DYNAMICS OF NONLINEAR SNAP–THROUGH CHAINS
WITH APPLICATION TO ENERGY HARVESTING AND
WAVE PROPAGATION
By
Smruti Ranjan Panigrahi

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Mechanical Engineering – Doctor of Philosophy

2014

ABSTRACT
DYNAMICS OF NONLINEAR SNAP–THROUGH CHAINS
WITH APPLICATION TO ENERGY HARVESTING AND
WAVE PROPAGATION
By
Smruti Ranjan Panigrahi
There is much current research interest in nonlinear structures, smart materials, and
metamaterials, that incorporate bistable, or snap-through, structural elements. Various
applications include energy harvesting, energy dissipation, vibration absorption, vibration
isolation, targeted energy transfer, bandgap design and metamaterials.
In this dissertation, we explore snap-through structures with nonlinearity and negative
linear stiffness. We start with a study of a simple Duffing oscillator with snap-through
orbits around the separatrix. Multi-degree-of-freedom snap-through structures are known to
convert the low-frequency inputs into high-frequency oscillations, and are called twinkling
oscillators. A generalized two-degree-of-freedom (2-DOF) snap-through oscillator is shown to
have rich bifurcation structure. The steady-state bifurcation analysis uncovered two unique
bifurcations “star” and “eclipse” bifurcations, named due to their structures. The 2-DOF
twinkler exhibits transient chaos in the snap-through regime. A fractal basin boundary
study provides insight into the regions in the parameter space where the total energy level
is predictable in an unsymmetric twinkler.
Due to its capacity to convert low frequency to high-frequency oscillations, the snapthrough oscillators can be used to harvest energy from low-frequency vibration sources.
This idea has led us to explore the energy harvesting capacity of twinkling oscillators. Using
magnets and linear springs we built (in collaboration with researchers at Duke university)

novel experimental twinkling oscillators (SDOF and 2-DOF) for energy harvesting. When the
magnets exhibit high-frequency oscillations through the inducting coil, a current is generated
in the coil. This experiment shows promising results both for the SDOF and the 2-DOF
twinkling energy generators by validating the frequency up-conversion and generating power
from the low-frequency input oscillations. The experimental twinkling oscillator converted a
0.1 Hz input oscillation into 2.5 Hz output oscillation, a 25 times frequency up-conversion.
The second part of this dissertation focuses on the dispersive nature of the waves in
one dimensional nonlinear chains with weak nonlinearity. For metamaterial design, it is
important to study the wave dispersion properties in the material for channeling energy in a
desired direction or to build frequency-selective materials. In nonlinear structures there are
various design parameters that can be tuned to produce desirable properties. The motivation
of the wave propagation analysis is to understand the quadratic and cubic nonlinearity
effects on the wave propagation behavior in an uniform periodic chain. Here the dispersion
properties are studied through a multiple-scales perturbation approach for weakly nonlinear
periodic media. Wave speed, cut-off frequencies, and wave-wave interaction characteristics
are presented. The results show significant effect of quadratic nonlinearities in the dispersion
characteristics of the waves in the chain.

To the two strongest women in my life,
my beautiful and loving wife Jovy, and my loving and caring mother Sabitri

iv

ACKNOWLEDGMENTS

I would like to express my deepest appreciation and gratitude to my advisor, Prof. Brian
Feeny for guiding, supporting, and encouraging throughout my doctoral study at Michigan
State University. I would like to thank him for his unconditional availability and scientific
inputs in every step of the research. I am grateful for his efforts in providing me with
opportunities to present and participate at various conferences and symposia where I gained
a great deal of scientific exposure and knowledge. I deeply appreciate his time and patience
for numerous technical discussions which led to accomplish major milestones in my research.
Without his guidance and persistent help this dissertation would not have been possible.
Special thanks to my co-advisor, Prof. Alejandro Diaz for his support in letting me
choose a research topic of my interest and supporting me throughout my doctoral program.
His research inputs and advice throughout my graduate school have made this work possible.
I would like to deeply thank my dearest wife, Jovy Panigrahi for her unconditional love
and emotional support throughout the strenuous graduate school and keeping me focused
and grounded. I am grateful for her immeasurable selfless love and for being by my side
every step of the way. Her extraordinary support and selfless love and care has made the
past four years at Michigan State, the best years of my life.
I am very grateful to my family, my first teacher on Earth my mother Mrs. Sabitri Panigrahi, and my brothers Satya Bhusan Panigrahi, Chittaranjan Panigrahi and Rashmi Ranjan
Panigrahi for their unconditional love, invaluable support and encouragement throughout my
life. Especially, I will forever be grateful to my brother Satya, for his training that was instrumental in building a strong foundation in mathematics and physics, which helped shape
my educational attainment and career. Without my family I would not be what I am today.
v

I would also like to thank Prof. Brian Mann of Duke University, for his efforts in collaborative work and making his laboratory and equipments available to me to conduct experiments on energy harvesting. Special thanks to Dr. Brian Bernard, for helping set-up the
experiments at Duke University and sharing his knowledge of experimental work. It was my
great pleasure of working with and mentoring the undergraduate assistant, Lucas Steele in
the Mechanical Engineering department at Michigan State University.
I am thankful to my PhD committee members Prof. Steven Shaw, and Prof. Robert
McGough for their time and insightful thoughts which were invaluable for the completion
of this thesis. I would also like to thank Mrs. Aida Montalvo and Mrs. Suzanne Kroll, for
their outstanding administrative support and making my graduate study in the Mechanical
Engineering Department less hassle and more enjoyable experience.
Finally, I would like to thank the National Science Foundation (Grant No. CMMI1030377) for their financial support towards my graduate research in the Department of
Mechanical Engineering at Michigan State University.

vi

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Chapter 1 Introduction . . . .
1.1 Motivation . . . . . . . . .
1.2 Objective . . . . . . . . .
1.3 Background . . . . . . . .
1.3.1 Energy Transfer . .
1.3.2 Wave Management
1.4 Dissertation Organization

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Chapter 2 Harmonic Balance Analysis of Snap-Through Orbits
damped Duffing Oscillator . . . . . . . . . . . . . . . .
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Harmonic Balance Approximations . . . . . . . . . . . . . . . .
2.3.1 Three-Term Harmonic Balance Approximation . . . . . .
2.3.2 Fourier Series Limiting Case . . . . . . . . . . . . . . . .
2.4 Analytical and Numerical Results . . . . . . . . . . . . . . . . .
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 3 Bifurcations in Snap-Through Twinkling Oscillators . .
3.1 Equations of Motion and Stability Analysis . . . . . . . . . . . . . .
3.2 Examples of Bifurcations of Equilibria in Snap-Through Oscillators
3.3 Bifurcations in SDOF Snap-Through Oscillator . . . . . . . . . . .
3.3.1 Global Equilibria . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Local Normal Form and Stability Analysis . . . . . . . . . .
3.4 Bifurcations in 2-DOF Snap-Through Oscillators . . . . . . . . . . .
3.4.1 Global Equilibria . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Star Bifurcations . . . . . . . . . . . . . . . . . . . . . . . .
3.4.3 Eclipse Bifurcation . . . . . . . . . . . . . . . . . . . . . . .
3.4.4 Energy Levels and Global Perturbations . . . . . . . . . . .
3.5 Transient Chaos Analysis Through Fractals . . . . . . . . . . . . . .
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

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Chapter 4 Experimental Energy Harvesting Using Twinkling Oscillators . 63
4.1 Experimental Setup and Parameters for the Twinkling Energy Generator (TEG) 65
4.1.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.1.2 Parameter Values of the TEG . . . . . . . . . . . . . . . . . . . . . . 67
4.2 Mathematical Model of the Twinkling Energy Generator . . . . . . . . . . . 71
4.2.1 Force Due to Linear Springs . . . . . . . . . . . . . . . . . . . . . . . 72
4.2.1.1 Force Due to Linear Springs in SDOF TEG . . . . . . . . . 72
4.2.1.2 Force Due to Linear Springs in 2-DOF TEG . . . . . . . . . 73
4.2.2 Force Due to Magnets: Model I . . . . . . . . . . . . . . . . . . . . . 74
4.2.3 Force Due to Magnets: Model II . . . . . . . . . . . . . . . . . . . . . 75
4.2.4 Force Balance in Idle Steady State . . . . . . . . . . . . . . . . . . . 78
4.2.5 Governing Equations of Motion of the TEG . . . . . . . . . . . . . . 79
4.2.5.1 Equations of Motion of the SDOF TEG . . . . . . . . . . . 80
4.2.5.2 Equations of Motion of the 2-DOF TEG . . . . . . . . . . . 80
4.3 Numerical Simulation of the TEG . . . . . . . . . . . . . . . . . . . . . . . . 81
4.3.1 Numerical Simulation of the SDOF TEG . . . . . . . . . . . . . . . . 85
4.3.2 State-Space Representation of the 2-DOF TEG for Numerical Simulation 89
4.3.3 Energy and Power Computations . . . . . . . . . . . . . . . . . . . . 96
4.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.4.1 Experimental Results of the SDOF TEG . . . . . . . . . . . . . . . . 98
4.4.2 Experimental Results of the 2-DOF TEG . . . . . . . . . . . . . . . . 100
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Chapter 5 Wave Propagations in Discrete Periodic Weakly Nonlinear Chains104
5.1 Discrete Periodic Chain of Nonlinear Oscillators . . . . . . . . . . . . . . . . 105
5.2 Single Wave Dispersion Analysis Using Second-Order Multiple Scales . . . . 108
5.2.1 Traveling Wave Solutions in Strain Coordinates . . . . . . . . . . . . 108
5.2.2 Frequency Expression in Displacement Coordinates . . . . . . . . . . 113
5.2.3 Comparison in the Continuum Limit . . . . . . . . . . . . . . . . . . 114
5.3 Single Wave Dispersion Analysis in Continuum Limit . . . . . . . . . . . . . 115
5.3.1 Partial Differential Equations for Dispersion Analysis . . . . . . . . . 115
5.3.2 Second-Order Multiple Scales Analysis in Continuum Limit . . . . . . 117
5.4 Wave Dispersion Analysis Using Third-Order Multiple Scales with O(1) DC-Bias122
5.4.1 Analysis in Strain Coordinates with O(1) DC Term . . . . . . . . . . 122
5.4.2 Transforming Dispersion Relations from Strain into Displacement Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.5 Wave Dispersion Analysis Using Third-Order Multiple Scales with O( ) DC-Bias130
5.5.1 Analysis in Strain Coordinates with O( ) DC Term . . . . . . . . . . 130
5.5.2 Transforming Dispersion Relations from Strain into Displacement Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.6 Multi-Wave Interactions in an Infinite Nonlinear Chain Using First-Order
Multiple Scales Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.6.1 Multiple Scales Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.6.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.7 Analytical and Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 144
viii

5.8

5.7.1 Wave Dispersion in Nonlinear Periodic Chain . . . . . . . .
5.7.2 Wave-Wave Interactions in Periodic Nonlinear Chain . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8.1 Effects of Quadratic Nonlinearity in Single Wave Dispersion
5.8.2 Effects of Quadratic Nonlinearity in Wave-Wave Interactions

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Chapter 6 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . 158
6.1 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.1.1 Bifurcations and Transient Chaos in Twinkling Oscillators . . . . . . 158
6.1.2 Experimental Nonlinear Energy Harvesting . . . . . . . . . . . . . . . 159
6.1.3 Traveling Waves in Nonlinear Oscillators . . . . . . . . . . . . . . . . 160
6.2 Ongoing Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.2.1 Exploratory Study of Wave Propagation in Snap-Through Periodic
Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.2.2 Snap-Through Solitary Wave Equation Formulation . . . . . . . . . . 162
6.2.3 Analytical Wave Solutions . . . . . . . . . . . . . . . . . . . . . . . . 166
6.2.3.1 Solutions of Snapping Waves Using Harmonic Balance Method166
6.2.3.2 Dispersive Solitons and their Space-Time Modulation . . . . 168
6.2.3.3 Comparison with a Linear Chain . . . . . . . . . . . . . . . 170
6.2.4 Summary and Questions . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.3 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.3.1 Bandgaps in Weakly Nonlinear Chains with Quadratic and Cubic Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.3.2 Large Amplitude Waves in Snap-Through Twinkling Chains and its
Experimental Investigation . . . . . . . . . . . . . . . . . . . . . . . . 173
6.3.3 Inverse System Identification and Perturbation Analysis of the Experimental TEG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.3.4 Wave Propagation in Two–Dimensional Nonlinear Snap– Through Periodic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
APPENDICES . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . 175

Appendix AConditions for a General Quadratic Curve to be an Ellipse . . 176
Appendix BHarmonic Balance Solutions of the Snap–Through Wave Equation178
BIBLIOGRAPHY

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ix

LIST OF TABLES

Table 2.1

Table 2.2

Table 2.3

Table 4.1

Table 4.2

Table 4.3

Three-term harmonic balance solution look-up table for solutions
closer to the separatrix. . . . . . . . . . . . . . . . . . . . . . . . . .

25

Three-term harmonic balance solution look-up table for large amplitude oscillation farther from the separatrix . . . . . . . . . . . . . .

26

Numerical solutions look-up table for snap-through oscillation outside
the separatrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

Weight of the carts and spring parameter values used in the experimental twinkling energy generator (TEG). . . . . . . . . . . . . . .

71

Magnet parameter values used in the experimental twinkling energy
generator (TEG). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

Electrical circuit parameter values used in the experimental twinkling
energy generator (TEG). . . . . . . . . . . . . . . . . . . . . . . . .

73

Table B.1

Harmonic balance solutions look-up table for snap-through waves corresponding to equation 6.11 for solutions closer to the separatrix. . . 178

Table B.2

Harmonic balance solutions look-up table for snap-through waves corresponding to equation 6.11 close to the steady-state equilibrium. . . 179

x

LIST OF FIGURES

Figure 2.1

Nondimensional spring mass Duffing oscillator. . . . . . . . . . . . .

11

Figure 2.2

This plot approximates the solution very near and outside the separatrix. The period T is infinite on the separatrix. As the periodic
orbit approaches the separatrix, the peaks can be approximated by
Dirac delta dunctions in the derivation of the Fourier series. . . . . .

15

Various solutions using harmonic balance and numerical approaches
at x0 = 1.65062, x0 = 1.50675, and x0 = 1.45933 corresponding to
k5 = 0.01, k5 = 0.02 and k5 = 0.03 respectively. . . . . . . . . . . . .

17

Phase plot for x0 = 1.50675 corresponding to k5 = 0.02. The threeterm solution results in more accurate solution as compared to the
two-term solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

Frequency vs. the leading-order amplitude from harmonic balance
approximation with one, two, and three odd-harmonic terms. . . . .

19

Zoomed-in view of the frequency vs. the leading-order amplitude
from harmonic balance approximation with one, two, and three oddharmonic terms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

This plot shows a comparison of the amplitudes in one-, two-, and
three-term solutions as the frequency changes. Here the amplitudes
of the harmonics are shown in a logarithmic-scale. C1 is the amplitude of the first harmonic in one-term approximation, B1 , B3 are the
amplitudes of the first and third harmonic in two-term approximation, and A1 , A3 , and A5 are the amplitudes of the first, third and the
fifth harmonics in three-term approximation. As we approach from
the right on the frequency axis i.e. approaching the separatrix on the
phase space, we see that the minimum frequency that is captured
with the truncated Fourier series varies depending on the number
of harmonics in the HB solution. The limits on the frequencies reduce with added harmonics for which the approximate solution of the
Duffing oscillator can be found . . . . . . . . . . . . . . . . . . . . .

21

Figure 2.3

Figure 2.4

Figure 2.5

Figure 2.6

Figure 2.7

xi

Figure 2.8

Figure 2.9

Figure 3.1

Figure 3.2

Figure 3.3

Figure 3.4

This plot shows the relationship between the amplitude of the first,
third, and fifth harmonics versus the initial condition x0 . The blue
line corresponds to a solution with three odd harmonics and the black
line corresponds to a solution with two odd-harmonics. We see that
for an initial condition x0 greater than 1.6, the amplitude ratio of the
third harmonic (k3 ) doesn’t change much for the two-term harmonic
balance solution compared to that of the three-term solution. The
amplitude ratios of the fifth to the leading order harmonics (k5 ) shows
the weakening of the amplitude of the fifth harmonic as we go farther
away from the separatrix. . . . . . . . . . . . . . . . . . . . . . . .

22

The frequency ω is plotted versus the initial condition x0 for approximate solutions with one, two, and three odd harmonics. The
plot suggests a linear linear trend when x0 is large. The numerical
results (red dots) confirm the accuracy of the harmonic balance solutions (solid lines) with three odd harmonics. The ω limits show that
the three-term harmonic approximation captures solutions closer to
the separatrix, and of lower frequency, than the one- and two-term
solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

n-degree-of-freedom spring-mass chain connected by n masses, (n+1)
nonlinear springs, and n dash-pots. As shown in this figure the left
spring is fixed to a base and the right most spring is pulled quasistatically to a distance y0 . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

The characteristic spring force fi (si ) of the nonlinear spring as a
function of the spring deformation si , and the quasistatic pull as a
function of time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

SOF spring-mass chain connected by one mass, two nonlinear springs,
and one dash-pots. As shown in this figure the left spring is fixed to a
base and the right most spring is pulled quasistatically to a distance
y0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

The global bifurcation behavior of a SDOF system with respect to
the pull parameter y0 . (a) Bifurcations of equilibria in symmetric
system. (b) Bifurcations in symmetry breaking system with an applied perturbation a2 = a1 + δ and b2 = b1 + δ, for a1 = 0.5, b1 = 3,
and δ = 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

xii

Figure 3.5

Figure 3.6

Figure 3.7

Left figure is the bifurcation diagram for the symmetric case in the
transformed coordinate system. The bifurcation diagram shows qualitatively similar bifurcation behaviors at the bifurcation points B1
and B2 . The next two figures show two different configurations of
the global bifurcations in the perturbed SDOF system. The second
and third figure correspond to perturbed spring forces in the transformed system where the right hand side of the equation (3.15) are
(−u(4u2 − 4y1 p + 3p2 ) − ) and (−u(4u2 − 4y1 p + 3p2 ) − p + 2 )
respectively for = 0.1. The symmetry breaking at the bifurcation
points show the expected perturbations of a pitchfork bifurcation that
breaks into the saddle-node bifurcation. . . . . . . . . . . . . . . . .

38

These figures represent the total spring potential energy of the SDOF
system at the final equilibrium state. Two overlapping energy levels
from the symmetric case in the negative stiffness region are unfolded
with an applied perturbation in the right figure, where a2 = a +
and b2 = b + , for = 0.1. . . . . . . . . . . . . . . . . . . . . . . .

39

2-degree-of-freedom spring-mass chain connected by 2 masses, 3 nonlinear springs, and 2 dash-pots. The left spring is fixed to a base and
the right most spring is pulled quasistatically to a distance y0 . . . .

40

Figure 3.8

The bifurcation diagram for the equilibrium solutions of the lightly
damped symmetric 2-DOF system with respect to the pull parameter y0 , where B1 , B2 , . . ., B10 are the bifurcation points. The dashed
lines represent unstable solutions, and the solid lines represent the
stable equilibrium solutions (neutrally stable for the undamped system). The vertical dotted lines show infinitely many solutions at
y0 = a + b, where at the bifurcation points B7 − B10 , two of the
four eigenvalues are complex conjugates with zero real parts and the
other two are zeros for undamped system, whereas with light damping there is one zero, one purely real negative, and the other two are
complex conjugate eigenvalues with negative real parts. The bifurcation points B3 − B6 , are saddle-nodes with two zero and two complex
conjugate eigenvalues with zero real parts for undamped system, and
with light damping there are two complex conjugate eigenvalues with
negative real parts, one zero, and one purely real negative eigenvalues. 42

Figure 3.9

The bifurcation diagram for the equilibrium solutions (u1 , u2 ) in terms
of the pull parameter p are qualitatively similar to the Figure 3.8,
hence the stabilities of the solution curves and the degree of degeneracy of the bifurcation points are inferred. . . . . . . . . . . . . . .

xiii

43

Figure 3.10

Figure 3.11

Figure 3.12

Figure 3.13

Figure 3.14

Figure 3.15

The equilibrium solutions projected onto the u1 − u2 plane and noted
as Ri = (u1 , u2 )i . The solid and dashed ellipses and straight lines
satisfy g1 (u1 , u2 , p) = 0 and g2 (u1 , u2 , p) = 0 respectively. As p
approaches zero from both directions the equilibrium solutions R1 ,
R2 and R3 converge into R0 . For p < 0 the points R1 , R2 and R3
are unstable and become stable when p > 0. R0 changes from stable
to unstable as p goes from negative to positive. The stabilities of the
points R4 , R5 and R6 remain stable on both sides local to p = 0. . .

44

The vertical axis represents the eigenvalues λ at the equilibria in the
normal form at the star bifurcation local to pˆ = 0. The points R1 ,
R2 and R3 remain unstable whereas the stability of R0 changes from
stable to unstable as pˆ goes from negative to positive. . . . . . . . .

46

Symmetric case of the star bifurcation is shown on the top, and the
bottom figures show the symmetry breaking of the star bifurcation,
ˆ 1 + = 0, h
ˆ 2 = 0}. With reference to the symmetric case,
where {h
overlapping projected branches are revealed. . . . . . . . . . . . . .

47

ˆ 1 − = 0, h
ˆ 2 = 0}, where the
(top) The perturbation is such that {h
branches with positive slope on (ˆ
p, uˆ1 ) plane are both two distinct,
overlapping projected branches that is separated in the projection on
ˆ 1 − = 0, h
ˆ 2 + = 0}, where all the
the (ˆ
p, uˆ2 ) plane. (bottom) {h
branches in (ˆ
p, uˆ1 ) and (ˆ
p, uˆ2 ) planes are revealed unfolding the star
bifurcation into saddle nodes. . . . . . . . . . . . . . . . . . . . . . .

48

ˆ 1 + = 0, h
ˆ 2 − = 0}, star breaks into pitchfork
In top figures {h
bifurcations on both the uˆ1 and uˆ2 plane. Whereas in the bottom
ˆ 1 − = 0, h
ˆ 2 − = 0}.
figures it breaks into saddle-nodes, where {h
Here the symmetry breaking on (ˆ
p, uˆ1 ) plane is qualitatively similar
ˆ
ˆ 2 + = 0} shown in Figure 3.13.
to the perturbation case {h1 − = 0, h
However this presents a different symmetry breaking configuration on
(ˆ
p, uˆ2 ) plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

The state-variable eigenvalues α0 of the symmetric damped (with
two dampers) 2-DOF system near the star bifurcation in the firstorder form at equilibrium R0 . Single and double arrows represent
slow and fast approach respectively to show relative dynamics of the
eigenvalues. Solid squares and circles represent the initial positions
(i.e. for pˆ < 0) and the hollow squares and circles represent the final
positions (i.e. for pˆ > 0) of the eigenvalues as pˆ goes from negative
to positive in the direction of the arrows. Both the square and the
circle on the positive x-axis pass through the origin at pˆ = 0, hence
making pˆ = 0 a codimension-two bifurcation point. . . . . . . . . . .

50

xiv

Figure 3.16

Figure 3.17

Figure 3.18

Figure 3.19

Figure 3.20

Columns (a) and (b) represent eigenvalues at equilibria R0 and R1
respectively i.e. the real and imaginary parts of the eigenvalues at
equilibria R0 (α0 ) and R1 (α1 ) of the symmetric damped 2-DOF
system near the star bifurcation in the first-order form as pˆ goes from
negative to positive. Each equilibria has four different eigenvalues
that are designated as αji on the plot for j = 0, 1 and i = 1, 2, 3, 4.
The dashed and dotted curves represent two and four overlapping
eigenvalues respectively. . . . . . . . . . . . . . . . . . . . . . . . . .

51

Columns (a) and (b) represent eigenvalues at equilibria R2 and R3
respectively i.e. the real and imaginary parts of the eigenvalues at
equilibria R2 (α2 ) and R3 (α3 ) of the symmetric damped 2-DOF
system near the star bifurcation in the first-order form as pˆ goes from
negative to positive. Each equilibria has four different eigenvalues
that are designated as αji on the plot for j = 2, 3 and i = 1, 2, 3, 4.
The dashed and dotted curves represent two and four overlapping
eigenvalues respectively. . . . . . . . . . . . . . . . . . . . . . . . . .

52

The equilibrium solutions projected onto the (u1 , u2 ) plane and noted
as Rj = (u1 , u2 )j , for j = 0, 1, 2, · · · , 6. The solid and dashed ellipses
and straight lines satisfy h1 (u1 , u2 , q) = 0 and h2 (u1 , u2 , q) = 0 respectively. As q approaches zero from both directions, the two ellipses
coincide, resulting in an infinite number of solutions on the ellipse.
All the solutions on the ellipse are marginally stable. . . . . . . . . .

53

The vertical axis represents the eigenvalues at the equilibria in the
normal form at the eclipse bifurcation local to q = 0. For q < 0 the
points R1 , R4 and R6 are stable and become unstable when q > 0
where as R2 , R3 and R5 changes from unstable to unstable as q goes
from negative to positive and R0 remain unstable on both sides local
to q = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

Local eclipse bifurcations. The solid and dashed curves represent the
stable and unstable branches respectively and the dots represent the
bifurcation points. Left and right figures show the projections on u1 −
q, and u2 − q planes respectively. (a) Symmetric case for the eclipse
bifurcation where {h1 = 0, h2 = 0}. (b) Symmetry breaking case
where the eclipse bifurcation is unfolded into transcritical bifurcations
for the perturbation {h1 = 0, h2 − u1 u2 = 0}. . . . . . . . . . . . .

56

xv

Figure 3.21

Figure 3.22

Figure 3.23

Figure 3.24

Figure 3.25

Different symmetry-breaking bifurcations are presented where the
projections on u1 − q, and u2 − q planes shown. The solid and
dashed curves represent the stable and unstable branches respectively
and the dots represent the bifurcation points. (a) The eclipse bifurcation breaks into stiff subcritical pitchfork bifurcations for the
perturbation {h1 = 0, h2 − u2 = 0}. (b) The eclipse bifurcation
breaks into stiff supercritical pitchfork bifurcations for the perturbation {h1 = 0, h2 + u2 = 0}. . . . . . . . . . . . . . . . . . . . . . .

