the use of noise to transfer the energy from
a softening oscillator to a hardening one has been illustrated. An interesting phe-
nomenon for noise assisted shifting of energy from one oscillator to a neighboring has
been presented. A cascade of energy increase was shown to be possible by applying
noise to a localized response mode, when the excitation frequency is close to the left
boundary of a multi-stability region. However, near the right boundary, through
the numerical results, it is shown that the application of noise can destroy responses
of high amplitude oscillations in all oscillators at once, which is not in conformity
with the experimental results. The experimental results presented in the different
chapters for noise influenced responses of arrays of coupled nonlinear oscillators are
the first of their type.
6.2 Recommendations for Future Work
As discussed earlier, while the system model presented here can be used to
accurately predict the experimental behavior near the left boundary of a multi-
stability region, the same is not true for the right boundary of a multi-stability
region. Thus, a better model is needed to obtain noise influenced responses of
coupled oscillators arrays.
90
The focus of this work has been on arrays with monostable oscillators. How-
ever, the system responses can be more complicated if one were to consider other
types oscillators, for example bi-stable oscillators. Also, based on few trials in this
work, it is found that in the presence of imperfections and/or different types of
oscillators in the considered array, one can observe interesting changes in the dy-
namics. Further studies are recommended to study these aspects, as they would be
important for practical systems.
Here, control has been used in a limited sense to illustrate the potential of noise
to change the system dynamics. However, one needs to consider the application of
control in a broad sense. For example, it is still unclear as to what is the right
duration and level of needed noise input to reach the desired final state. This
prompts a recommendation to look into this further.
Some methods have been used to explore the possible potential for noise in-
fluence, including the basins of attraction and the length of the branches in the
frequency-response curves. However, through these methods, one is only accounting
for limited information. Work on other methods for high-dimensional systems is
recommended.
Finally, different questions also suggest additional directions for future work.
For example, how will the system behave with other (practical) noise types than
the white noise? How does the system respond to noise when damping coupling
is added? How will the results obtained after including the damping nonlinearities
in the system model compare with those from the current model? How can one
combine noise and controlled imperfections to influence the system dynamics?
91
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