57

For symmetric system at q = 0, infinitely many equilibria exist on √
the
2√ 2
.
overlapping ellipses that are contained within the circle of radius
3
This circle is used as a criterion to determine that these equilibria are
marginally stable. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

(a)-(b) The bifurcation diagram for the equilibrium solutions on the
entire p − u1 and p − u2 planes, in various symmetry breaking configurations. The dots represent the bifurcations that involve a stable
branch. The perturbation is applied such that the spring force of the
end spring is perturbed by making a3 = a + , b3 = b + on the
top row, a3 = a + , b3 = b − on the middle row, and a3 = a − ,
b3 = b + on the bottom row, for = 0.15. The star bifurcations at
p = 0 and p = 6 and the eclipse bifurcation at p = 3, are broken into
combinations of pitchfork and saddle-node bifurcations. Column (c)
shows the corresponding energy levels in the perturbed systems. . .

59

Fractal basin plots of the various energy levels at y0
symmetry breaking of a2 = a − and a3 = a + , with
= 0.2 for varying initial conditions of the second mass

= 2.7 with
a = 0.5 and
x20 and x˙ 20 .

60

Fractal basin plots of the various energy levels at y0 = 2.7 with
symmetry breaking of a2 = a − and a3 = a + , with a = 0.5 and
= 0.2 for varying the masses, m1 and m2 . of the 2-DOF oscillator.

61

Figure 4.1

Shown here is the TEG experimental setup on the air-track. (a) shows
the idle equilibrium state of the TEG consisting of the primary mass
(the cart), the magnets, the magnet mounts, and a linear spring. (b)
shows the TEG in its dynamic state when the magnets are detached
and the right mass oscillates inside the generator coil. The right
spring in (b) is pulled by a horizontal shaker of stroke-length 6 inches. 66

Figure 4.2

The experimental setup of a 2-DOF twinkling energy generator (TEG)
consisting of an air-track, carts, magnet mounts, magnets, linear
springs, and horizontal shaker. . . . . . . . . . . . . . . . . . . . . .

xvi

67

Figure 4.3

The SDOF twinkling oscillator schematic consisting of one cart (black),
magnet mounts (blue), magnets (red), beads (yellow) of length dm /2
to maintain a separation of dm between the magnets in the steady
state, and linear springs. Mass m is the sum of the weights of the cart,
the magnet mount, magnet, the separating-beads, and the springs. k
is the stiffness of the spring connected to the left base, and ke is the
stiffness of the end spring. On the magnets, N and S represent the
north and south poles of the cylindrical magnets respectively. When
the magnet passes through the coil with inductance RL it generates
current in the coil and powers the resistive load RL . x(t) is the dynamic displacement of the cart and y(t) is the horizontal pull input.

68

The 2-DOF twinkling oscillator schematic consisting of two carts
(black), magnet mounts (blue), magnets (red), beads (yellow) to
maintain a separation between the magnets in the steady state, linear springs, and horizontal pull input Y (t). Mass m is the sum of
the weights of the cart, the magnet mount, magnet, the separatingbeads, and the springs. k1 is the stiffness of the spring connected to
the left base, k2 is the stiffness of the spring between the carts 1 and
2, and ke is the stiffness of the end spring. On the magnets, N and
S represent the north and south poles of the cylindrical magnets respectively. When the magnet passes through the coil with inductance
RL it generates current in the coil and powers the resistive load RL .
x1 (t) and x2 (t) are the dynamic displacements of the cart 1 and 2
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

The electrical circuit in the energy generator consisting of an inductor
(L), a variable resistor with resistance (RL ), and internal resistance
of the circuit (Ri ). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

Figure 4.6

The horizontal shaker input y(t) used for the numerical simulation. .

78

Figure 4.7

The numerical output x(t) of the mathematical model of the SDOF
twinkler, after applying one cycle of 0.1Hz input oscillation using
the first analytical model. Here we have used the assumption tf ≈
T − 0.933∆t0idle , where T = 10, is the period of the input oscillation. 82

Figure 4.8

The numerical simulation result of the SDOF twinkler phase portrait
after one cycle of 0.1Hz input oscillation using the first analytical
model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

The SDOF twinkling energy generator voltage output V (t), from the
numerical simulation, after one cycle of 0.1Hz input oscillation at the
end spring using the first analytical model. . . . . . . . . . . . . . .

84

Figure 4.4

Figure 4.5

Figure 4.9

xvii

Figure 4.10

Figure 4.11

Figure 4.12

The numerical frequency content of the SDOF TEG voltage output
using the first analytical model. The snap-through has produced a
3.25Hz output, a ≈ 32 times the input frequency. . . . . . . . . . . .

85

The numerical frequency content in the displacement of the mass
using the first analytical model of the twinkling magnet oscillation
in the SDOF TEG over one period of input oscillation of frequency
0.1Hz. This plot shows the frequency up-conversion of the input
excitation. The displacement oscillation has a frequency of 3.2Hz, a
32 times up-conversion of the input frequency. . . . . . . . . . . . .

86

The numerical harvested energy after one cycle of 0.1Hz input oscillation using the first analytical model. . . . . . . . . . . . . . . . . .

87

Figure 4.13

The numerical simulation displacement result of the SDOF TEG for
the second analytical model. Here we have used the assumption tf ≈
T − 0.922∆t0idle , where T = 10, is the period of the input oscillation. 88

Figure 4.14

The numerical simulation phase portrait result of the SDOF TEG for
the second analytical model. . . . . . . . . . . . . . . . . . . . . . .

89

The numerical result for the voltage output in the SDOF TEG for
the second analytical model. . . . . . . . . . . . . . . . . . . . . . .

90

Figure 4.15

Figure 4.16

The numerical frequency content using the second analytical model
of the twinkling magnet oscillation in the SDOF TEG over one period
of input oscillation of frequency 0.1Hz. This plot shows the frequency
up-conversion of the input excitation. Two dominant frequencies in
the output voltage are 2.8Hz (28 times the input frequency), 6Hz (60
times the input frequency), and 8.33Hz (83 times the input frequency). 91

Figure 4.17

The numerical frequency content in the displacement of the mass
using the second analytical model of the twinkling magnet oscillation
in the SDOF TEG over one period of input oscillation of frequency
0.1Hz. This plot shows the frequency up-conversion of the input
excitation. The displacement oscillation has a frequency of 2.8Hz, a
28 times up-conversion of the input frequency. . . . . . . . . . . . .

92

The numerical simulation results using the second analytical model
of the SDOF TEG power output shown in mW for a single cycle of
input oscillation of frequency 0.1Hz. The total power at any given
time in a 2000Ω resistive load lie on the curve at the particular time.

93

Figure 4.18

xviii

Figure 4.19

Figure 4.20

Figure 4.21

The experimental voltage output in the SDOF TEG shown in volts.
Single harmonic of 0.1Hz input frequency is converted into more than
2.5Hz output. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

The experimental SDOF TEG harvested energy shown in mWs for a
single cycle of input oscillation of frequency 0.1Hz. The total energy
harvested at any given time in a 2000Ω resistive load lie on the curve
at the particular time. The total energy harvested in the SDOF TEG
after one period (T = 10sec) of input oscillation is 0.045mWs. . . .

98

The experimental frequency content of the twinkling magnet oscillation in the SDOF TEG over one period of input oscillation of frequency 0.1Hz. This plot shows the frequency up-conversion of the
input excitation. Two dominant frequencies in the output voltage
are 2.45Hz (25 times the input frequency) and 5Hz (50 times the
input frequency). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

Figure 4.22

The experimental voltage output in the 2-DOF twinkling energy generator shown in volts. Single harmonic of 0.1Hz input frequency is
converted into 2.5 Hz output. . . . . . . . . . . . . . . . . . . . . . . 100

Figure 4.23

The 2-DOF TEG experimental energy output shown in Milliwattsecond for a single cycle of input oscillation of frequency 0.1 Hz. The
total energy up to any given time in a 2000Ω resistive load lie on the
curve at that particular time. The total energy harvested in the coil
in the 2-DOF TEG after one period (T = 10) of input oscillation (at
time t = T = 10 seconds) is 0.096 mWs. . . . . . . . . . . . . . . . . 101

Figure 4.24

The frequency content of the twinkling magnet oscillation in the experimental 2-DOF TEG over one period of input oscillation of frequency 0.1 Hz. This plot shows the frequency up-conversion of the
input excitation with one dominant frequencies in the output voltage
at 2.5Hz which is 25 times the input frequency. In addition to the
dominant frequency there is a broad band of frequency between 2 Hz
to 8 Hz generated by the 2-DOF twinkler. . . . . . . . . . . . . . . . 102

xix

Figure 5.1

The characteristic spring force f (sj ), and spring potential V (sj ) of
the linear and weakly nonlinear spring as a function of the spring
deformation sj . Referring to the spring force with characteristics as
in equation (5.1), the solid (red) curve corresponds to α = 1, β = 0.57,
and γ = 0.09, Dashed (black) line corresponds to α = 1, β = 0, and
γ = 0, dot-dashed (blue) curve correspond to α = 1, β = 0, and
γ = 0.09, and the dotted (green) curve correspond to α = 0, β = 0,
and γ = 0.09. The potential well shown to be shifted to the left with
positive quadratic nonlinearity, hence indicates a negative DC-bias. . 105

Figure 5.2

Infinite mass chain. The unstretched position and displacement of
mass mj are denoted by xj and u˜j respectively. The springs are
cubic nonlinear as in Figure 5.1 with unstretched length h. . . . . . 107

Figure 5.3

The dynamics of the wave dispersion in an infinite chain with quadratic
and cubic nonlinearities. The nonlinear chain was simulated by injecting a sinusoidal wave, at the 0th location, of amplitude d0 = 0.8,
and frequency ω = 1.0 with the perturbation parameter = 0.3. The
chain consisting of 400 elements are simulated for 400 seconds. To
prevent the reflection of the waves from the other end, we consider
the dynamics at t = 320 seconds. The wave number is computed
through a numerical FFT in space at t = 320 seconds. The DC-bias,
B as in equation (5.113), is computed as B = ∆U
∆N The parameter
values for the nonlinear spring elements are α = 1, β = 1.9, and γ = 1.144

Figure 5.4

The DC-bais generated in the nonlinear chain with both quadratic
and cubic nonlinearities computed from the numerical simulations. The
∆U of the DC-bias
data points are found by computing the slope ∆N
in the displacement coordinate as shown in Figure 5.3 at the frequencies simulated between 0 and 2.0. The parameter values are d0 = 0.8,
= 0.3, α = 1, β = 1.9, and γ = 1. . . . . . . . . . . . . . . . . . . . 145

Figure 5.5

The FFT in space corresponding to the numerical simulation shown
in Figure 5.3 at injected frequency ω = 1.0 and amplitude d0 = 0.8
for the parameters = 0.3, α = 1, β = 1.9, and γ = 1. In addition
to the leading order wavenumber, the single wave creates super and
sub-harmonics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

xx

Figure 5.6

This is a comparison of the wave dispersion characteristics in linear
and nonlinear chains when a sinusoidal wave of amplitude d0 = 0.8
is propagated down a chain connected by nonlinear spring elements.
Here the x-axis represents the wave number (k) and the y-axis represents the frequency ω which is a function of both the wave number
k and the amplitude of oscillation d0 . The thick solid (blue) curve
corresponds to the quadratic nonlinear case where α = 1, β = 1.9,
and γ = 1. The dashed (red) curve corresponds to the cubic nonlinear case where α = 1, β = 0, and γ = 1. The thin solid (black) curve
corresponds to the linear case where α = 1, β = 0, and γ = 0. The
cross (green +) data points represent the solution when the DC-bias
is taken into consideration, and the triangular (red ∆) data points
correspond to the numerical simulation results. . . . . . . . . . . . . 147

Figure 5.7

The wave speed or the phase velocity, c, shown with respect to the
wave number, k, at an amplitude d0 = 0.8. The thick solid (blue)
curve corresponds to the quadratic nonlinear case where α = 1, β =
1.9, and γ = 1. The dashed (black) curve corresponds to the cubic
nonlinear case where α = 1, β = 0, and γ = 1. The thin solid (black)
curve corresponds to the linear case where α = 1, β = 0, and γ = 0.
The cross (green +) data points represent the analytical solution with
the DC-bias obtained from the numerical simulation results. . . . . . 148

Figure 5.8

The wave number, k, dependent group velocity, vg , at an amplitude
d0 = 0.8 shows the wave velocities in the linear and nonlinear chains.
TThe thick solid (blue) curve corresponds to the quadratic nonlinear
case where α = 1, β = 1.9, and γ = 1. The dashed (black) curve
corresponds to the cubic nonlinear case where α = 1, β = 0, and
γ = 1. The thin solid (black) curve corresponds to the linear case
where α = 1, β = 0, and γ = 0. The cross (green +) data points
represent the analytical solution with the DC-bias obtained from the
numerical simulation results. . . . . . . . . . . . . . . . . . . . . . . 149

Figure 5.9

The phase plot of the slow scale dynamics corresponding to equation (5.131) of a multi-wave propagation in an infinite nonlinear chain
with weak quadratic nonlinearity. The x-axis represents the phase φ
and the y-axis represents the amplitude b. The figure corresponds to
wave number µ = 0.5, = 0.1, E = 2, and σω = 1, and β = α = 1. . 150

xxi

Figure 5.10

This plot shows the phase plot corresponding to equation (5.131) and
a snap-shot of the time evolution of wave propagation in a chain. The
blue dots on the top figure represents the fixed points and the green
lines represent the phase trajectories for initial conditions other than
the fixed points. The time evolution snap-shot on the bottom plot
correspond to a point on the thick blue-green curve on the top figure.
The figures correspond to wave number µA = 0.5, = 0.1, E = 2,
and β = α = 1, and σω = 1. . . . . . . . . . . . . . . . . . . . . . . 151

Figure 5.11

Time evolution of slowly varying amplitudes. The initial conditions
are b0 = 0.1, φ0 = 0, and the parameter values for this particular plot
are µA = 0.5, E = 2, = 0.1, α = 1, β = 1, σω = 1. . . . . . . . . . . 152

Figure 5.12

Time evolution of slowly varying phases. The initial conditions are
b0 = 0.1, φ0 = 0, and the parameter values for this particular plot are
µA = 0.5, E = 2, = 0.1, α = 1, β = 1, σω = 1. . . . . . . . . . . . . . 153

Figure 5.13

Time evolution of slowly varying amplitudes. The initial conditions
are b0 = −0.1, φ0 = 0, and the parameter values for this particular
plot are µA = 0.5, E = 2, = 0.1, α = 1, β = 1, σω = 1. . . . . . . . . 154

Figure 5.14

Time evolution of slowly varying phases. The initial conditions are
b0 = −0.1, φ0 = 0, and the parameter values for this particular plot
are µA = 0.5, E = 2, = 0.1, α = 1, β = 1, σω = 1. . . . . . . . . . . 155

Figure 6.1

The force and the potential energy of each snap through spring elements in the strain coordinates zj where f (zj ) = −αzj + γzj3 , where
zj = uj+1 − uj . Dashed (black), Dot-dashed (blue) and solid (red)
curves correspond to (α = 1, γ = 0), (α = 0, γ = 1), and (α = 1,
γ = 1) respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Figure 6.2

The force and the potential energy in w-space for f (w) = αw − γw3 .
Dashed (black), Dot-dashed (blue) and solid (red) curves correspond
to (α = 1, γ = 0), (α = 0, γ = 1), and (α = 1, γ = 1) respectively. . 165

Figure 6.3

Solution of the Duffing equation (6.11) for frequency ω and amplitude
ratios with respect to initial condition w0 . . . . . . . . . . . . . . . 167

xxii

Figure 6.4

The top figure shows the harmonic balance solution of the Duffing
equation (6.11) based on equation (6.12) when the initial condition
w0 is very close to the separatrix. The bottom figure is the phase
portrait of the numerical solution compared to that of the three-term
harmonic balance solution, where w (ζ) = ∂w
∂ζ . These solutions are
valid for any wave speed, c, since the equation 6.11 is independent of
the wave speed c. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

Figure 6.5

(a) A possible arrangement of masses with magnets and linear spring
to produce a chain of bistable elements. The masses consist of magnets which are arranged in a way that they repel each other. Each
mass consist of 3 magnets (filled circles), two on the right (sleeve)
and one on the left (tongue) except for the first and the last mass.
Here y is the quasistatic pull distance of the end spring. (b) The repelling forces fm between the magnets on the tongue and the sleeves
is shown. S and N represent the south and the north pole of the
magnets respectively, h is the separation distance between the two
sleeves, and 2fmx is the resultant magnetic force in the direction of
motion of the tongue and sleeves. (c) Total nonlinear magnetic spring
force (solid curve) that can be achieved from combining the forces due
to the magnets (dashed curve) and the linear spring (dotted line). . 172

xxiii

Chapter 1
Introduction

1.1

Motivation

There are various sources of low-frequency vibrations in nature, for example low-frequency
ocean waves, tsunamis, and low-frequency seismic activities such as earthquakes, ocean waves
and tsunamis. These natural phenomena have a vast amount of energy. To put this into
perspective, the 9.0 magnitude earthquake in Japan in 2011 contained 3.8994 ∗ 1022 Joules
of energy [1], enough to have powered the whole United States for about 378 years and the
whole world for about 75 years. If such energy can be captured, even a fraction of it, it
would ease the growing energy needs of the twenty-first century.
Apart from these large-amplitude and low-frequency seismic waves, we humans, produce
a lot of small-amplitude and low frequency vibrations in our day-to-day activities such as
walking, jogging, running, hiking, biking, driving etc. If converted into usable electricity,
these energies can be used to power our indispensable mobile electronic devices such as smartphones, cameras etc. To put this into perspective, the total energy consumed in powering an
iPhone 5 is 9.6Wh over one charging cycle. Over one year, with one charge per day would
consume a total of 3.5kWh electricity. Approximately 170 million iPhones sold in a year,
would consume 595 million-KWh of electricity over one year period. The total electricity
consumed by all electronic devices we use in our day-to-day lives are of big concern for the
future of energy. The total US retail electricity sale was a staggering 3.7 billion-MWh [2] for
1

the year 2012. The energy demand keep increasing exponentially every passing year and we
need to continue our focus on possible ways to harvest sustainable ambient energy.
Metamaterials have been of growing interest for achieving certain desirable properties
such as band gaps, negative refractive indices, phonon tunneling, and phonon focusing [3].
Engineering applications of these properties are in waveguides, acoustic filters, acoustic mirrors, transducers, etc. [4–12]. These properties are important in building materials that can
disperse energy in a desired direction or path, or obstruct certain frequency content. It is
of interest to allow selective frequencies to pass through such materials. These frequencyselective structures can be used to obstruct undesired frequencies while optimizing the transfer of the desired frequencies.
It is worth doing research on possible ways to harvest energy from these ambient vibrations and building metamaterials of the future. Keeping such motivations in mind,
snap-through structures are proposed be used to induce high-frequency oscillations from
low-frequency ambient vibrations [13] to harvest or dissipate energy. Study of nonlinear
snap-through elements involving negative stiffness are of primary focus for our research.
The main motivating factor in using the negative stiffness structures are due to their applications in signal detection (as in a bullfrogs hearing mechanism), energy harvesting, energy
transfer, energy sinks, vibration absorption, vibration isolation, nonlinear waveguides, and
metamaterials design.

1.2

Objective

This research focuses on exploring the dynamics of nonlinear snap-through structures for
their energy harvesting and wave dispersion properties. As discussed in the previous sections
the usage of the nonlinear snap-through structures are in energy harvesting, targeted energy
2

transfer, energy absorption, and energy dissipation. This work is motivated by the idea
of using snap-through structures to induce high-frequency oscillations from low-frequency
vibrations, that we refer to as “twinkling”, for wave management and energy harvesting.
In this dissertation, our main objective is to explore the nonlinear dynamics of the snapthrough chains as well as wave propagation behavior in periodic chains with weak quadratic
and cubic nonlinearities.

1.3

Background

A stable state always corresponds to the lowest energy state. Linear structures always have a
single stable equilibrium state at which it has the lowest potential energy. Linear structures
present predictable behavior and have been studied thoroughly by researchers. Combination
of multiple linear structures could, however, present challenges in certain types of designs.
Also linear structures have a great deal of advantage because of their simplicity and ease
of design. Researchers have used linear mass-spring systems to harvest [14–18], dissipate,
isolate, and absorb energy [19–22] in various structures.
However, it is almost impossible to find strictly linear designs in nature. Researchers
have been inspired by nature [23–25] to get inherently nonlinear design ideas to implement
in various science and engineering applications [26–28]. The nonlinearities can be used to
our advantage in certain designs. Working with these types of designs requires in-depth
understanding of the nonlinear properties. A lot of the current research in dynamics focuses
on exploiting the nonlinear properties and creating nonlinear materials that can be used to
get better results in energy transfer [29–33], energy harvesting [34–38], waveguides, acoustic
filters, acoustic mirrors, transducers, and bandgap design [4–10].

3

1.3.1

Energy Transfer

Nonlinear spring-mass systems have the potential to harvest energy from a variety of lowfrequency sources. For example small amplitude ambient vibrations from day-to-day activities such as walking, jogging, running, hiking, biking and driving as well as large amplitude
seismic activities such as earthquakes, ocean waves and tsunamis. To be able to harvest such
energy we first need to understand the underlying dynamics of the coupled snap-through oscillators.
Several authors have studied the dynamics of various snap-through negative stiffness and
bistable systems [13, 39–45]. Vibration-based energy harvesting from linear systems [14, 15]
has been optimized experimentally [16–18] by tuning the forcing frequency to the natural frequency of the oscillator. Piezoelectric materials are used for successful experimental energy
harvesting from vibrating sources [14, 15, 34, 35, 46, 47] and fluctuating pressure load [48, 49].
Nonlinearity has been studied by various authors for energy management. For example,
essential nonlinearity has been used as a nonlinear energy sink (NES) for energy harvesting [50–53], nonlinear energy pumping [54–57], and nonlinear targeted energy transfer (TET)
[29–33,58]. Novel experimental energy harvesting have been conceived for low-frequency ambient excitations [34–37, 59] and nonlinear oscillations of magnetic levitations [38].
Wireless energy transfer (WET) has recently become the subject of renewed research
[60–65], after the concept was first introduced almost one century ago by Nikola Tesla [66].
Both of these concepts of TET in NES systems and WET can be used to harvest energy
using piezoelectric materials, and transfer energy across devices and media respectively. The
snap-through structures, possibly in the scale of a MEMS device, can be installed in the
WET device that can be triggered by magnetic excitation from the WET source, to induce a
4

high-frequency oscillation in the snap-through system, thereby allowing the power harvesting
in the device while the energy transfer is performed by the WET source.
Recently there has been growing interest in designing materials that can exhibit multiple
stable equilibrium states. One way to achieve this is by using bistable elements. There are
two stable equilibrium states for every bistable element. There are structures in nature, such
as the hairs in bull-frog’s ears [67,68], that can exhibit bistable properties [69,70]. Though the
stability of a single degree-of-freedom bi-stable structure seems simple, when more bistable
elements are added, it forms complex dynamical systems and exhibits multiple equilibria
[71–74]. These bistable elements when combined, produce high-frequency oscillations. This
property of a bistable chain can be exploited in order to convert low-frequency vibrations
into high-frequency oscillations, which in turn can be used for energy harvesting and energy
transfer.

1.3.2

Wave Management

Due to the presence of band gaps in periodic media, many researchers have been focusing
on wave propagation in nonlinear periodic structures and its application to the design of
novel metamaterials [5, 12, 75–80]. Structures exhibiting band gaps prevent the propagation of waves at certain frequencies. These structures may be phononic (sonic) or photonic,
depending on their band-gap frequency range. Phononic or sonic band-gap structures can
be used as sensing devices based on resonators, acoustic logic ports and wave guides, and
frequency filters based on surface acoustic waves, while photonic band-gap structures have
applications in optics and microwaves. Synthesis of phononic materials with desired bandgap and wave-guiding characteristics has been achieved through the application of topology and material optimization procedures [9, 81, 82]. Various types of periodic media have

5

been studied, including one-dimensional undamped mass-spring chains [78–80, 83], strongly
nonlinear contact in beaded systems [84], kink dynamics [85], and weakly coupled layered
systems [86, 87]. The application of periodic plane grid structures as phononic materials
and its design optimization process has been presented in [88], where a limited number of
continuously varying parameters define the geometry of a predefined cellular topology that
deals with periodic structures of infinite size as well as demonstrate the validity of the results
to finite systems.
Periodic nonlinear structures with cubic nonlinearities are seen to exhibit bandgap behaviors [79, 80] which motivates us to study traveling wave behavior in nonlinear structures
with both quadratic and cubic nonlinearities [89].

1.4

Dissertation Organization

The dissertation is organized as follows:
• Harmonic Balance Analysis of Snap-Through Duffing Oscillator
In Chapter 2, using harmonic balance we formulate a solution method for the Duffing
oscillator for its snap-through behavior. The solutions outside the separatrix has been
analytically studied to show amplitude and frequency relations in up to fifth harmonic
approximation.
• Bifurcations and Chaos in Nonlinear Snap-Through Structures
In Chapter 3, we explore the dynamics of a nonlinear mass chain with cubic springs
that have three distinct roots in the characteristic. The chain is connected to a fixed
base at one end and to a pulled point at the other end. The end spring is pulled
quasistatically to investigate the bifurcations of equilibria in the snap-through regime

6

in single and two degree-of-freedom oscillators. The chaotic nature of the 2-DOF
system is characterized through fractal basin boundary study.
• Experimental Energy Harvesting Using Snap-Through Elements
In Chapter 4, we formulate novel experimental and analytical models for energy harvesting using snap-through elements. Attracting magnets and linear springs are used
form snap-through elements. The twinkling phenomenon (i.e. the frequency up-conversion),
and the energy harvesting capacity of the snap-through structures are validated in this
chapter.
• Wave Propagation in Weakly Nonlinear Chains
In Chapter 5, we examine the traveling wave behavior in weakly nonlinear periodic
structures. In the current study, the quadratic and cubic nonlinearities have been presented for non-snap-through structures. A periodic infinite chain connected by springs
that have linear, quadratic and cubic parts in their characteristics is considered for its
wave propagation properties. Second-order and third-order perturbation approach is
used to capture the effects of both the quadratic and the cubic nonlinearities. Also
in this chapter, we explored the interactions between two waves in a weakly nonlinear
chain and the effects of quadratic nonlinearity.
• Conclusion and Future Work
Finally, in Chapter 6 we present a summary of the research completed in this dissertation, the ongoing research on the wave propagation analysis in snap-through chains,
and propose future works for further advancement in the area.

7

Chapter 2
Harmonic Balance Analysis of
Snap-Through Orbits in an
Undamped Duffing Oscillator
In this chapter, using harmonic balance method we formulate establish relations between the
amplitudes and frequency of a truncated (up to the fifth harmonic) Fourier series solution
of the snap-through Duffing oscillator.

2.1

Introduction

Nonlinear vibrations and their solution methods are of great interest to the scientific community. The Duffing oscillator is a classic example of a nonlinear system, and can be used
to represent snap-through dynamics. Snap-through systems have been recently shown to
be useful for energy harvesting [35, 90–95]. Snap-through dynamics of a Duffing oscillator
involves strong nonlinearity. Different methods of solving strongly nonlinear equations have
been studied by various authors and summarized in a review article [96].

Duffing oscillators have been studied using various solution methods [97] such as the
method of nonlinear output frequency response functions [98], residue harmonic balance

8

method [99], simple point collocation method [100], and the method of Jacobi elliptic functions [101, 102], as well as the simple harmonic balance (HB) method [103–107], and HB
with up to two harmonics [108–110]. HB has also been used for a variety of engineering
applications such as nonlinear circuit analysis [106, 107], bifurcations and chaotic dynamics
in nonlinear systems [111, 112], and system identification [111–116].

Different types of Duffing oscillators have been studied using various solution methods
such as the method of nonlinear output frequency response functions [98], residue harmonic
balance method [99], simple point collocation method [100, 117], and the method of Jacobi elliptic functions [101, 102, 118, 119], as well as the simple harmonic balance method
[103–105,120,121]. A review of various free and forced Duffing oscillators is presented in [97].

The investigation in this article will provide insight for different energy states in the
snap-through regions. The energy states for single- and two-degree-of-freedom snap-through
equilibria are presented in [73, 74].

Though the Duffing oscillator and the harmonic balance methods are well known, we
have found no literature on a comparison study of solutions using different number of harmonics for high-amplitude oscillations of a snap-through system, i.e. when the oscillator goes
through the negative-stiffness region.

In this article we explore the dynamics of a mass with a cubic spring that has three
equilibria. Here we wish to solve the undamped, unforced case using a harmonic balance
approximation and evaluate two-term and three-term harmonic approximations. We will
focus our attention on solutions outside the separatrix, that is solutions that snap-through.
9

We will also establish the relationship between the frequency and leading order amplitudes
of these solutions. While the undamped Duffing oscillator is an idealization, an idealization
can be useful for obtaining insight.

Although the snap-through orbits of the symmetric Duffing oscillator have an analytical
solution involving an elliptic integral, a truncated Fourier approximation has some advantages. It is easily interpreted, amenable to post processing and signal analysis, and the solution approach can be adapted if additional nonlinear terms are added. As such, a truncated
harmonic-balance approximation, with an evaluation of its accuracy and characteristics, is
of value.

In sections 2 and 3, we present the nonlinear equation of motion, with two- and three-term
harmonic solutions. In section 4, we show the comparison of the harmonic balance analysis
with the numerical solution. Finally in section 5, we conclude with our final remarks.

2.2

Equation of Motion

The equation of motion of a single-degree-of-freedom spring-mass snap-through system is
M y + ∆y − Ky + Γy 3 = 0

(2.1)

where M is the mass, K > 0 is the linear coefficient, Γ > 0 is the nonlinearity constant, ∆
is the damping coefficient, and y = y(τ ) is the displacement of the mass. By substituting
τ =

M/K t, ∆ = K −1/2 M 3/2 δ, and y =

equation of motion as

K/Γ x, we obtain the nondimensionalized

x¨ + δ x˙ − x + x3 = 0

10

(2.2)

x

f (x)

1
ζ
Figure 2.1 Nondimensional spring mass Duffing oscillator.
The undamped equation, i.e. with δ = 0, is of the form x¨+f (x) = 0, where f (x) = −x+x3
is the nonlinear restoring force of the spring. To make our computations simpler, without
loss of generality, we can choose initial conditions x(0) = x0 and x(0)
˙
= 0. We then apply
harmonic balance method in order to solve the above nonlinear differential equation.

2.3

Harmonic Balance Approximations

We aim to establish a periodic solution outside the separatrix using the harmonic balance
method. We assume a Fourier series solution of the form

∞

x(t) = A0 +

An cos(nωt + φn )

(2.3)

n=1

Since f (x) is odd in x and since the solution outside the separatrix is centered at zero
and symmetric about both x-axis and x-axis,
˙
A0 can be discarded [122].

We can assume, without loss of generality, that the phases are zero (since x(0)
˙
= 0),
i.e. φn = 0 for all n. Also due to symmetry we have x(t + π) = −x(t). So the even
harmonic components in Equation (2.3) are zero [123]. Therefore, we only need to consider
the odd-harmonics from the above Fourier series. Hence equation (2.3) is simplified to

11

∞

A2n−1 cos((2n − 1)ωt)

x(t) =

(2.4)

n=1

For limited number of harmonics in the solution we only need to consider a truncated
Fourier series from equation (2.4). For a one-term harmonic solution, we know that the
phase portrait will be a circle giving us no information about the separatrix or details of the
snap-through region. So in the next subsection we will consider the two-term and three-term
truncated solutions in order to approximate the snap-through behaviour.

2.3.1

Three-Term Harmonic Balance Approximation

We aim to look at two and three term approximate solutions. Two-term computations are
a special case of three-term expressions. Thus, we present the three term analysis.
The three-term harmonic balance approximation of the solution to equation (2.2) is
written as
x = A1 cos(ωt) + A3 cos(3ωt) + A5 cos(5ωt)

(2.5)

where A1 , A3 and A5 are the amplitudes of the first, third, and fifth harmonics respectively,
and ω is the frequency of oscillation.
Differentiating x twice with respect to time t we get
x¨ = −ω 2 A1 cos(ωt) − 9ω 2 A3 cos(3ωt) − 25ω 2 A5 cos(5ωt)

(2.6)

Plugging the equations (2.5) and (2.6) into (2.2) we obtain
− ω 2 A1 cos(ωt) + 9ω 2 A3 cos(3ωt) + 25ω 2 A5 cos(5ωt)
− [A1 cos(ωt) + A3 cos(3ωt) + A5 cos(5ωt)]
+ [A1 cos(ωt) + A3 cos(3ωt) + A5 cos(5ωt)]3 = 0

12

(2.7)

Equating like terms from the above equation, i.e. coefficients of cos(ωt), cos(3ωt) and
cos(5ωt), and applying the initial condition we get the following set of equations:
3
1 + ω 2 − A21 1 + k52 + k32 (2 + k5 ) + k3 (1 + 2k5 ) = 0
4
1
1 + 9ω 2 k3 − A21 1 + 3k33 + 3k5 + 6k3 (1 + k5 + k52 ) = 0
4
3
1 + 25ω 2 k5 − A21 k3 + k32 (1 + 2k5 ) + k5 (2 + k52 ) = 0
4

A1 + A3 + A5 = x0

(2.8)

(2.9)

where
A3
= k3 ,
A1

A5
= k5
A1

(2.10)

are the amplitude ratios of the third harmonic to that of the first harmonic, and the fifth
harmonic to that of the first harmonic.

A21 =
ω2 =

32k3
3
+ 51k3 − 3k5 − 6k3 k5 + 54k32 k5 + 27k33 k5 + 48k3 k52
−1 − 3k3 + 3k32 + 3k33 − 3k5 − 6k3 k5 + 6k32 k5 + 3k33 k5
−1 + 21k3 + 27k32 + 51k33 − 3k5 − 6k3 k5 + 54k32 k5 + 27k33 k5 + 48k3 k52
−1 + 21k3 + 27k32

(2.11)

Solving the first two equations in equation (2.8) for A21 and ω 2 in terms of k3 and k5 we
get the relationships as in equation (2.11). From the third equation in equation-set (2.8),
equation (2.9), equation (2.10), and equations (2.11), one can solve for ω, A1 , k3 , and x0 for
a given ratio k5 .
Plugging the expressions from equation (2.11) into the third equation in equation-set (2.8)
we solve for k3 in terms of k5 . An assumed value of k5 will give us a possible value of k3 , and
this in-turn can be used to get values of A21 and ω 2 . So plugging the values of A1 , k3 and k5

13

back into the fourth equation A1 (1 + k3 + k5 ) = x0 from equations (2.9) and (2.10), we find
the value of x0 . In this way, choosing k5 instead of the value of x0 as the given parameter
value makes the computation more efficient.
For two-term solutions we plug k5 = 0, into equations (2.11) and drop the third equation
in the equation set (2.8) as there is no fifth-harmonic to balance. We then denote the
2 in
amplitude as B1 , the amplitude ratio as k˜3 , and the frequency as ωB , to get B12 and ωB

terms of k˜3 :
B12 =
2
ωB

=

32k˜3
−1 + 21k˜3 + 27k˜32 + 51k˜33
−1 − 3k˜3 + 3k˜2 + 3k˜3
3
˜
˜
−1 + 21k3 + 27k32

(2.12)

3

+ 51k˜33

We notice more complex equations with three harmonics as compared to two harmonics.
One can solve for ωB , B1 and k˜3 for a given initial condition x0 . Plugging in k5 = 0 into
equation (2.9) we get B1 (1 + k˜3 ) = x0 and use B1 from equation (2.12) to solve for k˜3 in
terms of x0 . So for a given initial condition x0 we can find k3 and which can then be used
to solve for B12 and ω 2 in the equation (2.12).
For comparisons between the two and three-term appoximations, once we get the value
of x0 from the three-term harmonic balance method we use that in the two-term harmonic
2 from equation (2.12).
solution to solve for k˜3 and the corresponding values of B12 and ωB

2.3.2

Fourier Series Limiting Case

The exact solution of the separatrix of the Duffing oscillator in equation (2.2) is given as
√
√
(x, x)
˙ = (± 2 sech t, ∓ 2 tanh t sech t) [124]. An approximation to a solution very near
the separatrix is shown in Figure 2.2.
The Fourier series representation of a solution near the separatrix is
14

1.5

2

1.0

0

0.5

T

0.0

2
T

200

150

100

50

0

50

1.5

1.0

0.5

t 0

time
Figure 2.2 This plot approximates the solution very near and outside the separatrix. The
period T is infinite on the separatrix. As the periodic orbit approaches the separatrix, the
peaks can be approximated by Dirac delta dunctions in the derivation of the Fourier series.

∞

x(t) =

Am cos(mω0 t)

(2.13)

n=0

where ω0 = 2π
T and m is an integer.
For arbitrary reference time t0 , as the trajectory approaches the separatrix, the coefficients Am can be denoted as Am∞ and found as
T

1 t0 + 2
Am∞ = lim
x(t) cos(mω0 t) dt
T →∞ T t0 − T
2

(2.14)

With reference to Figure 2.2, as the trajectory approaches the separatrix, the local peaks
in x(t) become more distantly spread, and sharper relative to the time interval of period T .
In the limit, the response can be idealized as a series of infinitely spaced Dirac delta-functions
of the strength xˆ, where

15

xˆ =

T
2
lim
T →∞ − T
2

x(t) dt =

T
2
lim
T →∞ − T
2

√
T
= lim 2 2 tan−1 tanh
2
T →∞

√
2 sech t dt
(2.15)

√
π
= 2 2 tan−1 1 = √ .
2

Therefore, the solution near the outer side of the separatrix is idealized in terms of a
series of Dirac delta functions as
∞

(−1)i δ t −

x(t) = xˆ

i=−∞

iT
2

(2.16)

Integrating this limiting form over an interval that includes the spikes (Figure 2.2) at
t = 0 and t = T /2, and using ω0 T = 2π and the properties of the δ-function, the limiting
Fourier coefficients can be evaluated as
T

T
1 t0 + 2
Am∞ =
xˆ δ(t) cos(mω0 t) − δ t −
cos(mω0 t) dt
T t0 − T
2
2




0,
m = 0, 2n
T
1
= xˆ cos(0) − cos mω0
=

T
2


 4ˆx , m = 2n − 1
T

(2.17)

From equation (2.17) we find that in the limit of the separatrix all the amplitudes of
x
the odd harmonics are equal i.e. A2n−1 = 4ˆ
T . Thus, more harmonics are expected to be

needed as we approach the separatrix. Since T → ∞ in this limit, the amplitudes of these
harmonics approach zero, i.e. A2n−1 → 0, but the infinite sum may be non-zero.

16

1.5

Numerical

2 term HB

3 term HB

1.0

Velocity

0.5
Separatrix at x0

0.0

2

0.5
1.0
1.5
1.5

1.0

0.5

0.0

0.5

1.0

1.5

Displacement
Figure 2.3 Various solutions using harmonic balance and numerical approaches at x0 =
1.65062, x0 = 1.50675, and x0 = 1.45933 corresponding to k5 = 0.01, k5 = 0.02 and
k5 = 0.03 respectively.

2.4

Analytical and Numerical Results

We plot the results in one phase-plot to compare with the numerical solutions based at the
same x0 . Figure 2.4 shows the phase plot for assumed amplitude ratio k5 = 0.02. For this
value of k5 the corresponding value of x0 is 1.50675.
Figure 2.5 shows the frequency ω and the leading-order amplitude A1 of the harmonic
balance solution with one, two, and three odd-harmonic terms. The frequency and the
leading-order amplitude are almost linearly related farther away from the separatrix. In
the lower-frequency region in Figure 2.6, we observe that, the lower frequency solutions can
be captured with three odd-harmonic terms in the approximate solution that can not be
observed for two-term approximation.
In Figures 2.5 and 2.6, there exist a cut-off amplitude for the leading-order amplitude A1

17

Numerical

2 term HB

3 term HB

1.0

Velocity

0.5

0.0

0.5

1.0
1.5

1.0

0.5

0.0

0.5

1.0

1.5

Displacement
Figure 2.4 Phase plot for x0 = 1.50675 corresponding to k5 = 0.02. The three-term solution
results in more accurate solution as compared to the two-term solution.
since we are looking for solutions outside the separatrix in the phase space. These cut-off
amplitudes differ for solutions with one, two, and three odd-harmonics. Recall that A1 = x0
for one-term, A1 +A3 = x0 for 2-term and A1 +A3 +A5 = x0 for three-term harmonic balance
approximations. Since for two and three-term solutions the amplitudes of the third and fifth
harmonic contribute to a particular initial condition x0 , the leading-order amplitudes for the
three-term harmonic balance can approach lower values as compared to the two and oneterm harmonic approximations as seen in Figures 2.5 and 2.6. The true behavior would show
a frequency that approaches zero as A1 approaches a limiting value at which the solution
reaches the separatrix. In this harmonic balance expansion, the calculation breaks down
below the minimum value plotted.
From the phase plots in Figure 2.3 we see that the harmonic balance approximation
matches the numerical solution better for larger amplitude orbits than for orbits closer to

18

Frequency Ω

2.0
1.5
1.0

••
••••
•
•
•
•••
••••
•
•
•
••
••••
•
•
•
••
••••
•
•
•
••
••••
•
•
•
•
••••
•
•
•
••
•••
•
•
•
•••
•••

1 term HB
2 term HB
3 term HB

0.5
0.0
1.0

1.5

2.0

2.5

Leading Order Amplitude A1
Figure 2.5 Frequency vs. the leading-order amplitude from harmonic balance approximation
with one, two, and three odd-harmonic terms.
the separatrix. Also we see examples of the extent of improvement of the phase portrait
when going from a two-term to a three-term approximation. For larger initial conditions
(here x0 = 1.65062) the solution with a three-term harmonic approximation matches very
closely with the numerical results in Figure 2.3. As we come closer to the separatrix, for
example for the initial condition x0 = 1.45933, the three-term harmonic solution seems
to separate from the numerical solution because we need more harmonics to represent the
solutions closer to the separatrix. Also here we see that the two-term approximation is a
closer match to the numerical solution farther away from the separatrix than close to the
separatrix.
For two- and three-term harmonic approximations the amplitudes of the harmonics are
shown in a logarithmic scale in Figure 2.7 to show the change of amplitudes when we consider solutions as we go closer to the separatrix. The amplitudes of the harmonics of the
Duffing oscillator tend to converge as the number of harmonics increases. Though it is not

19

0.9

Frequency Ω

0.8
0.7

1 term HB
2 term HB
3 term HB

••

••

•
••

••

•

•
••

••

•

0.6
0.5
0.4
0.3
0.2
1.1

1.2

1.3

1.4

1.5

Leading Order Amplitude A1
Figure 2.6 Zoomed-in view of the frequency vs. the leading-order amplitude from harmonic
balance approximation with one, two, and three odd-harmonic terms.
observable in Figure 2.7 for three-term approximation because of the frequency limits, we
predict from the equation (2.17) that all of the amplitudes converge to a single curve as ω
becomes very small, with all A2n−1 → 4ˆ
x/T → 0 as ω → 0. A direct comparison of the solution of the Duffing oscillator for its snap-through orbits with one, two, and three harmonic
approximations shows the minimum frequencies captured for these solutions in Figure 2.7.
As we approach the separatrix, the period T of the snap-through orbit increases and
in the limit becomes infinite on the separatrix. Hence the frequency steadily decreases,
approaching zero.
Figure 2.8 shows that as the orbit goes farther from the separatrix, as reflected by the
larger value of x0 , the harmonic amplitude ratio k3 decreases and the predicted ratios from
the two- and three-term expansions become similar. Given, for example, x0 = 1.5, there is
about 12% difference between the amplitude ratio of the third harmonic when only a twoterm harmonic balance solution is obtained as compared to a three-term harmonic balance

20

Log amplitude of harmonics

1

•
•• • • • • • • • • • •
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•••••••••••••••••••••••
•••••••••••••••••••••••••••••••••••••

0

Ω1

term

•

•

limit

1
2
3

Ω2

loge C1

term

limit

loge B1
loge B3

4

• • • loge A1
loge A3

5
0.0

Ω3

term

limit

loge A5
0.5

1.0

1.5

2.0

Frequency Ω

Figure 2.7 This plot shows a comparison of the amplitudes in one-, two-, and three-term
solutions as the frequency changes. Here the amplitudes of the harmonics are shown in
a logarithmic-scale. C1 is the amplitude of the first harmonic in one-term approximation,
B1 , B3 are the amplitudes of the first and third harmonic in two-term approximation, and
A1 , A3 , and A5 are the amplitudes of the first, third and the fifth harmonics in three-term
approximation. As we approach from the right on the frequency axis i.e. approaching the
separatrix on the phase space, we see that the minimum frequency that is captured with
the truncated Fourier series varies depending on the number of harmonics in the HB solution. The limits on the frequencies reduce with added harmonics for which the approximate
solution of the Duffing oscillator can be found
solution. This error increases as we go much closer to the separatrix, as we would need more
harmonics to accurately represent the solution. However, we can still consider the two-term
harmonic solution when we are not close to the separatrix to approximately represent the
solution to the snap-through Duffing equation. Given a harmonic ratio k3 , however, the error
in the associated initial condition is small.
Figure 2.8 also shows the relationship between the initial condition, x0 , and the relative
strength, k5 , of the first and fifth harmonics. The amplitude ratio of the fifth harmonic
decreases as the initial condition x0 goes farther away from the origin in the phase space.
This is an indication that as we go farther away from the separatrix, fewer harmonics are
21

2.0

2 term HB
3 term HB

Initial Condition x0

1.9

Numerical

1.8
1.7
1.6

k5

k3

1.5
1.4
0.00

0.05

0.10
0.15
Amplitude Ratios k3 , k5

0.20

Figure 2.8 This plot shows the relationship between the amplitude of the first, third, and
fifth harmonics versus the initial condition x0 . The blue line corresponds to a solution with
three odd harmonics and the black line corresponds to a solution with two odd-harmonics.
We see that for an initial condition x0 greater than 1.6, the amplitude ratio of the third
harmonic (k3 ) doesn’t change much for the two-term harmonic balance solution compared
to that of the three-term solution. The amplitude ratios of the fifth to the leading order
harmonics (k5 ) shows the weakening of the amplitude of the fifth harmonic as we go farther
away from the separatrix.
needed to represent the solution accurately.
Numerical results are shown in Figures 2.8 and 2.9 as red dots and are computed by first
solving the differential equation for a particular value of initial velocity, i.e. when x˙ 0 = 0 and
x0 = 0 and then computing the Fourier coefficients in MATLAB. The Fourier coefficients A1 ,
A3 , and A5 are computed using Fourier integral over one period of the numerical solution
x(t). The period, T , and the frequency, ω, are computed by finding the third zero-crossing
of the solution x(t) on the time axis for the initial conditions x˙ 0 = v0 and x0 = 0. This
process is repeated for various values of v0 and results are obtained for the corresponding
Fourier coefficients, amplitude ratios of the harmonics, and the frequencies.
22

1.4

Frequency Ω

1.2
1.0
0.8
Ω1

0.6
Ω3

0.4

term limit

Ω2

term

term

limit

limit

1 term HB
2 term HB
3 term HB

0.2

Numerical

0.0

1.4

1.5

1.6
1.7
1.8
Intial Condition x0

1.9

2.0

Figure 2.9 The frequency ω is plotted versus the initial condition x0 for approximate solutions
with one, two, and three odd harmonics. The plot suggests a linear linear trend when x0
is large. The numerical results (red dots) confirm the accuracy of the harmonic balance
solutions (solid lines) with three odd harmonics. The ω limits show that the three-term
harmonic approximation captures solutions closer to the separatrix, and of lower frequency,
than the one- and two-term solutions.
Figure 2.9 shows the frequency as a function of the initial condition x0 . For x0 < 1.414,
a non-snap-through solution exists but requires a formulation that includes a constant term
with both even and odd harmonics. A finite-term harmonic balance cannot capture very
low frequencies occurring near the separatrix, and relate them to the response amplitude.
The three-term expansion captures lower frequency orbits than the two-term expansion.
The calculation breaks down when x0 is chosen below the minimum value plotted. The
numerical solution shows the frequencies below that at which the truncated solution breaks
√
down, approaching zero as x0 → 2. Figure 4 shows that at large amplitudes, the frequency
varies nearly linearly with amplitude. This is consistent with the relationship predicted by
a single-harmonic expansion when applied to a Duffing oscillator with positive linear and

23

cubic stiffness coefficients [109].
Figures 2.8 and 2.9 can be used to write out approximate solutions at any given initial
condition x0 or frequency ω. For example, if an initial condition is given for the Duffing
equation (2.2) then one can obtain the values of k3 , k5 , and ω from the figures 2.8 and 2.9,
respectively. Once k3 and k5 are obtained from the plots one can compute the corresponding value of A1 using the fourth equation in the equation-set (2.8). Similarly, given the
frequency, ω, the complete truncated solution parameters can be determined, or when one
of the amplitude ratios, say k3 , is known, then the corresponding initial condition x0 , the
amplitude ratio k5 , and the frequency ω can be found.
In essence, the Figures 2.8 and 2.9 can be used as look-up charts in order to find threeterm approximate solutions of the Duffing oscillator for different initial conditions and frequencies. For example, in non-dimensional coordinates, if x0 = 1.5, then from Table 2.1,
x(t) = 1.2749 cos(ω t) + 0.1983 cos(3ω t) + 0.0268 cos(5ω t) with ω = 0.6909. In a mass-spring
system the approximate solution becomes y(τ ) =
0.1983 cos(3Ω τ ) + 0.0268 cos(5Ω τ )] with y0 = x0

Γ/K x(t) =

Γ/K [1.2749 cos(Ω τ ) +

Γ/K = 1.5 Γ/K, and Ω = ω

K/M =

0.6909 K/M . Tables 2.1 and 2.2 serve as look-up tables for solutions very close to the separatrix and far from the separatrix respectively. These look-up tables make it easy to write
out the solutions of three-term harmonic approximation using linear interpolations. Also
a look-up table of the numerical results for up-to fifth-order harmonics obtained through
Fourier coefficient computation in MATLAB are tabulated in the Table 2.3.

24

Table 2.1 Three-term harmonic balance solution look-up table for solutions closer to the
separatrix.
x0
1.55518
1.54320
1.53266
1.52331
1.51497
1.50746
1.50002
1.49104
1.48316
1.47619
1.46997
1.46438
1.45933
1.45473
1.45052
1.44664
1.44307
1.43975
1.43666
1.43378
1.43107
1.42852
1.42612
1.42384
1.42168
1.41963
1.41767

A1
1.35877
1.34181
1.32641
1.31230
1.29930
1.28724
1.27490
1.25941
1.24522
1.23211
1.21992
1.20852
1.19780
1.18767
1.17807
1.16894
1.16022
1.15187
1.14386
1.13615
1.12873
1.12156
1.11462
1.10790
1.10138
1.09505
1.08889

A3
0.176165
0.180054
0.183838
0.187521
0.191109
0.194604
0.198347
0.203291
0.208059
0.212663
0.217114
0.221422
0.225595
0.229640
0.233566
0.237379
0.241084
0.244687
0.248194
0.251608
0.254934
0.258176
0.261337
0.264421
0.267432
0.270371
0.273243

A5
0.0202457
0.0213348
0.0224163
0.0234902
0.0245568
0.0256160
0.0267729
0.0283368
0.0298853
0.0314189
0.0329379
0.0344428
0.0359340
0.0374117
0.0388764
0.0403283
0.0417678
0.0431951
0.0446105
0.0460142
0.0474066
0.0487878
0.0501580
0.0515175
0.0528664
0.0542051
0.0555336

ω
0.785547
0.766159
0.748636
0.732681
0.718061
0.704586
0.690908
0.673901
0.658496
0.644438
0.631523
0.619591
0.608511
0.598176
0.588496
0.579397
0.570817
0.562701
0.555003
0.547685
0.540710
0.534049
0.527676
0.521567
0.515702
0.510062
0.504630

In the frequency vs. initial condition plot in Figure 2.9(b), the frequency ω of the harmonic balance solution decreases as the initial condition x0 comes closer to the separatrix in
the phase space. For the three-term harmonic approximation, there exist a cut-off frequency
ωmin = 0.47 corresponding to x0 = 1.41422. However from the numerical solutions (shown
in red dots in Figure 2.9(b)) we see that the frequency decreases down to ωmin = 0. This is
accurate as we land on the separatrix itself and start from x0 at t = 0. That is because start-

25

Table 2.2 Three-term harmonic balance solution look-up table for large amplitude oscillation
farther from the separatrix
x0
2.66285
2.36287
2.18357
2.06285
2.00733
1.97536
1.90872
1.87576
1.79129
1.73200
1.68787
1.65363
1.62622
1.60373
1.58492
1.56894

A1
2.51130
2.21777
2.04032
1.91936
1.86315
1.83056
1.76199
1.72768
1.63816
1.57340
1.52367
1.48382
1.45087
1.42295
1.39882
1.37763

A3
0.143765
0.137112
0.134886
0.134657
0.135054
0.135468
0.136867
0.137884
0.141822
0.146170
0.150643
0.155116
0.159527
0.163847
0.168063
0.172169

A5
0.0077850
0.0079840
0.0083653
0.0088291
0.0091294
0.0093359
0.0098671
0.0101933
0.0113033
0.0124299
0.0135607
0.0146899
0.0158145
0.0169331
0.0180447
0.0191490

ω
2.00838
1.71666
1.53549
1.40924
1.34967
1.31485
1.24083
1.20341
1.10444
1.03165
0.97512
0.92948
0.89158
0.85939
0.83157
0.80718

ing from x0 it would take infinite time (i.e. zero frequency) to reach the origin of the phase
plot when we are on the separatrix. From the analytical solution on the separatrix [124] the
√
corresponding initial condition is x0 = 2.

2.5

Summary

In this chapter, we have presented a harmonic balance analysis of snap-through Duffing oscillator under large amplitude vibrations, i.e. for solutions outside the separatrix in the phase
space. The phase plots show a direct comparison of the numerical solution to that of the
two-term and three-term harmonic approximations. We have demonstrated the relationship
between the frequency, leading order amplitudes, and the initial conditions when the system
exhibits snap-through.

26

Table 2.3 Numerical solutions look-up table for snap-through oscillation outside the separatrix
x˙ 0
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
1.20
1.25
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
1.75
1.80
1.85
1.90
1.95
2.00

x0
1.4146
1.4170
1.4217
1.4278
1.4351
1.4442
1.4535
1.4657
1.4776
1.4905
1.5047
1.5199
1.5356
1.5506
1.5660
1.5843
1.5988
1.6181
1.6331
1.6528
1.6672
1.6876
1.7045
1.7204
1.7408
1.7575
1.7719
1.7930
1.8112
1.8266
1.8421
1.8627
1.8811
1.8972
1.9112
1.9283
1.9479
1.9661
1.9828
1.9975

ω
0.3324
0.3901
0.4347
0.4734
0.5088
0.5422
0.5740
0.6048
0.6347
0.6638
0.6923
0.7203
0.7477
0.7745
0.8009
0.8268
0.8523
0.8773
0.9019
0.9261
0.9499
0.9733
0.9964
1.0190
1.0413
1.0633
1.0850
1.1063
1.1273
1.1481
1.1685
1.1887
1.2086
1.2282
1.2476
1.2667
1.2856
1.3043
1.3227
1.3409

A1
0.8252
0.9254
0.9928
1.0452
1.0889
1.1269
1.1610
1.1923
1.2214
1.2488
1.2750
1.3002
1.3245
1.3481
1.3712
1.3937
1.4157
1.4374
1.4587
1.4797
1.5004
1.5208
1.5409
1.5608
1.5805
1.5999
1.6191
1.6381
1.6569
1.6755
1.6939
1.7122
1.7302
1.7481
1.7659
1.7834
1.8008
1.8181
1.8352
1.8521

A3
0.37640
0.34423
0.31656
0.29298
0.27279
0.25543
0.24043
0.22744
0.21615
0.20632
0.19774
0.19021
0.18362
0.17781
0.17269
0.16816
0.16415
0.16059
0.15742
0.15460
0.15207
0.14982
0.14781
0.14601
0.14439
0.14294
0.14165
0.14049
0.13944
0.13851
0.13768
0.13693
0.13626
0.13567
0.13515
0.13469
0.13429
0.13392
0.13362
0.13336

27

A5
0.137460
0.103690
0.082815
0.068228
0.057428
0.049153
0.042663
0.037483
0.033291
0.029858
0.027017
0.024645
0.022647
0.020951
0.019501
0.018253
0.017171
0.016231
0.015404
0.014678
0.014036
0.013464
0.012955
0.012496
0.012087
0.011716
0.011378
0.011073
0.010795
0.010542
0.010309
0.010097
0.009902
0.009723
0.009556
0.009399
0.009256
0.009126
0.009001
0.008884

k3
0.456120
0.372000
0.318860
0.280310
0.250520
0.226660
0.207080
0.190760
0.176970
0.165210
0.155080
0.146300
0.138630
0.131890
0.125940
0.120660
0.115950
0.111720
0.107910
0.104480
0.101350
0.098517
0.095921
0.093546
0.091358
0.089344
0.087486
0.085762
0.084159
0.082668
0.081276
0.079975
0.078755
0.077608
0.076534
0.075524
0.074569
0.073662
0.072811
0.072004

k5
0.166570
0.112050
0.083416
0.065277
0.052739
0.043617
0.036746
0.031439
0.027257
0.023908
0.021189
0.018955
0.017098
0.015541
0.014222
0.013097
0.012129
0.011291
0.010560
0.009919
0.009355
0.008853
0.008407
0.008006
0.007648
0.007323
0.007028
0.006760
0.006515
0.006292
0.006086
0.005897
0.005723
0.005562
0.005411
0.005270
0.005140
0.005020
0.004905
0.004797

The analysis for the limiting case predicts the convergence of the amplitudes of the
harmonics to a common diminishing value as the orbit approaches the separatrix. It is shown
that there are limits on the minimum frequencies that can be predicted by the approximate
solutions. These limiting frequencies vary with the number of harmonics considered in the
harmonic balance solution.
We also saw how the predicted amplitude of the third harmonic changes when we consider two- and three-term harmonic solutions to approximately represent the solution of the
snap-through Duffing equation. We examined the solutions for both two- and three-term
expansions corresponding to a particular initial condition x0 , and showed that the two-term
solution is a close match to that of the three-term solution when we go farther away from
the separatrix.
The approach we have taken can be used to further investigate solutions with more
harmonics. However, with more harmonic terms one would face more difficulty in solving
the equations to get relevant results. The relationships shown in Figures 2.8 and 2.9 can be
used as look-up charts to write out solutions up to three terms for different values of initial
conditions, frequencies, and amplitudes.
The harmonic balance method studied here is currently being extended to investigate
large-amplitude wave behavior in an infinite mass-spring chain connected by snap-through
elements to supplement small amplitude perturbation studies done previously [79, 89, 125,
126].

28

Chapter 3
Bifurcations in Snap-Through
Twinkling Oscillators
We explore the dynamics of a nonlinear mass chain with cubic springs that have three distinct
roots in the characterisitcs. The chain is connected to a fixed base at one end and to a pulled
point at the other end, as shown in Figure 3.1. The end spring is then pulled quasistatically
to a certain distance y to observe the “twinkling phenomenon” in the chain [71–74]. As
this pull parameter is changed we see high-frequency oscillations about different equilibrium
states.
The system exhibits twinkling phenomena and comes to an equilibrium state due to small
applied damping, with a total energy that was not present in the system before the applied
quasistatic pull. This residual energy will be of importance to energy harvesting and WET.
Final equilibria of this type of twinkler may have differing energy states. Predicting the final
energy of the system requires analysis of transients, which are influenced by the structure of
both stable and unstable equilibria. This would be relevant to energy harvesting and energy
dissipation, for instance, in crash worthiness. We expect that energy harvesting can be done
on these oscillators by using permanent magnets and coils.
First, we construct the nonlinear equation of motion (EOM), and discuss the conditions
for the stability of equilibria. Then we perform a coordinate transformation to obtain a

29

	
  

x1

f1 (x1 )

f 2 (x 2 − x1 )

m1	
  

c1
€

	
  

€ €

x2

f 3 (x 3 − x 2 ) f n (x n − x n −1 )

m2	
  

c2

€ €

	
  

xn

mn	
  

c3

cn

€

	
  

f n +1 (y − x n )

y

€ €

Figure 3.1 n-degree-of-freedom spring-mass chain connected by n masses, (n + 1) nonlinear
springs, and n dash-pots. As shown in this figure the left spring is fixed to a base and the
€
right most spring is pulled quasistatically to a distance y0 .

€

€

€

€

normal form and solve for equilibria of the global equations of motion in SDOF and 2-DOF
systems. Next, we identify the snap through region and perform a coordinate transformation
to obtain normal forms of the local bifurcations in both SDOF and 2-DOF systems. We then
study the local bifurcation behavior of a one-mass two-spring system and extend our analysis
to two-mass chain.
In the 2-DOF system, we look closely at the two new complicated highly degenerate
local bifurcations “star” bifurcations [71, 73] and “eclipse” bifurcation [72, 74], so called
because of the structures of their equilibria branches, which occurs in the snap-through
configuration. One possible way to experimentally realize this system is by using permanent
magnets arranged on the masses such that they create a negative stiffness or bistable system.
Similar study has been performed by Mann and Sims [38] for a single-degree-of-freedom
model using permanent magnets to obtain a Duffing-like oscillator capable of harvesting
energy by inducing current in the coils wrapped around the moving magnet. A speculative
schematic of our idea was laid out in [73, 74].
We present the stability analysis, and the related degree of degeneracy using the concept
of codimension, and show how the codimension of the system changes with damping. Next,
the snap-through region is studied for transient chaos analysis using numerical fractal basin
boundary of equilibria energy in the varying parameter region of an unsymmetric 2-DOF

30

twinkler. Finally, we present our concluding remarks and future research works in the area.

3.1

Equations of Motion and Stability Analysis

The equations of motion (EOM) of an n-degree-of-freedom (n-DOF) snap-through system
as shown in the Figure 3.1, is written using the Newton’s second law of motion, as

m1 x¨1 + c1 x˙ 1 − c2 (x˙ 2 − x˙ 1 ) = f2 (x2 − x1 ) − f1 (x1 )
mi x¨i + ci (x˙ i − x˙ i−1 ) − ci+1 (x˙ i+1 − x˙ i ) = fi+1 (xi+1 − xi ) − fi (xi − xi−1 ) ;
(3.1)
for i = 2, 3, . . . , n − 1
mn x¨n + cn (x˙ n − x˙ n−1 ) = fn+1 (y − xn ) − fn (xn − xn−1 )

ai

0

bi
Quasistatic Pull : y t

Spring Force : fi si

1
1
2
3
4
5
6
1

0

1

2

3

6

y0
4
2
0
0

Spring Deformation si

100

200

300

400

500

Time t

Figure 3.2 The characteristic spring force fi (si ) of the nonlinear spring as a function of the
spring deformation si , and the quasistatic pull as a function of time.
where mi is the ith mass and xi is its displacement, fi (xi − xi−1 ) is the elastic force in the
ith spring, ci is the damping of the ith spring and y is the quasistatic pull as shown in the
Figure 3.2, applied to the end spring. Dividing by the masses, the undamped and damped
31

system can be written in the form

(3.2)

¨ + Cx˙ = f (x)
x

where x is an n × 1 array of mass displacements and f (x) = −Kx + fnl with (−Kx) as the
linear and fnl as the nonlinear part of the spring force. We consider a cubic spring with the
spring force fi (si ) to be of the form

fi (si ) = γi si (si − ai ) (si − bi ) ;

for i = 1, 2, . . . , n + 1

(3.3)

where γi , ai , and bi are positive real numbers. Referring to Figure 3.2, for the end spring held
at y = y0 , we examine the equilibria. At equilibrium, the accelerations and the velocities of
all the masses are zero. Hence we get the following array of equilibrium equations for the
n-DOF structure:
f2 (x2 − x1 ) = f1 (x1 )
fi+1 (xi+1 − xi ) = fi (xi − xi−1 ) ;

for i = 2, 3, . . . , n − 1

(3.4)

fn+1 (y0 − xn ) = fn (xn − xn−1 )
As we solve equation (3.4) we find the equilibria exhibit bifurcation behavior with respect
to y0 . The stabilities of the equilibria of the set of second-order ordinary differential equations (3.2) can be found by computing the eigenvalues of the Jacobian of f . For a system of
second-order undamped equations the stabilities are determined as shown in the following
equations:
˜ or x
˜ =0
¨ = [Df ] x = −Kx
¨ + Kx
x
Assuming a response of the form x = e±iωt φ, the eigenvalue problem becomes
32

(3.5)

˜ − λI φ = 0 ⇒ λ = ω 2
K

(3.6)

where ω are the eigenfrequencies, λ are the eigenvalues and φ are the eigenvectors. If λ < 0
there is an exponential solution that blows up as time t increases. Neutrally stable oscillations
(stable with small added damping) exist for λ > 0, and neutrally stable solutions for λ = 0:
√

√

x = A1 e −λt + A2 e− −λt
x = A1

√
i
e λt

+ A2

∀λ<0
(3.7)

√
−i
λt
e

∀λ>0

Thus, in the second-order undamped form, a bifurcation is indicated as a single eigenvalue passes through zero, corresponding to a transition of a second-order oscillation to a
second-order saddle (in a two-dimensional phase space). Equivalently, it corresponds to an
effective stiffness going from positive to negative.

For analysis of the codimension, the second-order ordinary differential equations for the
n-DOF system are converted into first-order differential equations by defining a 2n × 1 state
˙ x]T yielding unforced equations of motion of the form
vector z = [x,

Az˙ = Lz + ˜
f

(3.8)

where




 M 0 
A=
,
0 I





 −C −K 
L=
,
I
0





 fnl 
˜
f =

0

(3.9)

A and L are 2n × 2n matrices and ˜
f is 2n × 1 vector. We use the above equation (3.8) for
the twinkler to obtain the numerical solution in a SDOF and 2-DOF system and study the

33

bifurcation behaviors with respect to the final pull position y0 . The state-variable system
can be linearized about an equilibrium and put in the form z˙ = Tz. The eigenvalues, α,
from the eigenvalue problem Tu = αu indicate stability from the signs of the real parts of
αj , for j = 1, 2, . . . , 2n.
In the following analyses, we use λ to indicate eigenvalues from the undamped secondorder-set formulation, and α to represent the eigenvalues from the state-variable description.

3.2

Examples of Bifurcations of Equilibria in SnapThrough Oscillators

The cubic nonlinearity in the twinkler gives rise to multiple stable equilibria when the twinkler is pulled into the negative stiffness region and snaps through. We study the bifurcations
of these equilibria first for the SDOF system and then perform an analytical solution backed
by numerical simulations for the 2-DOF oscillator.
We used a coordinate transformation to simplify the form for the analysis of the bifurcation events. This provides something like a normal form for the system of second-order
equations. For the n-DOF twinkler we obtain the equilibrium solutions from the equation (3.4). For the symmetric case, all the spring forces are identical. Hence γi = γ, ai = a
and bi = b. We assume mi = m, ci = c, and γ = 2m. For all computations, we have used
γ = 2.0, a = 0.5, and b = 3.0. The values of a and b are chosen to represent a stiffness
function of a spring that is preloaded to be somewhat near snap through.
The equilibrium equations are first solved analytically to understand the bifurcation behavior of the snap-through twinkling chain of one and two masses. The equilibrium solutions
to the global and local equations are presented in this chapter.
34

3.3
3.3.1

Bifurcations in SDOF Snap-Through Oscillator
Global Equilibria

The analytical solution of the SDOF system, at equilibrium, is given by f1 (x) = f2 (y0 − x),
where f1 and f2 are defined as in equation (3.3). Referring to Figure 3.3, the equilibrium
equation is written as

γ1 x(x − a1 )(x − b1 ) = γ2 (y0 − x)((y0 − x) − a2 )((y0 − x) − b2 )

	
  

x

f1 (x)
m	
  

c

(3.10)

f 2 (y − x)

y

	
  

€

€
Figure 3.3 SOF spring-mass
chain €
connected by one mass, two nonlinear springs, and one
dash-pots. As shown in this figure the left spring is fixed to a base and the right most spring
is pulled quasistatically to a distance y0 .

€

€

For the symmetric case, both springs in the SDOF chain are identical, and hence have
identical spring forces i.e. γ1 = γ2 = γ, a1 = a2 = a and b1 = b2 = b.
For γ = 2, solving the equation (3.10) for the symmetric case, we get three distinct roots
as functions of y0 , given by

x=

y0 1
,
2 2

y0 −

−4ab + 4(a + b)y0 − 3y02 ,

1
2

y0 +

−4ab + 4(a + b)y0 − 3y02
(3.11)

The numerical simulation is used to find the stability of the above equilibria using the
Jacobian and plotted in the Figure 3.4. To simplify the solution we now change coordinates
35

by applying the following transformation. First we move the origin to the bifurcation point,
y
and then rotate the x = 20 line to coincide with the horizontal axis:

y0 = p + yb ,
x

3

Stable
Unstable

3

1

B1

0

a2
b2

1
0

1

2

3

1
0

a1
b1

4

Stable
Unstable

2

B2

x

x

2

(3.12)

p + yb
= u +
2

a2
b2

1
5

0

y0

1

2

3

∆
∆

a1
b1
4

5

y0

Figure 3.4 The global bifurcation behavior of a SDOF system with respect to the pull parameter y0 . (a) Bifurcations of equilibria in symmetric system. (b) Bifurcations in symmetry
breaking system with an applied perturbation a2 = a1 + δ and b2 = b1 + δ, for a1 = 0.5,
b1 = 3, and δ = 0.2.
Now that the origin is moved and the axes are rotated, the bifurcation point coincides
y

with the origin, and the axis x = 20 coincides with the horizontal axis in the transformed
system. Also, as the oscillation in the chain settles down, we get y˙ = y˙0 = p˙ = 0, x˙ = u˙ and
x¨ = u¨. Plugging in the above transformations into our original equation (3.10) we get the
transformed damped equation of motion

m¨
u + cu˙ = − u 4u2 + 3p2 − (4(a + b) − 6yb ) p + 4ab − 4(a + b)yb + 3yb2

(3.13)

At equilibrium u˙ = 0 and u¨ = 0. Also after the transformation, one of the bifurcation
points lies at origin. Hence plugging (u, p) = (0, 0) into equation (3.13) at equilibrium, we
36

get the following quadratic equation in yb that gives us two values of yb from which we
consider the smaller value as below
3yb2 − 4(a + b)yb + 4ab = 0
2
a+b−
∴ yb =
3

(3.14)

a2 − ab + b2

We would consider the other solution if we were to move our new origin to the other bi√
furcation point. Letting y1 = a2 − ab + b2 , then yb = 32 (a + b − y1 ). Hence the transformed
non-dimensional EOM of the SDOF system is

u¨ + cu˙ = −u(4u2 − 4y1 p + 3p2 )

(3.15)

and the equilibrium solutions to the SDOF twinkler in terms of y1 and p are

u=

3.3.2

0,

−

1
2

4y1 p − 3p2 ,

1
2

4y1 p − 3p2

(3.16)

Local Normal Form and Stability Analysis

Local to p = 0 we find the normal form by neglecting the p2 term in equation (3.15) and
2

1

letting pˆ = (4 3 y1 p) and replacing (4 3 u) by our new u

u¨ + cu˙ = −u(u2 − pˆ)

(3.17)

Therefore the solutions to the normal form at equilibrium

u = 0, −
37

pˆ,

pˆ

(3.18)

B1

0

B2

u

u

1

1

2

2

1

1

0

0

u

2

1

2

Stable
1

0

1

2

Unstable
2

3

1

4

Stable
1

0

1

p

Unstable
2

3

4

2

Stable
1

0

p

1

Unstable
2

3

4

p

Figure 3.5 Left figure is the bifurcation diagram for the symmetric case in the transformed
coordinate system. The bifurcation diagram shows qualitatively similar bifurcation behaviors
at the bifurcation points B1 and B2 . The next two figures show two different configurations
of the global bifurcations in the perturbed SDOF system. The second and third figure
correspond to perturbed spring forces in the transformed system where the right hand side
of the equation (3.15) are (−u(4u2 − 4y1 p + 3p2 ) − ) and (−u(4u2 − 4y1 p + 3p2 ) − p + 2 )
respectively for = 0.1. The symmetry breaking at the bifurcation points show the expected
perturbations of a pitchfork bifurcation that breaks into the saddle-node bifurcation.
To understand the dynamics at the bifurcation point we need to examine the degree
of degeneracy. In order to do so we will compute the codimension of the system from the
eigenvalues λ. In first-order form we can represent the normal form of the SDOF system as:




 z˙1 



 −c
=


 z˙ 
1
2






+ pˆ  z1 



 z 
0
2

−3z22

(3.19)

T
where, [ z1 , z2 ]T = [ u,
˙ u ] . The characteristic equation then becomes,

λ2 + λc − pˆ − 3z22 = 0

On the solution curve u = z2 = 0, the eigenvalues are λ = 21 −c ±

(3.20)

c2 + 4ˆ
p . For

p < 0, both the eigenvalues are purely imaginary for undamped case i.e. the solutions are
neutrally stable however in presence of light damping we get complex conjugate eigenvalues
with negative real part which means a stable solution. We get saddles for both damped

38

√
and undamped case for p > 0. On the solution curves u = z2 = ± p, the eigenvalues
become, λ = 12 −c ±

c2 − 8ˆ
p , where p > 0. Therefore for undamped system there are

two purely imaginary eigenvalues which means the solutions are centers i.e. neutrally stable
and in presence of light damping we get complex conjugate eigenvalues with negative real

5

a2
b2

0

a1
b1

Potential Energy

Potential Energy

parts which imply that the solutions are stable.

B1

5
10

B2

15

Stable
Unstable

20
0

2

4

5

a2
b2

0

8

10
15

Stable
Unstable
0

y0

∆
∆

5

20
6

a1
b1

2

4

6

8

y0

Figure 3.6 These figures represent the total spring potential energy of the SDOF system at
the final equilibrium state. Two overlapping energy levels from the symmetric case in the
negative stiffness region are unfolded with an applied perturbation in the right figure, where
a2 = a + and b2 = b + , for = 0.1.

Since at the local bifurcation point the coordinates are (p, u) = (0, 0), the characteristic
equation at the bifurcation point becomes λ (λ + c) = 0. Then the eigenvalues of the first
order equation (3.19) are λ = {0, −c}.
In case of undamped system we get two zero eigenvalues, which implies that the undamped SDOF system has a codimension-2 bifurcation. However in presence of one damping element as shown in Figure 3.3, it is clear from the equation (3.20) that one of the
eigenvalues is zero, confirming a codimension-1 bifurcation point for a lightly damped SDOF
system. The bifurcations for the symmetry breaking case involve the usual perturbation
from local pitchfork bifurcations to saddle node bifurcations, and transcritical bifurcations
which eventually change to saddle node bifurcations.
39

Two local pitchfork bifurcations are observed at B1 and B2 as shown in the Figure 3.4
and 3.5, when the symmetric system snaps through, providing a parameter interval, i.e. the
negative stiffness region, in which two stable and one unstable equilibria exist.
Breaking the symmetry typically breaks the pitchforks into saddle-node bifurcations,
preserving the interval of two coexisting stable equilibria. For the lightly damped symmetric
system, one zero eigenvalue at the bifurcation point can be referred to as a codimension
one bifurcation, which however is a codimension two bifurcation in the undamped system as
both the eigenvalues go to zero. The global symmetry breaking bifurcation behavior reveals
the presence of multiple energy levels in the negative stiffness region (Figure 3.6).
	
  

x1

f1 (x1 )

f 2 (x 2 − x1 )

m1	
  

c1
€

m2	
  

c2

	
  

€ €

x2

€

f 3 (y − x 2 )

y

	
  

€

Figure 3.7 2-degree-of-freedom spring-mass chain connected by 2 masses, 3 nonlinear springs,
and 2 dash-pots. The left spring is fixed to a base €
and the right most spring is pulled
quasistatically to a distance y0 .

€

3.4
3.4.1

€

Bifurcations in 2-DOF Snap-Through Oscillators
Global Equilibria

We study the dynamics of the 2-DOF system to understand the system behavior as the
oscillations settle down to an equilibrium configuration. At equilibrium f1 (x1 ) = f2 (x2 − x1 )
and f2 (x2 − x1 ) = f3 (y0 − x2 ). Therefore the equilibrium equations are written as

40

γ1 x1 (x1 − a1 )(x1 − b1 ) = γ2 (x2 − x1 )((x2 − x1 ) − a2 )((x2 − x1 ) − b2 )
γ2 (x2 − x1 )((x2 − x1 ) − a2 )((x2 − x1 ) − b2 ) = γ3 (y0 − x2 )((y0 − x2 ) − a3 )((y0 − x2 ) − b3 )
(3.21)
The numerical solution of the equilibrium for this 2-DOF spring mass system results
in the bifurcation curves as Figure 3.8. In making Figure 3.8, we numerically determine
stability from the eigenvalues of the Jacobian of equation (3.1), for the 2-DOF case, about
the equilibrium. In doing so we find that at least one of the eigenvalues at points B1 and
B2 is zero indicating bifurcation points.
We transform the coordinates to simplify the EOM. The change of coordinates is done
by first moving the origin to the bifurcation point B1 , located at y = yb , and then rotating
y

2y

p+yb
3 ,

and x2 = u2 +

the lines x1 = 30 and x2 = 30 to coincide with the horizontal axis, using y0 = p + yb ,
x1 = u1 +

2(p+yb )
,
3

where yb is the bifurcation value corresponding to

point B1 in Figure 3.8.
The value(s) of yb is calculated by knowing that the Jacobian at point B1 , has a zero
eigenvalue. The Jacobian of


f (x) = 
y


f2 (x2 − x1 ) − f1 (x1 )
f3 (y0 − x2 ) − f2 (x2 − x1 )

(3.22)




2y

on the line x1 = 30 , and x2 = 30 (Figure 3.8) is

2
−2
J = (3ab − 2ay0 − 2by0 + y02 ) 
3
1


1

−2

(3.23)

Now making determinant of above Jacobian zero and solving for y0 (denote it as yb ) we
get

41

4

Unstable
Neutrally Stable
Stable
B8

3

B6

2

1

x2

x1

B4

B5

B4

B2
B5
B9
B7

B1

0

B3

B3

0

B6

B10

2

B9

B1

Neutrally Stable
B8

4

B2

B10

1
0

Unstable
Stable

6

B7

2

2
4

6

8

0

y0

2

4

6

8

y0

Figure 3.8 The bifurcation diagram for the equilibrium solutions of the lightly damped symmetric 2-DOF system with respect to the pull parameter y0 , where B1 , B2 , . . ., B10 are
the bifurcation points. The dashed lines represent unstable solutions, and the solid lines
represent the stable equilibrium solutions (neutrally stable for the undamped system). The
vertical dotted lines show infinitely many solutions at y0 = a + b, where at the bifurcation
points B7 − B10 , two of the four eigenvalues are complex conjugates with zero real parts
and the other two are zeros for undamped system, whereas with light damping there is one
zero, one purely real negative, and the other two are complex conjugate eigenvalues with
negative real parts. The bifurcation points B3 − B6 , are saddle-nodes with two zero and
two complex conjugate eigenvalues with zero real parts for undamped system, and with light
damping there are two complex conjugate eigenvalues with negative real parts, one zero, and
one purely real negative eigenvalues.
yb2 − 2(a + b)yb + 3ab = 0
∴ yb = a + b ±

a2

(3.24)

− ab + b2

Since we are moving our coordinate system to the bifurcation point B1 , here we consider
the smaller value of yb . The other solution for yb corresponds to the bifurcation point B2 .
Since both B1 and B2 are qualitatively similar we only consider to study the behavior at
B1 .
To simplify equations (3.21), we transform the coordinates [71, 73] and subsequently
y

introduce non-dimensional variables (p, u1 , u2 , τ, ζ), such that y0 = p 31 + yb , x1 = (3u1 +
√
√
√
yb
2yb
y1
y1
2
c
p) 9 + 3 , x2 = (3u2 + 2p) 9 + 3 , τ = ty1 3 , and m = ζy1 32 , where y1 = a2 − ab + b2 ,

42

Stable
Unstable

B8

2

u1

Stable

Neutrally Stable
B10

1 B4

B6

0

B1

1 B3

u2

3

B2
B5

B9

2

B7
0

2

Neutrally Stable

Unstable
3
B8
2
B10
1
B4
0
B1
1 B
B9
3
2
B

B2

B6
B5

7

4

0

6

2

p

4

6

p

Figure 3.9 The bifurcation diagram for the equilibrium solutions (u1 , u2 ) in terms of the pull
parameter p are qualitatively similar to the Figure 3.8, hence the stabilities of the solution
curves and the degree of degeneracy of the bifurcation points are inferred.
√
yb = a + b − a2 − ab + b2 , to obtain nondimensionalized global equations of motion for the
twinkling oscillator as

u¨1 + 2ζ u˙ 1 − ζ u˙ 2 = (u2 − 2u1 ) A(u1 , u2 , p) − 3u2 + pu2

(3.25)

u¨2 − ζ u˙ 1 + ζ u˙ 2 = (u1 − 2u2 ) A(u1 , u2 , p) + 3u1 − pu1

where A(u1 , u2 , p) =

2
d2 u
du
u21 − u1 u2 + u22 − 2p + p3 , u˙ i = dτi , and u¨i = 2i . At equilibrium
dτ

u¨1 = u¨2 = 0, and u˙ 1 = u˙ 2 = 0. Therefore the nondimentional equilibrium condition for the
2-DOF twinkler is
g1 (u1 , u2 , p) = (u2 − 2u1 ) A(u1 , u2 , p) − 3u2 + pu2 = 0

(3.26)

g2 (u1 , u2 , p) = (u2 − 2u1 ) A(u1 , u2 , p) − 3u2 + pu2 = 0
Hence the equilibrium solutions to the above equations (3.26) are
u1 , u2 = (0, 0), (−2r1 , −r1 ), (−2r2 , −r2 ),
(r1 , −r1 ), (r2 , −r2 ), (r2 , 2r2 ), (r1 , 2r1 )
43

(3.27)

0
R4

R2 R1

R3

R6

R0

R5
4

3

2

1 0

1

2

3

4
3
2
1
0
1
2
3

p

0
R4

R6

u2

p

u2

u2

4
3
2
1
0
1
2
3

R0
R5
4

3

2

1 0

u1

1

2

4
3
2
1
0
1
2
3

3

p

0
R4

R3

R6

R0
R1

R2

R5
4

3

2

u1

1 0

1

2

3

u1

Figure 3.10 The equilibrium solutions projected onto the u1 − u2 plane and noted as Ri =
(u1 , u2 )i . The solid and dashed ellipses and straight lines satisfy g1 (u1 , u2 , p) = 0 and
g2 (u1 , u2 , p) = 0 respectively. As p approaches zero from both directions the equilibrium
solutions R1 , R2 and R3 converge into R0 . For p < 0 the points R1 , R2 and R3 are unstable
and become stable when p > 0. R0 changes from stable to unstable as p goes from negative
to positive. The stabilities of the points R4 , R5 and R6 remain stable on both sides local to
p = 0.
where r1 = 61 p − 3 +

9 + 18p − 3p2

and r2 = 16 p − 3 −

9 + 18p − 3p2 , which

have elliptical solution curves as shown in Figure 3.9 where, the narrow elliptical curve of
equilibria are actually two overlapping elliptical curves. These expressions were obtained
using the “Solve” command in Mathematica. The stabilities of the equilibria curves are
determined from the eigenvalues, λ, as discussed in equations (3.5) - (3.7).
As p increases, the baseline solution (the horizontal line in Figure 3.9, and the diagonal
straight line in Figure 3.8) destabilizes in a bifurcation at p = 0 that features a collision
of four branches of equilibria on either side of the bifurcation point. Three branches are
visible on either side of the bifurcation point in each figure because the vertical axes of
these figures are projections of a higher dimensional phase space, and due to symmetry some
curves overlap in each projection. In both plots in Figure 3.9, the narrow elliptical curve of
equilibria is actually two overlapping elliptical curves.
At p = 0 bifurcation point B1 eight curves emanate from the bifurcation point, in which
a neutrally stable (stable when light damping is applied) curve collides simultaneously with
three unstable curves to produce four unstable curves. Similar phenomenon occurs at p = 6
44

at bifurcation point B2 . Due to their structure we call the bifurcations at B1 , and B2 as
the “star bifurcation”. The star bifurcation occurs when the twinkler enters the negative
stiffness region with respect to the quasistatic pull parameter p.
The equilibria on the (u1 , u2 ) plane are shown in Figure 3.10. The dashed and solid
ellipses and straight lines in Figure 3.10 satisfy g1 (u1 , u2 , p) = 0 and g2 (u1 , u2 , p) = 0
respectively of equation (3.26). Points of intersection between a dashed and solid curves
satisfy both equations, and hence are equilibrium points (u1 , u2 ), denoted by R0 , R1 ,...,
R7 . As p increases, at p = 3, the solid and dashed ellipses in Figure 3.10 on the (u1 , u2 )
plane overlap generating infinite number of equilibria at this particular value of p (details
are discussed later in this section). At p = 3 the equilibria on the elliptical curves undergo
an exchange of stability without intersection at B7 -B10 as seen in Figure 3.9. Hence we call
this bifurcation as “eclipse bifurcation”.
We examine the structure of these these new type of bifurcations star and eclipse bifurcations in the normal form next. The degree of degeneracy of these bifurcations are discussed
as well.

3.4.2

Star Bifurcations

In order to study the bifurcation behavior and degree of degeneracy of the star bifurcation,
we find a normal form near B1 . Local to the origin near p = 0, let p, u1 , and u2 be of order
. Hence neglecting the cubic terms i.e. O( 3 ) from equation (3.26), we get the undamped
equations

u¨1 = h1 (u1 , u2 , p) = (2u1 − u2 ) 3u2 + 2p
u¨2 = h2 (u1 , u2 , p) = (u1 − 2u2 ) 3u1 − 2p
45

(3.28)

2

1

1

0

Λ1
Λ2

1
2

Λ 1, 2 at R1

Λ 1, 2 at R0

2

0

Λ1

1
2

0.4 0.2 0.0 0.2 0.4

Λ2

0.4 0.2 0.0 0.2 0.4

p

p
2

Λ1

1

Λ 1, 2 at R3

Λ 1, 2 at R2

2

0

Λ2

1
2

1
0

Λ1

1
2

0.4 0.2 0.0 0.2 0.4

Λ2

0.4 0.2 0.0 0.2 0.4

p

p

Figure 3.11 The vertical axis represents the eigenvalues λ at the equilibria in the normal form
at the star bifurcation local to pˆ = 0. The points R1 , R2 and R3 remain unstable whereas
the stability of R0 changes from stable to unstable as pˆ goes from negative to positive.
u
ˆ

u
ˆ

Letting p = p2ˆ , u1 = 31 , and u2 = 32 , the above equations are simplified to obtain a
normal form as

ˆ 1 (ˆ
u¨ˆ1 = h
u1 , uˆ2 , pˆ) = (2ˆ
u1 − uˆ2 )(ˆ
u2 + pˆ)

(3.29)

ˆ 2 (ˆ
u¨ˆ2 = h
u1 , uˆ2 , pˆ) = (ˆ
u1 − 2ˆ
u2 )(ˆ
u1 − pˆ)
The equilibrium solutions to the above normal form are

uˆ1 , uˆ2 = (0, 0), (−2ˆ
p, −ˆ
p), (ˆ
p, −ˆ
p), (ˆ
p, 2ˆ
p)

(3.30)

ˆ 1 (ˆ
ˆ 2 (ˆ
The solutions of h
u1 , uˆ2 , pˆ) = 0 and h
u1 , uˆ2 , pˆ) = 0 correspond qualitatively to
46

u1

0.1
0.0
0.1

0.2

R1

R2 , R3

0.1

R0

u2

0.2

R0
R2 , R3

0.0
0.1

R1

0.2
0.10 0.05 0.00 0.05 0.10

R1 , R2

R0
R0
R1 , R2

R3

0.2
0.10 0.05 0.00 0.05 0.10

p

p

0.2

0.2

0.1

0.1

u2

u1

R3

0.0

0.0

0.1

0.1

0.2
0.10 0.05 0.00 0.05 0.10

0.2
0.10 0.05 0.00 0.05 0.10

p

p

Figure 3.12 Symmetric case of the star bifurcation is shown on the top, and the bottom
ˆ 1 + = 0, h
ˆ 2 = 0}.
figures show the symmetry breaking of the star bifurcation, where {h
With reference to the symmetric case, overlapping projected branches are revealed.
the equilibria R0 , R1 , R2 and R3 in Figure 3.10. As pˆ passes through zero, these four
equilibria are seen to collide at the origin, and then separate again. This is part of the star
bifurcation. For the stability analysis of these equilibria, the Jacobian of the second-order
ˆ 1 (ˆ
ˆ 2 (ˆ
ˆ where h
ˆ= h
form is computed from J = −Dh,
u1 , uˆ2 , pˆ), h
u1 , uˆ2 , pˆ)


T

, such that



u2 + pˆ)
(2ˆ
u2 − 2ˆ
u1 + pˆ)
 −2(ˆ
J=

(2ˆ
u2 − 2ˆ
u1 + pˆ)
2(ˆ
u1 − pˆ)

(3.31)

The two eigenvalues λi of the undamped normal form are computed from the above
Jacobian, at the equilibria obtained in equation (3.30), and are illustrated in Figure 3.11.
The stabilities are determined by these eigenvalues λi of the Jacobian, J, evaluated at the
47

0.2

0.1

0.1

u2

u1

0.2

0.0

0.0

0.1

0.1

0.2
0.10 0.05 0.00 0.05 0.10

0.2
0.10 0.05 0.00 0.05 0.10

p

0.2

0.2

0.1

0.1

u2

u1

p

0.0

0.0

0.1

0.1

0.2
0.10 0.05 0.00 0.05 0.10

0.2
0.10 0.05 0.00 0.05 0.10

p

p

ˆ 1 − = 0, h
ˆ 2 = 0}, where the branches
Figure 3.13 (top) The perturbation is such that {h
with positive slope on (ˆ
p, uˆ1 ) plane are both two distinct, overlapping projected branches
ˆ 1 − = 0, h
ˆ 2 + = 0},
that is separated in the projection on the (ˆ
p, uˆ2 ) plane. (bottom) {h
where all the branches in (ˆ
p, uˆ1 ) and (ˆ
p, uˆ2 ) planes are revealed unfolding the star bifurcation
into saddle nodes.
fixed points. Since it is a 2-DOF system, the two eigenvalues λi in the second-order form
result in four eigenvalues αj in the first-order form. Therefore four eigenvalues αj go to
zero as pˆ passes through pˆ = 0. The system with four eigenvalues αj in the first-order form
simultaneously going to zero is referred to as a codimension-four bifurcation [127, 128]. The
bifurcation point B2 is qualitatively similar to B1 but from the opposite direction. Hence for
the undamped symmetric 2-DOF system there exist two codimension-four “star bifurcations”
B1 and B2 .
However when we apply one of the dampers, i.e. making the damping coefficient from

48

0.2

0.1

0.1

u2

u1

0.2

0.0

0.0

0.1

0.1

0.2
0.10 0.05 0.00 0.05 0.10

0.2
0.10 0.05 0.00 0.05 0.10

p

0.2

0.2

0.1

0.1

u2

u1

p

0.0

0.0

0.1

0.1

0.2
0.10 0.05 0.00 0.05 0.10

0.2
0.10 0.05 0.00 0.05 0.10

p

p

ˆ 1 + = 0, h
ˆ 2 − = 0}, star breaks into pitchfork bifurcations
Figure 3.14 In top figures {h
on both the uˆ1 and uˆ2 plane. Whereas in the bottom figures it breaks into saddle-nodes,
ˆ 1 − = 0, h
ˆ 2 − = 0}. Here the symmetry breaking on (ˆ
where {h
p, uˆ1 ) plane is qualitatively
ˆ 1 − = 0, h
ˆ 2 + = 0} shown in Figure 3.13. However this
similar to the perturbation case {h
presents a different symmetry breaking configuration on (ˆ
p, uˆ2 ) plane
one of the dashpots to be zero, we get three zero eigenvalues αj and one negative purely
real eigenvalue; a codimension-three bifurcation. In the presence of both the dashpots,
two of the four eigenvalues are zero and the other two are negative and purely real, hence
a codimension-two bifurcation occurs, (One such example is shown in Figure 3.15 on the
complex plane for equilibrium R0 ). Therefore depending on the nature of damping we get
either codimension-two or three star bifurcation at pˆ = 0 for a damped system.
Now to reveal the degeneracy at the star bifurcation point, we apply a small perturbation
to the normal form, which results in the breaking and unfolding of the star bifurcation. The

49

0.2

Im�Α0 �

0.1

0.0

�0.1

�0.2
�0.3

�0.2

�0.1

Re�Α0 �

0.0

0.1

Figure 3.15 The state-variable eigenvalues α0 of the symmetric damped (with two dampers)
2-DOF system near the star bifurcation in the first-order form at equilibrium R0 . Single and
double arrows represent slow and fast approach respectively to show relative dynamics of
the eigenvalues. Solid squares and circles represent the initial positions (i.e. for pˆ < 0) and
the hollow squares and circles represent the final positions (i.e. for pˆ > 0) of the eigenvalues
as pˆ goes from negative to positive in the direction of the arrows. Both the square and
the circle on the positive x-axis pass through the origin at pˆ = 0, hence making pˆ = 0 a
codimension-two bifurcation point.
bifurcations are shown in both the (ˆ
p, uˆ1 ) and (ˆ
p, uˆ2 ) planes through the local normal form
under various types of perturbations (Figures 3.12, 3.13 and 3.14). We observed pitchfork
bifurcations under specific perturbations and saddle nodes with general perturbations to the
star bifurcations.
The case with two damping elements shows that two of the eigenvalues go through zero
as pˆ increases as in Figures 3.16 and 3.17 for all four equilibria R0 , · · · , R3 . In fact, both of
these eigenvalues go through zero simultaneously at pˆ = 0.
The equilibrium point R0 has all four eigenvalues with negative real parts for pˆ < 0 indicating a stable equilibrium. As pˆ increases and passes through zero two of these eigenvalues
(α01 , α03 ) collide at the origin and the other two (α02 , α04 ) become purely real and negative
as shown in the zoomed in part in Figure 3.16(a). Eventually for pˆ > 0, all four of these
50

0.2
0.1

Α01

Α02

Α03

0.0
0.1

Α03

Α04

Α02

0.1

Α13

Α11
Α12

0.0

Re Α1

Re Α0

0.2

Α01

0.2

0.1
0.2

0.02
0.00
0.3
0.02
0.04
0.4
0.06

Α11 Α12

0.02
0.00
0.02
0.04

Α14

0.3

Α04

0.0005
0.0000
0.0005
0.0010
0.0015
0.0020
Α
14

0.4

0.003
0.002
0.001
0.000
0.001
0.002
0.02
0.01

0.00

0.01

0.02

0.010

0.005

p
0.2

Α01
Α01
Α03

0.0
0.1

Α02
Α04

Α02

0.02

0.0

0.2
0.01

0.00

0.010

Α11
Α13
Α13
Α14

Α11
Α14

0.1

Α04

0.2

0.005

0.1

Im Α1

Im Α0

0.1

0.000

p

Α03

0.2

Α13

0.01

0.02

Α12
Α12
0.010

0.005

p

0.000

0.005

0.010

p

(a)

(b)

Figure 3.16 Columns (a) and (b) represent eigenvalues at equilibria R0 and R1 respectively
i.e. the real and imaginary parts of the eigenvalues at equilibria R0 (α0 ) and R1 (α1 ) of the
symmetric damped 2-DOF system near the star bifurcation in the first-order form as pˆ goes
from negative to positive. Each equilibria has four different eigenvalues that are designated
as αji on the plot for j = 0, 1 and i = 1, 2, 3, 4. The dashed and dotted curves represent two
and four overlapping eigenvalues respectively.
eigenvalues (α01 -α04 ) become purely real with two of them positive (α01 , α03 ) and the other
two negative (α02 , α04 ) indicating hyperbolic saddle points for equilibrium R0 .
However at the other three equilibria Rj (for j = 1, 2, 3), for pˆ < 0 two of the four
eigenvalues (Figures 3.16(a), 3.17(a)-(b)) are complex conjugates (αj1 , αj2 ) with negative
real parts, and the other two are purely real (one positive (αj3 ) and one negative(αj4 )),
hence are unstable. For pˆ > 0 the eigenvalues αj2 and αj3 become complex conjugates
with negative real parts, and the other two are purely real (one positive (αj1 ) and one
51

negative(αj4 )), hence are unstable equilibria. We also observe that there is an interval of pˆ
where all of these eigenvalues become purely real with one of them being positive through-out
the interval indicating hyperbolic saddle points, hence unstable.
0.2

Α21

0.1

Α23

0.0

Α21 Α22

0.1

Α22

0.02
0.1
0.00
0.02
0.2
0.04
0.06

Α23

Α32
Α31
Α34

0.0010
0.0005
0.0000
0.0005

Α34

0.4
0.005

0.000

0.005

0.010

0.010

0.005

p
0.2

Α21

0.2

Α23

Α23
Α24

Α21
Α24

Im Α3

Im Α2

0.005

0.010

Α31

Α33

0.1

0.1
0.2

0.000

p

0.1
0.0

Α33

0.02
0.00
0.02
0.04
0.06

Α32

0.3

Α24
0.010

0.1
0.2

Α24

0.3 0.0004
0.0002
0.0000
0.0002
0.0004
0.4

Α31

Α33

0.0

Re Α3

Re Α2

0.2

Α22

0.2
0.005

0.000

0.005

Α33
Α34

Α31
Α34

0.1

Α22
0.010

0.0

0.010

Α32

Α32
0.010

0.005

p

0.000

0.005

0.010

p

(a)

(b)

Figure 3.17 Columns (a) and (b) represent eigenvalues at equilibria R2 and R3 respectively
i.e. the real and imaginary parts of the eigenvalues at equilibria R2 (α2 ) and R3 (α3 ) of the
symmetric damped 2-DOF system near the star bifurcation in the first-order form as pˆ goes
from negative to positive. Each equilibria has four different eigenvalues that are designated
as αji on the plot for j = 2, 3 and i = 1, 2, 3, 4. The dashed and dotted curves represent two
and four overlapping eigenvalues respectively.

3.4.3

Eclipse Bifurcation

Here we formulate a non-dimensionalized normal form local to p = 3 in order to study the
eclipse bifurcation behavior of the 2-DOF system in detail. While the system goes through
52

the eclipse bifurcation, it exhibits infinitely many equilibria whose stability and related degree
of degeneracy will be determined using the concept of codimension [127, 128].
Let’s consider the equations of motion local to p = 3 for the degenerate exchange of
stability. We make a coordinate transformation for the eclipse bifurcation as we consider
√
solutions very close to the bifurcation points B7 − B10 by letting p = 3 + 3q, where q is
√
√
√
√
of order , τ˜ = 3τ , u1 = 3˜
u1 , u2 = 3˜
u2 , and ζ = 3ζ˜ in equation (3.25). Eventually
neglecting O( 2 ) term (q 2 /3) and dropping the tilde, we get simplified non-dimensional
normal form near the eclipse bifurcation:

u¨1 + 2ζ u˙ 1 − ζ u˙ 2 = h1 (u1 , u2 , q) = (u2 − 2u1 ) u21 − u1 u2 + u22 + qu2 − 1
u¨2 − ζ u˙ 1 + ζ u˙ 2 = h2 (u1 , u2 , q) = (u1 − 2u2 )

1.0

R5

R4

0.5

u2

0
R1

R0

0.0
0.5

R3

R2

1.0

R6

1.5
1.5

1.0

0.5 0.0

0.5

1.0

1.5

1.5
1.0
0.5
0.0
0.5
1.0
1.5

q

0

− u1 u2 + u22

R5
R4
R1

R0
R3

u2

q

u2

1.5

u21

R2
R6

1.5 1.0 0.5 0.0 0.5 1.0 1.5

u1

u1

(3.32)

− qu1 − 1

1.5 q 0
R5
1.0
R4
R1
0.5
R0
0.0
0.5 R3
R6
R2
1.0
1.5
1.5 1.0 0.5 0.0 0.5 1.0 1.5

u1

Figure 3.18 The equilibrium solutions projected onto the (u1 , u2 ) plane and noted as
Rj = (u1 , u2 )j , for j = 0, 1, 2, · · · , 6. The solid and dashed ellipses and straight lines
satisfy h1 (u1 , u2 , q) = 0 and h2 (u1 , u2 , q) = 0 respectively. As q approaches zero from both
directions, the two ellipses coincide, resulting in an infinite number of solutions on the ellipse.
All the solutions on the ellipse are marginally stable.
At equilibrium both of these equations represent a straight curve and an ellipse in u1 , u2
space. The dashed and solid ellipses and straight lines in Figure 3.18 satisfy h1 (u1 , u2 , q) = 0
and h2 (u1 , u2 , q) = 0 of equation (3.32) respectively. The intersections between a solid curve
and a dashed curve satisfy both of equations (3.32), and are therefore the equilibria (u1 , u2 ),
53

denoted by R0 , R1 ,..., R6 . At q = 0, both ellipses overlap perfectly in the symmetric system,
producing an elliptic locus of infinitely many equilibria. We refer to this as an “eclipse
bifurcation”. The equilibrium solutions to equations (3.32) are
For stability of these equilibria, the Jacobian of the second-order undamped (ζ = 0) form
of equations (3.32) is computed from J = −Dh, where h = [h1 (u1 , u2 ), h2 (u1 , u2 )]T . We
eliminate q from the Jacobian at equilibrium by using the equations h1 = 0, and h2 = 0 to
represent the Jacobian as functions of (u1 , u2 ) as



2(u2 − 2u1
2(u1 − 2u2 )(u2 − 2u1 )

J=

2(u1 − 2u2 )(u2 − 2u1 )
2(u1 − 2u2 )2
)2

u1 , u2 = R0 : (0, 0); R1 : (−2r2 , −r2 ); R2 : (r1 , −r1 ); R3 : (−2r1 , −r1 );

(3.33)

(3.34)

R4 : (r2 , −r2 ); R5 : (r1 , 2r1 ); R6 : (r2 , 2r2 )
√
√
where r1 = 61 q + 2 3 and r2 = 61 q − 2 3 .
This Jacobian is implicitly dependent on q through the relationships in equation (3.34)
and has the following characteristic equation:

λ2 − 2λ(5u21 − 8u1 u2 + 5u22 ) = 0

(3.35)

The eigenvalues at the equilibria from equation (3.34) are λ1 = 0, and λ2 = 0, 18r22 , 36r12 ,
18r12 , 36r22 , 18r12 , 18r22 for R0 to R6 respectively as shown in Figure 3.19. With λ1 = 0
and λ2 ≥ 0 at all of these equilibria we conclude the solutions to be marginally stable. The
bifurcations of the equilibria in the unperturbed system are shown in Figure 3.20(a).
We examine the stabilities of the infinitely many equilibria lying on overlapping ellipses at

54

Λ 1, 2 at R3 & R5

Λ 1, 2 at R1 & R6

20
15
10
5
0
5

Λ2
Λ1
0.2 0.1 0.0 0.1 0.2

20
15
10
5
0
5

Λ2
Λ1
0.2 0.1 0.0 0.1 0.2

q

q
40

Λ2

30

Λ 1, 2 at R4

Λ 1, 2 at R2

40
20
10

Λ1

0

Λ2

30
20
10

Λ1

0

0.2 0.1 0.0 0.1 0.2

0.2 0.1 0.0 0.1 0.2

q

q

Figure 3.19 The vertical axis represents the eigenvalues at the equilibria in the normal form
at the eclipse bifurcation local to q = 0. For q < 0 the points R1 , R4 and R6 are stable and
become unstable when q > 0 where as R2 , R3 and R5 changes from unstable to unstable as
q goes from negative to positive and R0 remain unstable on both sides local to q = 0.
q = 0. These equilibria satisfy u1 u2 = u21 +u22 −1 (from rewriting the ellipse factor in either
of (3.32) with q = 0), which is inserted into equation (10) to obtain a new characteristic
equation, specifically for equilibria at the eclipse bifurcation, as

8
=0
λ2 + 6λ u21 + u22 −
3

(3.36)

the solutions to which are λ1 = 0, and λ2 = −6(u21 + u22 − 83 ).
It is seen that the eigenvalue λ2 lies on a circle of radius

√
2√ 2
3

on (u1 , u2 ) plane. Therefore

λ2 > 0 if the equilibrium solutions (u1 , u2 ) on the overlapping ellipses lie inside the circle of
√
√
√
radius 2√ 2 . Since the major and minor radius of the ellipses are 2 and √2 respectively (see
3

3

Appendix) and both are less than the radius of the circle, all of the solutions on the ellipses

55

R1
R2 , R5

R0

u2

u1

1.5
1.0
0.5
0.0
0.5
1.0
1.5

R4 , R6

R3

1.5
1.0
0.5
0.0
0.5
1.0
1.5

R5

R1 , R4

R0
R2 , R3

R6

0.10 0.05 0.00 0.05 0.10

0.10 0.05 0.00 0.05 0.10

q

q

1.5
1.0
0.5
0.0
0.5
1.0
1.5

u2

u1

(a)

1.5
1.0
0.5
0.0
0.5
1.0
1.5

0.10 0.05 0.00 0.05 0.10

0.10 0.05 0.00 0.05 0.10

q

q
(b)

Figure 3.20 Local eclipse bifurcations. The solid and dashed curves represent the stable
and unstable branches respectively and the dots represent the bifurcation points. Left and
right figures show the projections on u1 − q, and u2 − q planes respectively. (a) Symmetric
case for the eclipse bifurcation where {h1 = 0, h2 = 0}. (b) Symmetry breaking case
where the eclipse bifurcation is unfolded into transcritical bifurcations for the perturbation
{h1 = 0, h2 − u1 u2 = 0}.
√

are contained inside the circle of radius 2√ 2 (Figure 3.22). Hence we have λ1 = 0 and λ2 > 0
3

at q = 0, i.e. all of these infinitely many solutions are marginally stable. This is always true
for q = 0 since all the equations are in non-dimensional form and do not involve any other
parameters. If cast in state-variable form (discussed in [71, 73]) there are four eigenvalues,
two of which are zero (since λ1 = 0 in the second-order form) at q = 0. The symmetric
system at q = 0, therefore exhibits a codimension-two [127, 128] eclipse bifurcation.
We apply a small perturbation to the local normal form, to reveal the degeneracy at
the eclipse bifurcation points, which results in the breaking and unfolding of the eclipse
56

u2

u1

1.5
1.0
0.5
0.0
0.5
1.0
1.5
0.10 0.05 0.00 0.05 0.10

1.5
1.0
0.5
0.0
0.5
1.0
1.5
0.10 0.05 0.00 0.05 0.10

q

q

1.5
1.0
0.5
0.0
0.5
1.0
1.5

u2

u1

(a)

1.5
1.0
0.5
0.0
0.5
1.0
1.5

0.10 0.05 0.00 0.05 0.10

0.10 0.05 0.00 0.05 0.10

q

q
(b)

Figure 3.21 Different symmetry-breaking bifurcations are presented where the projections
on u1 − q, and u2 − q planes shown. The solid and dashed curves represent the stable
and unstable branches respectively and the dots represent the bifurcation points. (a) The
eclipse bifurcation breaks into stiff subcritical pitchfork bifurcations for the perturbation
{h1 = 0, h2 − u2 = 0}. (b) The eclipse bifurcation breaks into stiff supercritical pitchfork
bifurcations for the perturbation {h1 = 0, h2 + u2 = 0}.
bifurcation. The local normal form is studied for various types of perturbations on both the
(q, u1 ) and (q, u2 ) planes as shown in Figures 3.20(b) and 3.21. It is observed from the various
perturbations that the eclipse bifurcation can change to stiff pitchfork and transcritical
bifurcations depending on the perturbation type, which can be further reduced to saddlenodes with additional perturbations. A general arbitrary perturbation will take the system
straight into saddle-node bifurcations. Also the overlapping branches at u1 and u2 equal to
±0.5, are revealed with the perturbations. In the presented examples of symmetry breaking,
the locus of neutrally stable equilibria at q = 0 of the symmetric case is disrupted, and some
57

1.5
1.0

u2

0.5
0.0
0.5
1.0
1.5
1.5

1.0

0.5

0.0

0.5

1.0

1.5

u1
Figure 3.22 For symmetric system at q = 0, infinitely many
√ equilibria exist on the overlapping
2√ 2
ellipses that are contained within the circle of radius
. This circle is used as a criterion
3
to determine that these equilibria are marginally stable.
parts of the resulting branches are then changed to stable and some to unstable as seen in the
perturbations of the bifurcations of the normal form local to q = 0 (Figures 3.20 and 3.21).
These features are also evident in the global picture which is described next.

3.4.4

Energy Levels and Global Perturbations

The entire bifurcation set is presented in Figure 3.23, for three symmetry-breaking examples,
revealing the overlapped branches on the (p, u1 ) plane. The locus of infinite equilibria at the
eclipse bifurcation is disrupted under the perturbation resulting in stiff pitchfork bifurcations
for the particular perturbations shown in Figure 3.23. These perturbed cases are observed
when we tweak the parameters, specifically the spring characteristics, of the end spring in
the 2-DOF twinkler. Under various perturbations, these global pictures show the very rich
dynamics of a 2-DOF twinkler as it passes through the negative stiffness regime. Column
(d) of Figure 3.20 shows a local perturbation of the eclipse bifurcation of the same type as
58

observed in the top row of Figure 3.23(a)(b).
5

Unstable
Stable

2
1

u2

1

u1

0

Potential Energy

2

0

0

1

1

2

2
0

2

4

2

0

1

1

2

2
4

4

0

6

2

4

12

6

8

10

12

6

8

10

12

20

6

Unstable
Stable
2

0

2

4

p

Potential Energy

1

u2

0

1

1

2

2
6

10

15

30

2

1

4

8

5

Unstable
Stable

0

6

5

p

2

Unstable
Stable
0

2

4

0
5
10
15
20

Unstable
Stable

25

6

2

0

2

p

p

(a)

2

10

25

Unstable
Stable

p

2

0

0

Potential Energy

0

0

2

p

1

u2

u1

6

2

1

2

Unstable
Stable

30

5

Unstable
Stable

2

0

20

p

p

u1

4

15

25

Unstable
Stable
0

6

5
10

(b)

4

p

(c)

Figure 3.23 (a)-(b) The bifurcation diagram for the equilibrium solutions on the entire p − u1
and p − u2 planes, in various symmetry breaking configurations. The dots represent the
bifurcations that involve a stable branch. The perturbation is applied such that the spring
force of the end spring is perturbed by making a3 = a + , b3 = b + on the top row,
a3 = a + , b3 = b − on the middle row, and a3 = a − , b3 = b + on the bottom row, for
= 0.15. The star bifurcations at p = 0 and p = 6 and the eclipse bifurcation at p = 3, are
broken into combinations of pitchfork and saddle-node bifurcations. Column (c) shows the
corresponding energy levels in the perturbed systems.
Multiple energy levels exist in the negative stiffness region for the symmetry breaking
cases as seen in Figure 3.23(c). At a particular value of the pull parameter p in the negative
stiffness regime, depending on initial conditions or parameter values such as the mass and
damping coefficients, the final energy lands on one of those levels.
59

Figure 3.24 Fractal basin plots of the various energy levels at y0 = 2.7 with symmetry
breaking of a2 = a − and a3 = a + , with a = 0.5 and = 0.2 for varying initial conditions
of the second mass x20 and x˙ 20 .

3.5

Transient Chaos Analysis Through Fractals

We have seen that, given a pull distance, there can be multiple equilibria in a twinkling
oscillator. If symmetry is broken, different equilibrium will have different potential energies,
influencing the available energy for harvesting. Thus, it is of interest to understand how the
equilibria may be reached. The twinkling oscillator is strongly nonlinear and the approach
to equilibria may be transiently chaotic.
We extend the study of the bifurcation analysis of the twinkling oscillator studied in this
Chapter, to understand the chaotic nature of the 2-DOF twinkling oscillator. We intend to
60

Figure 3.25 Fractal basin plots of the various energy levels at y0 = 2.7 with symmetry
breaking of a2 = a − and a3 = a + , with a = 0.5 and = 0.2 for varying the masses, m1
and m2 . of the 2-DOF oscillator.
establish a numerical computation approach for quantification of chaos in the 2-DOF twinkler. We find fractal basins using numerical variations of parameters to find corresponding
energy states in the snap-through regimes.
In this study we performed a numerical simulation, on the twinkling oscillators studied in
Chapter 3, with varying parameters, to estimate the parameter ranges for various attainable
energy levels in the snap-through regime. The EOM from 3 has been numerically simulated
for various initial conditions and various masses for total potential energy availability.
We have used ode23s as the nonlinear solver in MATLAB to solve the EOM for various
masses and initial conditions when a quasistatic pull has been applied to the end spring
61

of the twinkler. From both the fractal plots in Figures 3.24 and 3.25, there are parameter
regions where the system can be easily predicted for its equilibrium energy state as well as
parameter ranges where it becomes almost impossible to pin-point the energy state of the
system at steady state. For example, when x20 is between 1.5 and 2.5, the snap-through
twinkler is very chaotic with unpredictable equilibrium energy state.

3.6

Summary

In this chapter we studied the bifurcations of equilibria with respect to a pull to the end
spring in a simple 2-DOF nonlinear snap-through oscillator. Through the study of local
normal form, we discovered two new bifurcations, namely the star bifurcation and the eclipse
bifurcations, named due to their structures in the state-space.
Perturbation to the normal form has shown the braking of these bifurcations to more
common type of bifurcations such as saddle-node and pitch-fork bifurcations. These perturbations correspond to an nonuniform nonlinear spring elements in the physical system.
Various energy levels are present in the snap-through multi-stable regime of the oscillator.
With parameter variation it is possible to jump from one to the other energy level for particular displacement to the end spring in the snap-through regime. This was shown using a
numerically computed fractal basin boundary. The chaotic nature of the 2-DOF twinkler is
obvious from this study.

62

Chapter 4
Experimental Energy Harvesting
Using Twinkling Oscillators
Experimental energy harvesting is of particular interest to the current scientific community
in order to fill the energy demand by scavenging energy from various ambient sources. In
this decade, we have seen exponential rise in the consumption of electronic devices, for which
power consumption has risen steadily due to the usage of billions of mobile devices, laptops
and computers. To meet this demand we must have continued effort to tap energy sources
that are present in our surroundings and are otherwise unused. One such energy source is the
inherent vibrations produced during our day-to-day activities as well as from the vehicles and
machines that we use. These vibration energy sources are wasted. Here, we are proposing
vibration based models to harvest such wasted power to usable form.
In this chapter we set-up single-degree-of-freedom (SDOF) and two-degree-of-freedom
(2DOF) experimental energy harvester models using bistable systems. The experimental
set-up for the twinkling energy generator (TEG) consists of linear springs and attracting
cylindrical bar magnets. The cylindrical coil of enamel-coated magnet wire with certain
numbers of windings is used as the energy generators. The governing equations are formulated analytically and the experimental results of the frequency up-conversion and power
output are presented.

63

The main idea of this research was to up the frequency in order for us to take advantage
of our magnet-coil harvester system that performs best for fast oscillation of the magnet
inside the coil. The frequency up-conversion is done using snap-through phenomenon of a
bistable oscillator. Energy harvesting using snap-through oscillators have been investigated
by various authors [35, 90–95, 129, 130]. In this research the power harvesting has been done
using attracting magnets and linear springs in a novel fashion such that it forms a snapthrough system. It is demonstrated that using the snap-through phenomenon one can up
the frequency more than 20 times the input frequency.
The work is organized as follows. First we present the experimental set-up and the physical parameters involved. Keeping the experimental set-up in focus, we describe the system
mathematically and perform numerical simulations in order to solve the governing equations. We formulate two different mathematical models for analytical study and comparison
of the numerical simulation results with the experimental output. First model assumes a
point magnetic dipole interactions where as the second model considers more accurate forcing between the cylindrical magnets. The physics behind the experimental nonlinear energy
harvester has been investigated using the two proposed mathematical models. Using a low
frequency input excitation the up-conversion of the frequency in both SDOF and 2DOF
twinkling oscillators has been shown by the mathematical models. Finally the experimental
results for harvested power has been presented as well as it’s comparison with the analytical
models that confirm the power harvesting capacity, and the frequency up-conversion as well
as the broadening of the frequency spectrum.

64

4.1

Experimental Setup and Parameters for the Twinkling Energy Generator (TEG)

In this section we describe in detail our experimental set-up and discuss the parameter values
that are used in the experiments. These parameter values are later used for obtaining the
numerical simulation results. The experimental set-up serves as a building block for our
mathematical models.

4.1.1

Experimental Setup

The experimental rig was designed collaboratively, built at Duke University, and conducted
collaboratively at Prof. Brian Mann’s laboratory at Duke University.
For our experiment, we used an air-track for low mechanical damping, horizontal shaker
to induce low-frequency input, triangular shaped carts as the primary masses with 3Dprinted mounts for the magnets, combination of linear springs and cylindrical bar magnets
to induce bi-stability in the system as well as to facilitate power generation in a coil made of
enamel-coated magnet wire, as the primary set-up shown in Figure 4.1 and 4.2 to facilitate
the twinkling phenomenon. A single mass consisted of the cart, the magnet mount, and
the magnet. The mass was constructed in such a fashion that the magnets are fixed to the
magnet-mounts using nuts and bolts, and the magnets were fixed to the magnet-mounts.
The oscillation of the magnets are used to generate current in a harvester coil made of
enamel-coated magnet wire around the magnet connected to the end mass. The harvester
coil was mounted outside the air-track and a cylindrical bar magnet is passed through the
center of the coil to generate power.
We set-up an electrical circuit in order to capture the voltage output generated in the
65

(a)

(b)

Figure 4.1 Shown here is the TEG experimental setup on the air-track. (a) shows the idle
equilibrium state of the TEG consisting of the primary mass (the cart), the magnets, the
magnet mounts, and a linear spring. (b) shows the TEG in its dynamic state when the
magnets are detached and the right mass oscillates inside the generator coil. The right
spring in (b) is pulled by a horizontal shaker of stroke-length 6 inches.
harvester coil when the oscillating magnet passes through the inducting coil. Using MATLAB
code we ran the experiment and initiated the horizontal shaker and then connected the
generator coil in the circuit board that fed the output analog signal (voltage) into a data
acquisition board (DAB). This DAB converted the analog output into digital signal. The
output digital signal was the voltage output generated in the generator coil and was plotted
using the MATLAB routine.
We built two different experimental models, SDOF and 2-DOF twinklers, for power
66

Figure 4.2 The experimental setup of a 2-DOF twinkling energy generator (TEG) consisting
of an air-track, carts, magnet mounts, magnets, linear springs, and horizontal shaker.
harvesting from low-frequency input oscillation. Springs with different stiffnesses were used
in both SDOF and 2-DOF TEGs. Our experiments included various set-ups for both SDOF
and 2-DOF TEGs. As such, we investigated the generator models with both unsymmetric
springs with different stiffnesses and symmetric springs with almost identical stiffnesses.
We calculated the spring stiffness values from various measured data for all 6 springs
used in the SDOF and the 2-DOF TEG experiments. We measured the weight of the two
cars and the weight of each spring. One can also remove one magnet arm in the SDOF TEG
to make a linear SDOF chain to compare voltage output with nonlinear TEG.

4.1.2

Parameter Values of the TEG

Referring to Figure 4.3, the mechanical parameters in the SDOF TEG are mass m, stiffness k
of the spring connected to the base, stiffness ke of the end spring, the mechanical dissipation
constants c and ce , undeformed length ls of the spring connected to the base, the undeformed
length lse of the end spring, and the initial spring deformation ∆x0 of the linear springs.
Similarly, the mechanical parameters for the 2-DOF oscillator as seen in Figure 4.4 are
67

RL

x(t)

S N

S N

y(t)
L

k

m

ke

Figure 4.3 The SDOF twinkling oscillator schematic consisting of one cart (black), magnet
mounts (blue), magnets (red), beads (yellow) of length dm /2 to maintain a separation of
dm between the magnets in the steady state, and linear springs. Mass m is the sum of the
weights of the cart, the magnet mount, magnet, the separating-beads, and the springs. k is
the stiffness of the spring connected to the left base, and ke is the stiffness of the end spring.
On the magnets, N and S represent the north and south poles of the cylindrical magnets
respectively. When the magnet passes through the coil with inductance RL it generates
current in the coil and powers the resistive load RL . x(t) is the dynamic displacement of the
cart and y(t) is the horizontal pull input.
the masses m1 and m2 , the stiffnesses k1 , k2 , and ke , the mechanical dissipation constants
c1 , c2 , and ce , the un-stretched lengths of the springs ls1 , ls2 , lse , and the initial spring
deformations ∆x01 , and ∆x02 .
The electrical circuit and the magnet parameters are the same for both the SDOF and
the 2-DOF TEGs. The electrical circuit shown in Figure 4.5 parameters are resistance RL
of the resistive load, the internal resistance Ri of the electrical circuit, inductance L of the
enamel-coated magnet wire coil, the number of windings n in the coil, the inner radius rci
and the length lc of the coil respectively, and the transducer constant γ of the circuit. The
magnet parameters are the magnetizations M of the magnets, volume Vm of the magnets,
the outer and inner diameters do and di of the magnets respectively, the length lm of the
magnets, the distance dm between the magnets at idle equilibrium state.
We compute the masses m, m1 , and m2 , by adding the weights of the magnets, magnetmounts, the separating beads, the putty used to fix the beads to the magnets, and half the

68

S N

x1(t)

S N

RL

S N S N

L

k1

m1

k2

m2

x2(t)

y(t)

ke

Figure 4.4 The 2-DOF twinkling oscillator schematic consisting of two carts (black), magnet
mounts (blue), magnets (red), beads (yellow) to maintain a separation between the magnets
in the steady state, linear springs, and horizontal pull input Y (t). Mass m is the sum of
the weights of the cart, the magnet mount, magnet, the separating-beads, and the springs.
k1 is the stiffness of the spring connected to the left base, k2 is the stiffness of the spring
between the carts 1 and 2, and ke is the stiffness of the end spring. On the magnets, N
and S represent the north and south poles of the cylindrical magnets respectively. When
the magnet passes through the coil with inductance RL it generates current in the coil and
powers the resistive load RL . x1 (t) and x2 (t) are the dynamic displacements of the cart 1
and 2 respectively.
weight of each spring to the weight of the carts. The stiffness parameters can change for both
SDOF and 2-DOF twinklers depending on the type of springs used in the experiment. In the
experiment we had used two types of springs, namely, type-1 short springs and type-2 long
springs. All the spring stiffnesses are measured and tabulated in Table 4.1. In Table 4.2, the
measured parameter values associated with the magnets are presented. The inductor coil
and the electrical circuit parameter values are noted in Table 4.3.
The values of the parameters used in the SDOF twinkler are computed using the Tables 4.1, 4.2, and 4.3 and noted to be m = wc + 12 (ws1 + ws2 ) = 0.331 kg for long spring
between base and cart and short type-1 end spring, m = wc + 12 (2ws2 ) = 0.342 kg when
both springs are long type-2 springs, k = kl2 = 44.10 N/m, ke = 17.77 N/m for short spring,
ke = kl3 = 45.52 N/m for long spring, ∆x0 = ∆xswidle = 0.088 m, dm = dsm = 0.011 m for
strong magnets, and dm = dsm = 0.0145 m for weak magnets. We assume the mechanical
dissipation to be ce = c = 0.02 Ns/m. Since we have used identical fixed and oscillating

69

Ri

RL

I

L

Figure 4.5 The electrical circuit in the energy generator consisting of an inductor (L), a
variable resistor with resistance (RL ), and internal resistance of the circuit (Ri ).
magnets the magnet parameters for the SDOF TEG are Mo = Vo = M = 1.05 × 106 A/m,
Vo = Vf = Vm = 4.8264 × 10−6 m3 . The electrical circuit parameters used in the experiment
are L = 5 Nm/A2 , RL = 2000 Ω, Ri = 200 Ω.
In the 2-DOF twinkler, since we only have one harvester coil that generates power from
the oscillation of the mass attached to the end spring, L1 = RL1 = XL1 = γ1 = 0. We use
the same generator coil that was used in the SDOF TEG. Therefore, γ2 = γ, RL2 = RL ,
Ri2 = Ri . Since all the magnets used in the 2-DOF generator are identical, the magnetization
and the volume of the magnets are Mo1 = Mo2 = Mf1 = Mf2 = M = 1.05 × 106 A/m. and
Vo1 = Vo2 = Vf1 = Vf2 = Vm = 1.2066 × 10−6 m3 respectively.
The masses in the 2-DOF oscillator are computed using the parameter values listed in
Table 4.1 as m1 = wc1 + 12 (2ws2 ) = 0.391 kg, and m2 = wc2 + 12 (2ws2 ) = 0.342 kg. The
spring stiffnesses used in the 2-DOF TEG are k1 = 42.76N/m, k2 = 44.1 N/m, ke = 45.52
N/m. We assume the mechanical damping to be constant throughout the model, such that
c1 = c2 = ce = c = 0.01 Ns/m. The distance between the carts and the separation between
the magnets are ∆x01 = ∆x02 = ∆xswidle = 0.88 m and dm = dsm = 0.0145 m respectively.
The electrical circuit parameters used in the 2-DOF set-up are the same as the parameters
used for the SDOF TEG.
70

Table 4.1 Weight of the carts and spring parameter values used in the experimental twinkling
energy generator (TEG).
Parameter
wc (kg)
wc1 (kg)

Value
0.3060
0.3548

wc2 (kg)

0.3060

ws1 (kg)
ws2 (kg)
c (Ns/m)
ky (N/m)
kg (N/m)
kb (N/m)
kl1 (N/m)
kl2 (N/m)
kl3 (N/m)
lsy (m)
lsg (m)
lsb (m)
lsl1 (m)
lsl2 (m)
lsl3 (m)
∆xswidle (m)
∆xssidle (m)
A0 (m)
f0 (Hz)

0.0136
0.0361
0.02
18.33
17.21
52.27
42.76
44.10
45.52
0.137
0.094
0.095
0.238
0.238
0.238
0.088
0.0845
0.1524
0.1

4.2

Description
Weight of SDOF cart with the bead, magnet and putty
Weight of 2-DOF cart near fixed base with yellow bead,
magnet and putty
Weight of 2-DOF cart attached to the end spring with the
bead, magnet and putty
Weight of the type-1 spring with yellow, green or blue tag
Weight of the long type-2 springs
Mechanical damping or dissipation constant
Stiffness of the type-1 spring with yellow tag
Stiffness of the type-1 spring with green tag
Stiffness of the type-1 spring with blue tag
Stiffness of the long type-2 spring 1
Stiffness of the long type-2 spring 2
Stiffness of the long type-2 spring 3
Pre-stretched length of the type-1 spring with yellow tag
Pre-stretched length of the type-1 spring with green tag
Pre-stretched length of the type-1 spring with blue tag
Pre-stretched length of the long type-2 spring 1
Pre-stretched length of the long type-2 spring 2
Pre-stretched length of the long type-2 spring 3
Initial spring stretch in case of weak magnets
Initial spring stretch in case of strong magnets
Stroke-length of the horizontal shaker
Frequency of the input oscillation at the end spring

Mathematical Model of the Twinkling Energy Generator

We derive analytical models of the nonlinear twinkling energy generator in order to understand the dynamics and compare the results with the experiments. The equations of motion
of the TEG would consist of both forces due to the magnets as well as the linear springs.
In this section we show the spring forces and build two different analytical models to derive
the magnetic forces in the nonlinear spring elements.
71

Table 4.2 Magnet parameter values used in the experimental twinkling energy generator
(TEG).
Parameter
wm (kg)
lm (m)
do (m)
di (m)
Vm (m3 )
M (A/m)
dwm (m)
dsm (m)
dcw (m)
dcs (m)
x0coil (m)

Value
0.009
0.0127
0.0127
0.00635
1.2066 × 10−6
1.05 × 106
0.0145
0.011
0.326
0.3225
0.025

Description
Weight of the cylindrical Neodymium magnets
Length of the cylindrical Neodymium magnets
Outer diameter of the cylindrical Neodymium magnets
Inner diameter of the cylindrical Neodymium magnets
Volume of the cylindrical Neodymium magnets
Magnetization of the Neodymium magnets
Distance between weak magnets due to beads
Distance between strong magnets due to beads
Distance between carts in case of weak magnet
Distance between carts in case of strong magnet
Distance of the coil from magnet in idle equilibium

4.2.1

Force Due to Linear Springs

4.2.1.1

Force Due to Linear Springs in SDOF TEG

The spring force due to the linear springs in a SDOF twinkler with a low-frequency pull, as
shown in the figure 4.3, is
Fs = k(∆x0 + x(t))

(4.1)

Fse = ke (y(t) − x(t))
where Fs and Fse are the linear spring force of the spring attached to the base and the spring
force of the end spring respectively, y(t) is the low-frequency pull on the end spring, x(t) is
the displacement of the mass, ke is the stiffness of the end spring, k is the stiffnesses of the
spring connected to the fixed base, and ∆x0 is the initial stretch of the spring connected to
the fixed base in the stationary configuration.
The initial spring stretch (∆x0 ) in stationary configuration is computed from the following
relation:
∆x0 = dc − ls

72

(4.2)

Table 4.3 Electrical circuit parameter values used in the experimental twinkling energy generator (TEG).
Parameter

Value

Description

µ0 (H/m or N/A2 )
µ (H/m or N/A2 )
RL (Ω)
Ri (Ω)
n
rci (m)
lc (m)
L (H)
γmodel1 (As/m)
γmodel2 (As/m)

4π × 10−7
1.05
2000
200
2000
0.016
0.022
5
0.32
0.55

Permeability of free-space
Permeability of the Neodymium magnets
Resistance of the resistive load
Resistance in the inductor coil i.e. inductive reactance
Number of winds in the generator coil
Inner radius of the generator coil
Length of the enamel-coated magnet wire coil
Inductance of the generator coil
Transducer constant for SDOF analytical Model-I
Transducer constant for SDOF analytical Model-II

where dc is the distance between the carts, and ls is the undeformed pre-stretched length of
the spring attached to the fixed base.

4.2.1.2

Force Due to Linear Springs in 2-DOF TEG

The spring forces due to the linear springs in the 2-DOF TEG with a low-frequency pull, as
shown in the Figure 4.4, are

Fs1 = k1 [∆x01 + x1 (t)]
Fs2 = k2 [∆x02 + x2 (t) − x1 (t)]

(4.3)

Fse = ke [y(t) − x2 (t)]
where y(t) is the low-frequency pull on the end spring, x1 (t) and x2 (t) are the displacements
of the masses m1 and m2 , k1 , k2 and ke are the stiffnesses of the springs, and ∆x01 and
∆x02 are the initial stretch of the springs 1 and 2 respectively.
The initial stretches of the springs 1 and 2 in the stationary configuration are computed

73

from the following relation:
∆x01 = dc − ls1

(4.4)

∆x02 = dc − ls2
where dc is the distance between carts, and ls1 and ls2 are the undeformed pre-stretched
lengths of the springs 1 and 2 respectively.

4.2.2

Force Due to Magnets: Model I

In this model, the magnetic force is assumed to follow a point dipole approximation. This
approximation holds true when the lengths, l, of the cylindrical bar magnets are much smaller
compared to the separation, x, between them (l << x) which is often referred to as Gilbert’s
model. Keeping this assumption in mind, we will derive the magnetic forces.
The magnetic field strength, B, at location rf due to the magnet located at ro is

µ
B=− 0 ∇
4π

mo · rf /o
|rf /o |3

µ
=− 0
4π

3rf /o mo · rf /o
mo
−
|rf /o |3
|rf /o |5

(4.5)

where, µ0 = 4π × 10−7 H/m is the permeability of free space, mo is the magnetic moment of
the oscillating magnet at location ro , ∇ is the vector gradient, rf /o = rf − ro , is the vector
to a point of interest (to the fixed magnets in our case) from the oscillating magnet, and
|rf /o | is the distance between the oscillating and the fixed magnet.
In reference to the arrangements in the figure 4.3 mo = mo i, rf /o = −(dm + x)i, |rf /o | =
(dm + x), and mo · rf /o = (dm + x)mo , where dm is the distance between the magnets
due to attached beads and putty to the magnet-mounts to reduce the attracting force on
the magnets and maintain a separation between the magnets in the stable configuration.
This is to avoid direct impact between the magnets and to keep the magnets from impact

74

damages. Also the magnetic moment of a magnet is given as mo = Mo Vo , where Mo is the
magnetization and Vo is the volume of the magnet. Hence the magnetic strength is

µ
B=− 0
4π

mo i
3((dm + x) i)mo (dm + x)
µ0 mo
−
=
i
(dm + x)3
(dm + x)5
2π(dm + x)3

(4.6)

The potential energy of the fixed magnets at rf with magnetic moment mf = mf i, in
the field generated by the magnet at ro is

Um = mf · B =

µ0 mo mf
2π(dm + x)3

(4.7)

The total magnetic force in the longitudinal direction (along the oscillating magnet) is
then computed as
3µ0 mo mf
∂Um
=−
Fˆm = ∇Um =
∂x
2π(dm + x)4

4.2.3

(4.8)

Force Due to Magnets: Model II

In deriving the magnetic force in the model I, we assumed a Gilbert’s point dipole approximation for the force between the two attracting cylindrical magnets. However, in a recent
article [131], the forces between two cylindrical permanent magnets have been derived mathematically. The results there have shown an excellent convergence to the forcing obtained
through experiments for cylindrical magnets of length lm = O(0.01 m) while separated by a
distance of dm in the order of 0.01 m. In this TEG model, we will consider this promising
theoretical derivation [131] of the magnetic forces between two cylindrical bar magnets.
The attracting force between the cylindrical magnets is derived from the gradient of the
magneto-static interaction energy Em . Hence, for two cylindrical magnets aligned along the

75

x-axis, the force is derived as [132]

Fm = −∇Em = −

∂Em
3 ∂Jd
= 2πµ0 M 2 Rm
∂x
∂x

(4.9)

where µ0 is the permeability of vacuum, M is the magnetization of the cylindrical magnets,
Rm is the radius of the magnets, and Jd is the dipolar coupling integral [133]. The dipolar
coupling integral Jd for two cylindrical magnets with parallel axes with lateral distance r
can be found as
∞

Jd (τ1 , τ2 , x) = 2

0

rα J12 (α)
J0
sinh(ατ1 ) sinh(ατ2 ) e−αη dα
2
Rm
α

(4.10)

where α is an independent integration variable, τi = lmi /(2Rm ), i = 1, 2, are the aspect
ratios of the two cylindrical magnets, η = Rx + τ1 + τ2 =
m

2x+lm1 +lm2
2Rm

is the nondimesional

distance between the center of the magnets when separated by a distance x, J0 (α) is the
modified Bessel function of the order zero, and J1 (α) is the modified Bessel function of the
first kind.
Hence the magnetostatic energy between the two cylindrical magnets with their axes
separated by distance r, is

3
Em (τ1 , τ2 , x) = 4πµ0 M 2 Rm

∞

J0

0

rα J12 (α)
sinh(ατ1 ) sinh(ατ2 ) e−αη dα
Rm
α2

(4.11)

and the corresponding attracting force is the energy gradient and is mathematically represented as

Fmx (τ1 , τ2 , x) =

2
−4πµ0 M 2 Rm

∞
0

rα J12 (α)
sinh(ατ1 ) sinh(ατ2 ) e−αη dα
J0
Rm
α
76

(4.12)

When the magnets axes are aligned i.e. r = 0, the zero-order Bessel function becomes
unity i.e. J0 (0) = 1. Since the aspect ratios of the magnets used in the twinkling energy
generator are identical, we have τ1 = τ2 = τ . Hence the force between the attracting
cylindrical magnet is

2
Fmx (τ, x) = −4πµ0 M 2 Rm

∞
0

J12 (α)
sinh2 (ατ ) e−αη dα
α

(4.13)

When the magnets are far from each other, an approximation of the integral in the
equation (4.12) is obtain by expanding the Bessel functions around q = 0 [134] and the force
between the magnets is approximated to be

1
4
Fmx (τ1 , τ2 , x) ≈ − πµ0 M 2 Rm
4

1

1

3r2
1
−
2
2(x + ilm1 + jlm2 )2
i=0 j=0 x + ilm1 + jlm2
(4.14)
(−1)i+j

In the twinkling energy generators the magnets are identical with the magnetization
vectors aligned on their common axes i.e r = 0 and τ1 = τ2 = τ , results in the force between
the fixed and the oscillating magnets as

1
2
1
1
4
Fmx (x) ≈ − πµ0 M 2 Rm
+
−
2
2
4
x
(x + 2lm )
(x + lm )2

(4.15)

With experimental investigation [131] the above magnetic force has shown to be an
excellent approximation when the separation between the magnets, x, is in the order of the
length of the magnets lm . It is to be noted that the above magnetic force is not only between
a fixed and a moving magnet but it is the force between two cylindrical magnets in their
dynamics state, fixed or moving.
Since the TEG experiments have used magnets with a hole in the center, we modify the
77

0.16
0.14
Input y(t) (meter)

0.12
0.1
0.08
0.06
0.04
0.02
0
0

5

10

15
Time (sec)

20

25

30

Figure 4.6 The horizontal shaker input y(t) used for the numerical simulation.
magnetic force such that the force is equal to the magnetic force due to interaction of solid
cylindrical magnets with diameters do less the magnetic force due to interaction of solid
cylindrical magnets with diameters di . Using this modification to the above magnetic force,
the TEGs with hollow cylindrical magnets separated by a distance dm (equal to dwm or
dsm ), the resulting magnetic force can be written as

πµ0 M
F˜m (x) = −

4.2.4

2

d4o − d4i
64

1
1
2
+
−
2
2
(x + dm )
(x + dm + 2lm )
(x + dm + lm )2

(4.16)

Force Balance in Idle Steady State

At the idle steady state condition, the total magnetic energy and the spring (connected to
the left base or the left cart) potential are non-zeros. Hence it would take certain amount of
work in order to detach the magnets. This work is done by the stretching of the end spring.
To mimic the experimental results, the mathematical models need to balance the energy
stored in the magnets and springs to the work done by the pull of the end spring. The total
displacement of the end spring needed for the magnet to detach is given by
78

The time it would take to detach the springs can be found by using the low frequency
input displacement. In our experiment the pull of the end spring is approximated as y(t) =
A0
2 (1

− cos(2πf0 t)), shown in Figure 4.6, where A0 = 6 inch is the stroke length of the

horizontal shaker, and f0 = 0.1 Hz is the frequency of the input oscillation by the shaker.
Hence the time taken by the horizontal shaker to detach the magnets from its idle equilibrium
state is

∆t0idle =

2∆xe
1
cos−1 1 −
2πf0
A0

(4.17)

This formulation can be directly used in the SDOF model. However, in the 2-DOF case
one need to be more careful of determining which of the two nonlinear spring elements would
detach first.

4.2.5

Governing Equations of Motion of the TEG

The force between the fixed and oscillating magnets of a twinkler with attracting magnets
is first derived using the dipole moments generated by the magnets. Combined with linear
springs, the interaction between the magnets forms a bi-stable system. The complete model
is then formulated using the electrical circuit consisting of an inducting coil, a transducer,
and a resistive load.
The generator coil will induce an electrical damping in the SDOF twinkler. The generator
will be manipulated by the magnets and the springs to scavenge energy induced by the lowfrequency pull to the end spring.

∆xe =

1
Fm0 + Fs0
ke
79

(4.18)

where ∆xe is the initial deformation of the end spring needed to detach the magnets, ke is
the stiffness of the end spring, and Fm0 and Fs0 are the magnetic force and the spring force
respectively of the nonlinear spring attached to the end cart.

4.2.5.1

Equations of Motion of the SDOF TEG

The governing equations of motion of the SDOF twinkler, as shown in the Figure 4.3, directly
powering a resistive load is
m¨
x + cx˙ + ce (y˙ − x)
˙ + Fs − Fse + Fm − γI = 0

(4.19)

LI˙ + (RL + Ri )I + γ x˙ = 0
where m is the combined mass of the magnet and the mount, c and ce are the mechanical
dissipation constants, y is the low-frequency pull, I is the electrical current, RL is the
resistance of the resistive load, Ri is the internal resistance of the electrical circuit, L ≈
2 /l is the inductance of the harvester coil where n, r and l are the number of
µ0 n2 πrci
c
c
ci

windings in the coil, inner radius of the coil and the length of the coil respectively, and γ is
a transducer constant.
The magnetic force Fm is replaced by Fˆm (x(t)) when we approximate the magnetic force
using model I and is replaced by F˜m (x(t)) when we use the magnetic force derived in the
model II.

4.2.5.2

Equations of Motion of the 2-DOF TEG

Using the formulation of the magnetic and spring forces, the governing equations of motion
of the 2-DOF twinkling energy scavenger as shown in the Figure 4.4 is formulated as

80

m1 x¨1 + c1 x˙ 1 − c2 (x˙ 2 − x˙ 1 ) + Fs1 + Fm1 − Fs2 − Fm2 − γ1 I1 = 0
L1 I˙1 + (RL1 + Ri1 )I1 + γ1 x˙ 1 = 0

(4.20)

m2 x¨2 + c2 (x˙ 2 − x˙ 1 ) − ce (y˙ − x˙ 2 ) + Fs2 + Fm2 − Fse − γ2 I2 = 0
L2 I˙2 + (RL2 + Ri2 )I2 + γ2 (x˙ 2 − x˙ 1 ) = 0
where m1 and m2 are the combined masses of the magnets and the mounts, c1 , c2 , and c3
are the mechanical dissipations, x1 , and x2 are the dynamic displacements of the masses
m1 and m2 respectively, y is the low-frequency pull to the end spring, I1 and I2 are the
electrical currents in the coils 1 and 2 respectively, L1 and L2 are the inductances of the
harvester coils, RL1 and RL2 are the resistances in the electrical circuit due to the resistive
loads, Ri1 and Ri2 are the internal resistances in the electrical circuits, and γ1 and γ2 are the
transducer constants, which can be derived from Faraday’s law of induction, which couples
the mechanical and electrical systems. Fs1 , Fs2 and Fs3 are the spring forces of the springs
1, 2 and 3 respectively, and Fm1 and Fm2 are the forces due to magnetic interactions between
masses 1 and 2.
Since all the magnets are identical in the TEGs, the magnetic forces Fm1 and Fm2 in
the 2-DOF TEG are replaced by Fˆm1 x1 (t) and Fˆm2 x2 (t) − x1 (t) respectively when
we approximate the magnetic force using the first model and is replaced by F˜m1 x1 (t) and
F˜m2 x2 (t)−x1 (t) respectively when we use the magnetic force derived in the second model.

4.3

Numerical Simulation of the TEG

The first mathematical model of the magnetic force in the twinkling energy generator in
Section 4.2.2 is derived with an assumption that the magnetic force follows a point dipole

81

0.08

Displacement (m)

0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0

1

2

3

4

5
Time (sec)

6

7

8

9

10

Figure 4.7 The numerical output x(t) of the mathematical model of the SDOF twinkler,
after applying one cycle of 0.1Hz input oscillation using the first analytical model. Here we
have used the assumption tf ≈ T − 0.933∆t0idle , where T = 10, is the period of the input
oscillation.
approximation which is true when the length of the cylindrical magnet (lm ) is much smaller
compared to the separation between the magnets (dm ). However, in idle equilibrium state
condition the distance between the two magnets, e.g. in the weak magnet case, is dm =
dwm = 0.0145 m where as the length of the magnets is lm = 0.0127 m. In the idle steady
state it is clear that the condition lm << dm is not satisfied. Hence we would expect a
deviation of the numerical solutions of the analytical model from the experimental results.
In the second mathematical model of the magnetic force in the tinkling energy generator
in Section 4.2.3 is derived using a more accurate representation of the magnetic interaction
forces. This model is an excellent model to mimic the experiment since both the lengths
and the radii of the magnets are in the order of 0.01 meters and the minimum distance dm
between the magnets is in the order of the length lm of the magnets. Using the magnetic
force, from equation (4.16), between the fixed and oscillating magnets of the twinkler and
the spring forces from equations (4.1) and (4.3) the governing equations of motion of the

82

0.8
0.6

Velocity (m/s)

0.4
0.2
0
ï0.2
ï0.4
ï0.6
ï0.8
0

0.01

0.02

0.03
0.04
0.05
Displacement (m)

0.06

0.07

0.08

Figure 4.8 The numerical simulation result of the SDOF twinkler phase portrait after one
cycle of 0.1Hz input oscillation using the first analytical model.
nonlinear energy generator is represented for both the SDOF and the 2-DOF nonlinear
oscillator.
In this section we present the solutions obtained through both the mathematical models
we have developed in the previous section. We solve the equations of motion using numerical
simulation tools in MATLAB using the ode45 solver. We use the measured parameter values
from the experiment to solve the nonlinear equations.
With the application of the periodic low frequency input, the magnets snap and detach
after certain time, ∆t0idle , when the spring force at the end spring is just enough to overcome
the total force due to the magnets and the left-spring. To facilitate this phenomenon in the
numerical simulation we multiply the governing state variables x and x˙ by a Heaviside
function heaviside t − ∆t0idle .
Also, the magnets do not pass through the coil until x goes beyond x0coil , where x0coil
is the distance between the magnet at the idle equilibrium state and the left end of the coil
spool as shown in Figures 4.3 and 4.4. Also the current is generated only when the magnet

83

0.4

Voltage Output (Volt)

0.3
0.2
0.1
0
ï0.1
ï0.2
ï0.3
ï0.4
0

1

2

3

4

5
Time (sec)

6

7

8

9

10

Figure 4.9 The SDOF twinkling energy generator voltage output V (t), from the numerical simulation, after one cycle of 0.1Hz input oscillation at the end spring using the first
analytical model.
pass inside of the coil. No current is generated once the moving magnet pass beyond the
length of the coil. To mimic this delayed snap behavior of the magnets, we multiply, in the
numerical simulation, the velocity term in the generator equation and the current term in
the displacement equations in the above governing equations of motion, with a Heaviside
function heaviside x − x0coil − heaviside x − x0

coil −lc

, where lc is the length of the

coil.
Since the Heaviside function has a sharp transition between 0 and 1, in some cases it is
better to use a stiff hyperbolic tangent functions. As such tanh 200 x − x0coil

can be

used instead of heaviside x − x0coil .
For the numerical simulation we use the measured experimental parameter values such
that m = 0.342 kg, c = 0.01 Ns/m, ce = 0.01 Ns/m, k = 44.10 N/m, ke = 45.52 N/m,
M = 1.05 × 106 A/m, Vm = 1.2066 × 10−6 m3 , dm = 0.0145 m, L = 5 H, RL = 2000 Ω, and
Ri = 200 Ω. A sinusoidal low frequency, f0 = 0.1 Hz, input oscillation of amplitude 3 inch
A

corresponding to horizontal shaker stroke length of A0 = 6 inch, y(t) = 20 (1−cos(2πf0 t)) =
84

Voltage FFT Amplitude

0.05
0.04
0.03
0.02
0.01

0

2

4

6
Frequency (Hz)

8

10

12

Figure 4.10 The numerical frequency content of the SDOF TEG voltage output using the
first analytical model. The snap-through has produced a 3.25Hz output, a ≈ 32 times the
input frequency.
0.0762 (1 − sin(0.2πt)), is used and shown in figure 4.6.
Since the experimental value for the transducer constant, γ, was not measured, we have
used two different values of γ for the numerical simulations corresponding to the two models.
For model I and model II the transducer constants used are γmodel1 = 0.32 As/m, and
γmodel2 = 0.55 As/m, respectively.

4.3.1

Numerical Simulation of the SDOF TEG

By changing the end spring and the distance between magnets, we have performed various
experimentations for comparison of the power output. To keep the analytical and numerical
simulations simple, we will take one such experimental case and study the governing equations of motion of the SDOF TEG. We consider both the springs to be type-2 long springs,
and use a weak magnetic force by increasing the separation distance between the magnets
in the idle stable equilibrium state. Substituting the spring and magnetic force for identical

85

Displacement FFT Amplitude

0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
0

2

4

6
Frequency (Hz)

8

10

12

Figure 4.11 The numerical frequency content in the displacement of the mass using the first
analytical model of the twinkling magnet oscillation in the SDOF TEG over one period of
input oscillation of frequency 0.1Hz. This plot shows the frequency up-conversion of the input
excitation. The displacement oscillation has a frequency of 3.2Hz, a 32 times up-conversion
of the input frequency.
magnets in equation (4.19), we deduce the equation of motion of the SDOF energy generator.
The state-space equations used in the numerical simulation are then written as
uˆ˙ = vˆ
1
vˆ˙ = −
k∆x0 + (c + ce )ˆ
v + (k + ke )ˆ
u + Fm (ˆ
u) − γ Iˆdelayed − ce y(t)
˙ − ke y(t)
m
1
˙
(RL + Ri )Iˆ + γˆ
vdelayed
Iˆ = −
L

(4.21)

where
Iˆdelayed = tanh u − x0coil − tanh u − x0coil − lc

Iˆ

vˆdelayed = tanh u − x0coil − tanh u − x0coil − lc

vˆ

(4.22)

A

with input oscillation y(t) = 20 (1 − cos(2πf0 t)) and y(t)
˙
= πA0 f0 sin(2πf0 t). A0 is the
stoke-length of the horizontal shaker, f0 is the frequency of the input oscillation, and T is
the period of the input oscillation y(t). Also,

86

Total Energy (mWsec)

0.05

0.04

0.03

0.02

0.01

0
0

1

2

3

4

5
Time (sec)

6

7

8

9

10

Figure 4.12 The numerical harvested energy after one cycle of 0.1Hz input oscillation using
the first analytical model.

uˆ = t > ∆t0idle & t < tf u
vˆ = t > ∆t0idle & t < tf v
Iˆ = t > ∆t0idle & t < tf I

(4.23)

u˙ = t > ∆t0idle & t < tf uˆ˙
v˙ = t > ∆t0idle & t < tf vˆ˙
˙
I˙ = t > ∆t0idle & t < tf Iˆ
where tf is the time at which the magnets snap-back together and get attached to each other.
This usually happens before the input pull y(t) completes one full period of oscillation. In
our numerical simulation we assumed that tf ≈ T − ∆t0idle , where T is the period of the
input oscillation y(t). However, the computation of tf needs more analytical formulation.
The condition t > ∆t0idle & t < tf

means that the ode solver in MATLAB solves the

state-space equations only when both the conditions t > ∆t0idle and t < tf are satisfied,
otherwise the integration is skipped and the solver moves to the next time step. Due to this
87

0.06

Displacement (m)

0.05
0.04
0.03
0.02
0.01
0
0

1

2

3

4

5
Time (sec)

6

7

8

9

10

Figure 4.13 The numerical simulation displacement result of the SDOF TEG for the second
analytical model. Here we have used the assumption tf ≈ T − 0.922∆t0idle , where T = 10,
is the period of the input oscillation.
approximation in tf , there is a constant displacement fot t > 7 seconds in the numerical
displacement results in the Figures 4.7 and 4.13.
The magnetic force is defined for the two models as
2
3µ0 M 2 Vm
Model I : Fm (ˆ
u) = Fˆm (ˆ
u) = −
2π(dm + uˆ)4

Model II : Fm (ˆ
u) = F˜m (ˆ
u) =
πµ0 M 2 d4o − d4i
−
64

(4.24)

1
1
2
+
−
(ˆ
u + dm )2 (ˆ
u + dm + 2lm )2 (ˆ
u + dm + lm )2

With the 0.1Hz input, the simulation results using the magnetic interaction as in model
I, derived in Section 4.2.2, are presented for the dynamic displacement of the magnet in
Figure 4.7, the phase portrait in Figure 4.8, the voltage generated in the harvester coil in
Figure 4.9, the frequency characteristics of the voltage output in Figure 4.10, the frequency
characteristics of the displacement output in Figure 4.11, and the total harvestable energy
in Figure 4.12. The results of the mathematical model using the magnetic interaction as

88

0.4
0.3

Velocity (m/s)

0.2
0.1
0
ï0.1
ï0.2
ï0.3
ï0.4
0

0.01

0.02

0.03
Displacement (m)

0.04

0.05

0.06

Figure 4.14 The numerical simulation phase portrait result of the SDOF TEG for the second
analytical model.
in model II, derived in Section 4.2.3, are shown in Figures 4.13 to 4.18. For comparison of
the numerical results to that of the experiments, we choose one of the many experimental
studies done using various springs.
The numerical displacement and voltage results clearly show the up-conversion of the
input frequency by 32 times using Model I and 28 times using the Model II. From the
results of the first model, the voltage FFT plot in Figure 4.10 and the displacement FFT
plot Figure 4.10 show good agreement on the frequency up-conversion, and show the same
frequencies (3.2 Hz) in both the plots. Similarly, in the second model Figures 4.16 and 4.17
show the same primary frequency of 2.8Hz.

4.3.2

State-Space Representation of the 2-DOF TEG for Numerical Simulation

The governing equations of motion of the 2-DOF TEG are much more complex than that
of the SDOF case. Here we present all possible scenarios in the 2-DOF TEG and formulate
89

0.4

Voltage Output (Volt)

0.3
0.2
0.1
0
ï0.1
ï0.2
ï0.3
ï0.4
0

1

2

3

4

5
Time (sec)

6

7

8

9

10

Figure 4.15 The numerical result for the voltage output in the SDOF TEG for the second
analytical model.
the state-space equations for all the cases for numerical simulation. There are three possible
cases that arise when the 2-DOF TEG shown in Figure 4.4 is studied for its dynamics under
the application of a low-frequency pull to the end spring.
Case I: Using the previously defined parameters along with the magnetic force derived
in equations (4.8) and (4.16), and the spring force shown in equation (4.3), we write the
governing equations of motion of the 2-DOF TEG in state-space representation. When both
masses are snapped and oscillate we call this the case I, and represent the equations as

90

Voltage FFT Amplitude

0.06
0.05
0.04
0.03
0.02
0.01
0
0

2

4

6
Frequency (Hz)

8

10

12

Figure 4.16 The numerical frequency content using the second analytical model of the twinkling magnet oscillation in the SDOF TEG over one period of input oscillation of frequency
0.1Hz. This plot shows the frequency up-conversion of the input excitation. Two dominant
frequencies in the output voltage are 2.8Hz (28 times the input frequency), 6Hz (60 times
the input frequency), and 8.33Hz (83 times the input frequency).

uˆ˙ 1 = vˆ1
uˆ˙ 2 = vˆ2
1
vˆ˙ 1 = −
k ∆x + (c1 + c2 )ˆ
v1 − c2 vˆ2 + (k1 + k2 )ˆ
u1 − k2 uˆ2
m1 1 01
Case I :

+ Fm1 (ˆ
u1 ) − Fm2 (ˆ
u2 − uˆ1 )
1
vˆ˙ 2 = −
k ∆x − c2 vˆ1 + (c2 + ce )ˆ
v2 − k2 uˆ1 + (k2 + ke )ˆ
u2
m2 2 02
+ Fm2 (ˆ
u2 − uˆ1 ) − γ Iˆdelayed − ce y(t)
˙ − ke y(t)
1
˙
Iˆ = −
(RL + Ri )Iˆ + γ vˆ2delayed − vˆ1
L

91

(4.25)

Displacement FFT Amplitude

0.03
0.025
0.02
0.015
0.01
0.005
0
0

2

4

6
Frequency (Hz)

8

10

12

Figure 4.17 The numerical frequency content in the displacement of the mass using the
second analytical model of the twinkling magnet oscillation in the SDOF TEG over one
period of input oscillation of frequency 0.1Hz. This plot shows the frequency up-conversion
of the input excitation. The displacement oscillation has a frequency of 2.8Hz, a 28 times
up-conversion of the input frequency.
where
Iˆdelayed = tanh u2 − x0coil − tanh u2 − x0coil − lc

Iˆ

vˆ2delayed = tanh u2 − x0coil − tanh u2 − x0coil − lc

vˆ2

(4.26)

The equation 4.25 is valid only when both the masses in Figure 4.4 snap-through and
all the magnets are detached. However, due to unsymmetry, both the masses do not snapthrough at the same time and do not snap-back together at the same time. Two cases
arise from this and require separate state-space representation to accommodate the possible
dynamics when the 2-DOF system behaves as a SDOF system. Below are the two cases:
Case II: In reference to Figure 4.4, there are time intervals when only the end mass
(m2 ) snaps-through and oscillates while the first mass (m1 ) is still attached through the
magnets. In this case, the 2-DOF oscillator behaves as a SDOF system with a total mass of
m2 . This results in the equations that are identical to the SDOF state-space representation

92

Total Energy (mWsec)

0.05

0.04

0.03

0.02

0.01

0
0

1

2

3

4

5
Time (sec)

6

7

8

9

10

Figure 4.18 The numerical simulation results using the second analytical model of the SDOF
TEG power output shown in mW for a single cycle of input oscillation of frequency 0.1Hz.
The total power at any given time in a 2000Ω resistive load lie on the curve at the particular
time.
in equation (4.21) and is shown below in equation (4.27)

uˆ1 = 0,

uˆ˙ 1 = 0,

vˆ1 = 0,

vˆ˙ 1 = 0

uˆ˙ 2 = vˆ2
1
Case II : vˆ˙ 2 = −
k ∆x + (c2 + ce )ˆ
v2 + (k2 + ke )ˆ
u2 + Fm (ˆ
u2 )
m2 2 02

(4.27)

− γ Iˆdelayed − ce y(t)
˙ − ke y(t)
1
˙
Iˆ = −
(RL + Ri )Iˆ + γˆ
v2delayed
L
Case III: In reference to Figure 4.4, there also exist time intervals where the first mass
(m1 ) detaches while the end mass (m2 ) is still attached to the first mass (m1 ), i.e. the
oscillation takes place such that the whole 2-DOF system acts like a SDOF oscillator with
a total mass of (m1 + m2 ). The state-space equations for this case are shown below in

93

equation (4.28)

uˆ˙ 2 = 0,

uˆ2 = 0,

vˆ2 = 0,

vˆ˙ 2 = 0

uˆ˙ 1 = vˆ1
Case III : vˆ˙ 1 = −

1
k ∆x + (c1 + ce )ˆ
v1 + (k1 + ke )ˆ
u1 + Fm (ˆ
u1 )
(m1 + m2 ) 1 01

(4.28)

− γ Iˆdelayed − ce y(t)
˙ − ke y(t)
1
˙
(RL + Ri )Iˆ + γˆ
v1delayed
Iˆ = −
L
Let the time intervals when the above three equations (4.25), (4.27), and (4.28) are used
be between t11 and t12 for case I, between t21 and t22 for case II, and between t31 and t32
for case III. Then we can write
uˆ1 = (t11 < t < t12 & t31 < t < t32 ) u1
uˆ2 = (t11 < t < t12 & t21 < t < t22 ) u2
vˆ1 = (t11 < t < t12 & t31 < t < t32 ) v1
vˆ2 = (t11 < t < t12 & t21 < t < t22 ) u2
Iˆ = (t11 < t < t12 & t21 < t < t22 & t31 < t < t32 ) I

(4.29)

u˙ 1 = (t11 < t < t12 & t31 < t < t32 ) uˆ˙ 1
u˙ 2 = (t11 < t < t12 & t21 < t < t22 ) uˆ˙ 2
v˙ 1 = (t11 < t < t12 & t31 < t < t32 ) vˆ˙ 1
v˙ 2 = (t11 < t < t12 & t21 < t < t22 ) vˆ˙ 2
˙
I˙ = (t11 < t < t12 & t21 < t < t22 & t31 < t < t32 ) Iˆ
where the condition, for example, (t11 < t < t12 & t31 < t < t32 ) means that the ode
solver in MATLAB solves the state-space equations only when both the conditions t11

94

< t < t12 and (t31 < t < t32 ) are satisfied, otherwise the integration is skipped and the
solver moves to the next time step.
The magnet forces are derived using the dipole moment formulation in model I and the
cylindrical permanent magnet forcing in model II in the previous section and are shown next.
The magnetic forces on masses 1 and 2 of the 2-DOF twinkling energy generator (TEG) are
defined for the two models as

Model I :
Fm1 (ˆ
u1 ) = Fˆm (ˆ
u1 ) = −

2
3µ0 M 2 Vm
2π(dm + uˆ1 )4

Fm2 (ˆ
u2 − uˆ1 ) = Fˆm (ˆ
u2 − uˆ1 ) = −

2
3µ0 M 2 Vm
2π(dm + uˆ2 − uˆ1 )4

Model II :
πµ0 M
Fm1 (ˆ
u1 ) = F˜m (ˆ
u1 ) = −
−

2

d4o − d4i
64

1
1
+
2
(ˆ
u1 + dm )
(ˆ
u1 + dm + 2lm )2

2

(ˆ
u1 + dm + lm )2
πµ0 M 2 d4o − d4i
Fm2 (ˆ
u2 − uˆ1 ) = F˜m (ˆ
u2 − uˆ1 ) = −
64
+

(4.30)

1
(ˆ
u2 − uˆ1 + dm )2

1
2
−
2
(ˆ
u2 − uˆ1 + dm + 2lm )
(ˆ
u2 − uˆ1 + dm + lm )2

At this point, we do not have the analytical formulation for t11 , · · · , t32 and will not
present any numerical simulation results for the 2-DOF TEG. However, using the above
set-up of equations for the numerical simulation, one can solve the 2-DOF TEG analytically
to compute the voltage, frequency and the harvestable energy.

95

4.3.3

Energy and Power Computations

The total energy measured in Joules, near the linear non-saturated regions of the magnetic
flux linkage and current relationship, stored by an inductor is equal to the amount of work
required to establish the current through the inductor and is given by
1
Estored = LI 2
2

(4.31)

where L is inductance and I is the current through the inductor.
In general the energy stored in an inductor that has initial current during a specific time
interval t1 and t2 is
1
1
E = LI(t2 )2 − LI(t1 )2
2
2

(4.32)

Since the electrical circuit in the TEG is such that it directly powers a resistive load, the
instantaneous power delivered to the electrical load is given by
P (t) = I(t) · V (t) = I(t)2 RL =

V (t)2
RL

(4.33)

The average power Pavg over a period of time T is given as

Pavg =

T
1 T
1
P (t)dt =
V (t)2 dt
T 0
T RL 0

(4.34)

The solution of the mathematical models are found using the numerical simulation.
Through the numerical simulation we solve for the current I passing through the coil. Using
the resistance we then compute the voltage V developed through the electrical circuit. This
voltage is used to compute the instantaneous power harvestable through the electrical circuit
as well as to compare the experimental voltage output.
96

0.4

Voltage (Volt)

0.3
0.2
0.1
0
ï0.1
ï0.2
ï0.3
ï0.4
0

5

10

15
Time (sec)

20

25

30

Figure 4.19 The experimental voltage output in the SDOF TEG shown in volts. Single
harmonic of 0.1Hz input frequency is converted into more than 2.5Hz output.

4.4

Experimental Results

In this section we present the results obtained from both the SDOF and 2-DOF experiments.
Since our experiments were motivated by the idea of frequency up-conversion and energy
harvesting, we only look at these aspects and discuss the results here. During the experiment
various springs were used. Though the results of all these cases are readily available, we show
only one such example from SDOF and one example from the 2-DOF experimental energy
harvesting.
During the experiments we found that softer springs are required in case of the 2-DOF
twinkler, as compared to the SDOF twinkler, in order to facilitate the snap through of
both the masses. However, for energy harvesting purposes, we had used one coil spool of
enamel-coated magnet wires for both the SDOF and the 2-DOF TEG placing it near the
end mass.
The harvested energy can be maximized by placing the harvester coil centered at the
high-frequency oscillations region of each mass. In the 2-DOF TEG we have used one coil
97

Total Energy (mWsec)

0.05
0.04
0.03
0.02
0.01
0
0

1

2

3

4

5
Time (sec)

6

7

8

9

10

Figure 4.20 The experimental SDOF TEG harvested energy shown in mWs for a single cycle
of input oscillation of frequency 0.1Hz. The total energy harvested at any given time in a
2000Ω resistive load lie on the curve at the particular time. The total energy harvested in
the SDOF TEG after one period (T = 10sec) of input oscillation is 0.045mWs.
to harvest the high-frequency oscillation energy from the mass pulled by the end spring. It
is possible to capture the energy of the other mass, close to the fixed base, by using another
generator coil along the center of the oscillation of the mass m1 . Other ways to maximize
the power output is by both choosing a longer coil and positioning the coil in such a way
that the harvester coil captures the whole oscillation of the oscillating magnet.

4.4.1

Experimental Results of the SDOF TEG

In the SDOF twinkler we had used two different types of spring. For both types of springs
we found high-frequency dynamics. We choose to present the results of one SDOF example,
that consisted of the type-2 long springs. When the magnets detach and the magnet on the
right side (refer to Figure 4.3) passes through the inductor coil, it generates current through
the coil and this current is used to power a resistive load which results in the voltage output.
Here we present the voltage output and perform a FFT on the voltage to find the frequency
of the SDOF TEG.
98

Voltage FFT Amplitude

0.04

0.03

0.02

0.01

0
0

2

4

6
Frequency (Hz)

8

10

12

Figure 4.21 The experimental frequency content of the twinkling magnet oscillation in the
SDOF TEG over one period of input oscillation of frequency 0.1Hz. This plot shows the
frequency up-conversion of the input excitation. Two dominant frequencies in the output
voltage are 2.45Hz (25 times the input frequency) and 5Hz (50 times the input frequency).
From the voltage we compute the total energy generated from time t = 0 till any given
time t within a period of time T (period of the input y(t) oscillation), by integrating the
instantaneous power P (t). For instance, the total energy generated in the harvester coil
by the SDOF TEG from time t = 0 till time t = 5 seconds, correspond to the point one
curve in the Figure 4.20 at t = 5 seconds. So, over one whole cycle of input oscillation the
total energy harvested is the point on the curve in Figure 4.20 corresponding to time t = 10
seconds.
The experimental voltage and energy output shown in Figures 4.19, and 4.20 respectively
are in good agreement with the numerical results obtained from the analytical Models I II.
However, the FFT of the experimental voltage output in Figure 4.21 shows that the model
II is a better analytical model than the model I. In particular, the frequency of the SDOF
TEG from the experiment was 2.45 Hz where as the frequencies from the numerical models
I and II are 3.2 Hz and 2.8 Hz respectively.

99

0.6

Voltage (Volt)

0.4
0.2
0
ï0.2
ï0.4
ï0.6
0

5

10

15
Time (sec)

20

25

30

Figure 4.22 The experimental voltage output in the 2-DOF twinkling energy generator shown
in volts. Single harmonic of 0.1Hz input frequency is converted into 2.5 Hz output.

4.4.2

Experimental Results of the 2-DOF TEG

From various experimental investigations of the 2-DOF TEG, we present one example to
show the twinkling nature and the frequency up-conversion. In this example we used type-2
long springs (same springs as in the SDOF experimental result presented). We keep the
generator coil at the same location as it was in the SDOF example case, i.e. near the second
mass m2 . Using the same input oscillation through the horizontal shaker, the current is
induced in the coil due to the movements of the magnet along the coil spool center axis. In
case of the 2-DOF TEG we obtain the voltage and power output as shown in the Figures 4.22
and 4.23 as well as compute the frequency content using FFT over measured experimental
voltage output as shown in Figure 4.24. The frequency output is 2.5 Hz and is almost
identical to that of the SDOF case. The experimental results show higher output voltage
and harvested energy in the 2-DOF case than that in the SDOF case.

100

Total Energy (mWsec)

0.1
0.08
0.06
0.04
0.02
0
0

1

2

3

4

5
Time (sec)

6

7

8

9

10

Figure 4.23 The 2-DOF TEG experimental energy output shown in Milliwatt-second for a
single cycle of input oscillation of frequency 0.1 Hz. The total energy up to any given time in
a 2000Ω resistive load lie on the curve at that particular time. The total energy harvested in
the coil in the 2-DOF TEG after one period (T = 10) of input oscillation (at time t = T = 10
seconds) is 0.096 mWs.

4.5

Summary

In this chapter, we validate the twinkling phenomenon by converting a 0.1 Hz input oscillation
into 2.5 Hz output frequency, a 25 times the input frequency. We have shown this by both
experimentally and numerically simulated results of the analytical models developed in this
chapter. Through this experimental energy harvesting study, we harvested energy using coils
and magnets by taking advantage of the frequency up-conversion.
To compare the experimental results to that of the numerical simulations, we use the
experimentally measured parameters for the numerical simulations when a theoretical value
of the same is not available in the literature. In the experimental set-up, one could change
the forcing by swapping springs of different stiffnesses or using magnets of different magnetic
strengths. The stiffer the springs the larger the amplitude of the low-frequency input needed
to build-up enough force to obtain the bi-stability of the twinkler. There is some room for
error in the experimental results as compared to the analytical and numerical results since
101

0.08

Voltage FFT Amplitude

0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0

2

4

6
Frequency (Hz)

8

10

12

Figure 4.24 The frequency content of the twinkling magnet oscillation in the experimental
2-DOF TEG over one period of input oscillation of frequency 0.1 Hz. This plot shows the
frequency up-conversion of the input excitation with one dominant frequencies in the output
voltage at 2.5Hz which is 25 times the input frequency. In addition to the dominant frequency
there is a broad band of frequency between 2 Hz to 8 Hz generated by the 2-DOF twinkler.
the horizontal shaker seems to be affected somewhat by the feedback restoring force from
the end spring which in-turn creates a deviation to the input signal. This results in a not
perfectly harmonic but a periodic input and feeds back to the circuit inducing a deviation
to the expected output. A perturbation analysis can perhaps provide an accurate analytical
representation of the experimental model, which is not included in this study.
The peak instantaneous power can be computed from the voltage plots in From the
Figures 4.19 and 4.22. In a 2000Ω resistive load, the SDOF TEG has a peak instantaneous
power of PSDOFmax ≈ 0.32 /2000 = 0.045mW and that for the 2-DOF is P2−DOFmax ≈
0.452 /2000 ≈ 0.15mW. The total cumulative energy produced in one cycle of low frequency
input oscillation, i.e. from t = 0 to t = 10 seconds for the SDOF example shown in figure 4.23
is 0.045mWs and for the 2-DOF example shown in figure 4.23 is 0.096mWs.
The numerical simulation models present a reliable way to estimate power output and
harvestable energy for given parameter values. By changing the electrical circuit parameters,
102

one might be able to harvest more energy. For example, keeping all the parameters same and
changing the electro-mechanical coupling parameter γ from 0.55 to 5, we get total energy of
3.3 mWs over one period of input oscillation, i.e. this much energy is harvested in 10 seconds.
So using the best parameter values one would be able to harvest the optimum amount of
energy. We will continue our research along this line, and find optimum parameter values to
harvest the maximum possible energy from the low-frequency pull.

103

Chapter 5
Wave Propagations in Discrete
Periodic Weakly Nonlinear Chains
In this chapter we study the wave propagation in nonlinear periodic mass-spring chain where
the springs have both quadratic and cubic nonlinearities along with the linear stiffness.
Amplitude-dependent dispersion and band-gap behavior have been explored in several discrete periodic systems characterized by cubic nonlinearities [79], where it was shown that the
boundary of the dispersion curve may shift with amplitude in the presence of a single plane
wave. as well as propagation of multiple harmonic plane waves that show the dispersion
properties of discrete, periodic, cubically nonlinear systems [80, 125]
While these studies address cubic nonlinear periodic media, there has been no research on
wave propagations in periodic media with strong quadratic nonlinearities. We look at small
amplitude waves propagating about an equilibrium configuration of a snap-through periodic
chain, and retain quadratic and cubic nonlinearity local to the equilibrium. A second-order
multiple scales perturbation analysis is applied for low-amplitude oscillations that capture
the quadratic effects. We obtain a nonlinear dispersion relationship from the theoretical
analysis and compare it to linear and cubically nonlinear cases. The amplitude dependence
of the dispersion relations shows that the mass-spring chains can be used as tunable acoustic
filters. The group and phase velocity dependence on wave number and amplitude shows the

104

Spring Force : f s j

3
2
1
0
1
2
3

Spring Potential : V s j

4

2
0
Spring Deformation s j

2

3
2
DC Bias

1
0
1

4

2
0
Spring Deformation s j

2

Figure 5.1 The characteristic spring force f (sj ), and spring potential V (sj ) of the linear and
weakly nonlinear spring as a function of the spring deformation sj . Referring to the spring
force with characteristics as in equation (5.1), the solid (red) curve corresponds to α = 1,
β = 0.57, and γ = 0.09, Dashed (black) line corresponds to α = 1, β = 0, and γ = 0,
dot-dashed (blue) curve correspond to α = 1, β = 0, and γ = 0.09, and the dotted (green)
curve correspond to α = 0, β = 0, and γ = 0.09. The potential well shown to be shifted to
the left with positive quadratic nonlinearity, hence indicates a negative DC-bias.
relevance of quadratic effects for applications in band gaps, event detection and nonlinear
waveguides [89].

5.1

Discrete Periodic Chain of Nonlinear Oscillators

We study the wave behaviors in an infinite uniform nonlinear mass-spring chain (Figure 5.2).
Due to the quadratic and cubic nonlinearity of the individual elements (Figure 5.1), it is
105

suspected that there will be a nonlinear wave phenomenon for an infinite element system.
In [73, 74], we studied the bifurcations of equilibria with respect to a quasistatic pull, for a
two-degree-of-freedom snap-through oscillator connected by bistable springs.
The mass-spring chain is arranged in a fashion such that each mass is separated by a
distance h from its nearest neighbor, where h is the un-stretched length of each spring. We
use the assumption that all the masses are equal (mj = m) and only the nearest neighbors
have direct effects on each other.
The spring characteristics with both the quadratic and cubic nonlinearity is give by

f (sj ) = αsj + βs2j + γs3j

(5.1)

where sj is the stretch or the deformation of the j th spring.
As such, we consider a cubic expansion of the spring force relative to the equilibrium
state. The equations of motion (EOM) can then be written as

mu¨˜j = α
˜ (˜
uj+1 − u˜j ) − (˜
uj − u˜j−1 ) + β˜ (˜
uj+1 − u˜j )2 − (˜
uj − u˜j−1 )2
(5.2)
+ γ˜ (˜
uj+1 − u˜j

)3

− (˜
uj − u˜j−1

)3

For small amplitude of oscillations we assume

u˜j = uj

(5.3)

α
˜ , β = β˜ , and γ = γ˜ , this transformation
where uj is of O(1) and u˜j is of O( ). Letting α = m
m
m

results in the undamped nonlinear equation of motion of the jth mass for weakly nonlinear
interactions with the adjacent elements as bellow,

106

u˜ j −1

€

x j −1

u˜ j +1

u˜ j
xj

€

€

x j +1

Figure 5.2 Infinite mass chain. The unstretched position and displacement of mass mj are
denoted by xj and u˜j respectively. The springs are cubic nonlinear as in Figure 5.1 with
unstretched length h.

€

€

€

β (uj+1 − uj )2 − (uj − uj−1 )2

u¨j = α (uj+1 − uj ) − (uj − uj−1 ) +

(5.4)
+

2γ

(uj+1 − uj

)3

− (uj − uj−1

)3

for j = · · · , −2, −1, 0, 1, 2, · · · . To simplify the equation we apply the following coordinate
transformation:
zj = uj+1 − uj

(5.5)

2
3
u¨j+1 = α zj+1 − zj + β zj+1
− zj2 + 2 γ zj+1
− zj3

(5.6)

2
3
u¨j = α zj − zj−1 + β zj2 − zj−1
+ 2 γ zj3 − zj−1

(5.7)

to get the following equations

where zj is the strain in the j th spring. Now subtracting equation (5.7) from equation (5.6),
we obtain the equation of motion in the z coordinates,
2
2
3
3
− 2 γ 2 zj3 − zj+1
− zj−1
z¨j = −α 2 zj − zj+1 − zj−1 − β 2 zj2 − zj+1
− zj−1

(5.8)
When

is small, the spring deformation is near a stable equilibria. With

as a book-

keeping parameter, the quadratic effect is more dominant than the cubic in equations (5.4)

107

and (5.8). With the small parameter , we will analyze these ordinary differential equations
by using the method of multiple scales next.

5.2

Single Wave Dispersion Analysis Using Second-Order
Multiple Scales

5.2.1

Traveling Wave Solutions in Strain Coordinates

We analyze the wave equation using the second-order method of multiple scales (MMS). We
assume,

zj (t) = zj0 (T0 , T1 , T2 ) + zj1 (T0 , T1 , T2 ) + 2 zj2 (T0 , T1 , T2 ) + · · ·
d
= D0 + D1 + 2 D2 + · · ·
dt
d2
= D02 + (2D0 D1 ) + 2 D12 + 2D0 D2 + · · ·
dt

(5.9)

∂ . Plugging into equation (5.8) yields
where Di = ∂T
i

D02 zj0 +

D02 zj1 + 2D0 D1 zj0 + 2 D02 zj2 + 2D0 D1 zj1 + (D12 + 2D0 D2 )zj0 =
− α 2 zj0 − zj+10 − zj−10
−

2
2
α 2 zj1 − zj+11 − zj−11 + β 2 zj2 − zj+1
− zj−1
0
0
0

3
3
− zj−1
− 2 α 2 zj2 − zj+12 − zj−12 + γ 2 zj3 − zj+1
0
0
0

+ 2 β 2 zj0 zj1 − zj+10 zj+11 − zj−10 zj−11
Equating the like powers of ,

108

(5.10)

0

: D02 zj0 + α 2 zj0 − zj+10 − zj−10 = 0

1

2
2
− zj−1
: D02 zj1 + α 2 zj1 − zj+11 − zj−11 = −2D0 D1 zj0 − β 2 zj2 − zj+1
0
0
0

2

: D02 zj2 + α 2 zj2 − zj+12 − zj−12 = −2D0 D1 zj1 − D12 + 2D0 D2 zj0
3
3
− 2β 2 zj0 zj1 − zj+10 zj+11 − zj−10 zj−11 − γ 2 zj3 − zj+1
− zj−1
0
0
0

(5.11)
Let us assume a traveling dispersive wave solution to solve for the 0 equation in the
above equation-set (5.11). Let
zj0 = yj0 + y¯j0 = Ae

where yj0 = Ae

i(kxj −ω0 T0 )

We use yj±10 = e±ikh Ae

i(kxj −ω0 T0 )

¯ −i(kxj −ω0 T0 )
+ Ae

i(kxj±1 −ω0 T0 )

. Then yj±10 = Ae

i(kxj −ω0 T0 )

(5.12)

. We also assume xj±1 = xj ±h.

= e±ikh yj0 , into the 0 equation in equations (5.11) we

get
ω02 = 2α(1 − cos kh)

(5.13)

Also, since A can be complex, we express A in terms of real amplitude and phase as A =
1 −iθ ,
2 ae

1 i(kxj −ω0 T0 −θ)
zj0 = yj0 + y¯j0 = ae
+ c.c.
2

(5.14)

The solution to 0 equation can be used in the 1 equation from equation-set (5.11) to
find the resonant terms that contribute to secularity or unbounded solutions. The resonant
i(kxj −ω0 T0 )

terms are those associate with e

secularity is −2D0 D1 zj0 = 2(iω0 )A e

. The solvability condition resulting from the

i(kxj −ω0 T0 )

i.e. a = 0 and θ = 0, such that

109

∂ ,
+ c.c.. Hence A = 0, where () = ∂T
1

a = a(T2 )

(5.15)

θ = θ(T2 )
i(kxj −ω0 T0 )

Removing the secular term i.e. coefficients of e

, we write the 1 equation as

2
2
− zj−1
D02 zj1 + α 2 zj1 − zj+11 − zj−11 = −β 2 zj2 − zj+1
0
0
0
2
1 −iθ
2i(kxj −ω0 T0 )
e
+ c.c.
= −β(2 − e2ikh − e−2ikh )
ae
2

(5.16)

βa2
2i(kxj −ω0 T0 −θ)
=−
(1 − cos 2kh)e
+ c.c.
2
The particular solution to the above equation will therefore be of the form,
zj1 = A1 e

2i(kxj −ω0 T0 −θ)

+ c.c.

(5.17)

In order to find A1 in terms of A, we plug-in the assumed solution z1 into the 1 equation.
2i(kxj −ω0 T0 −θ)

Balancing the coefficients of e

A1 =

leads to

βa2 sin2 kh
4 ω02 − α sin2 kh

(5.18)

The order- solution zj1 is therefore

zj1 =

βa2 sin2 kh
2i(kxj −ω0 T0 −θ)
e
+ c.c.
4 ω02 − α sin2 kh

(5.19)

To solve for the amplitudes a and phase θ, we examine the 2 equation. We note that
zj0 and zj1 are independent of slow time-scale T1 , since A is independent of T1 . Therefore
(−2D0 D1 zj1 ) and (D12 zj0 ) vanish in the 2 equation. Using the solutions of zj0 and zj1 , we
obtain the various terms of the right hand side of the 2 equation of the equation-set (5.11),
as below,
110

2D0 D2 zj0 = (a − aib )(−iω0 )e
zj0 zj1 =

βa3 sin2 kh
8 ω02 − α sin2 kh

zj±10 zj±11 =

e

i(kxj −ω0 T0 −θ)

3i(kxj −ω0 T0 −θ)

βa3 sin2 kh
8 ω02 − α sin2 kh

+ c.c.
i(kxj −ω0 T0 −θ)

+e

3i(kxj −ω0 T0 −θ)

e±3ikh e

+ c.c.

+ e±ikh e

i(kxj −ω0 T0 −θ)

a3 3i(kxj −ω0 T0 −θ)
i(kxj −ω0 T0 −θ)
+ c.c.
e
+ 3e
0
8
a3 ±3ikh 3i(kxj −ω0 T0 −θ)
i(kxj −ω0 T0 −θ)
3
+ c.c.
zj±1
=
e
e
+ 3e±ikh e
0
8

+ c.c.

zj3 =

(5.20)

∂a , and θ = ∂θ . Using the above simplifications in the 2 equation, we get
where, a = ∂T
∂T2
2

i(kxj −ω0 T0 −θ)

D02 zj2 + α 2 zj2 − zj+12 − zj−12 = − (a − aiθ )(−iω0 )e
− 2β

βa3 sin2 kh
8 ω02 − α sin2 kh

3i(kxj −ω0 T0 −θ)

(2 − e3ikh − e−3ikh )e

+ c.c.

i(kxj −ω0 T0 −θ)

+ (2 − eikh − e−ikh )e

+ c.c.

+ c.c.

(5.21)

a3
3i(kxj −ω0 T0 −θ)
−γ
(2 − e3ikh − e−3ikh )e
c.c.
8
i(kxj −ω0 T0 −θ)

+ 3(2 − eikh − e−ikh )e

+ c.c.

∂a , and θ = ∂b . The secular terms from the above equation are the coeffiwhere, a = ∂T
∂T2
2
i(kxj −ω0 T0 )

cients of e

. Eliminating the secular term results in

(iω0 a + ω0 aθ ) =

3γ
β 2 sin2 kh
+
4
2 ω02 − α sin2 kh

111

(1 − cos kh)a3

(5.22)

Equating the real and imaginary parts, the solvability condition can be written as

Im: a = 0
3γ
β 2 sin2 kh
Re: ω0 aθ =
+
4
2 ω02 − α sin2 kh

(5.23)
(1 − cos kh)a3

Using ω0 from equation (5.13), in equation (5.23), we get

Im: a = 0
2β 2 (1 + cos kh) 3ω0 2
Re: θ = γ +
a
3α(1 − cos kh) 8α

(5.24)

Solving the equation (5.24), we get

a = a0
2β 2 (1 + cos kh) 3ω0 2
a T + θc
θ= γ+
3α(1 − cos kh) 8α 0 2

(5.25)

We will neglect the integration constant θc without loss of generality. Also note that
T2 = 2 T0 . Hence the slowly varying amplitude (a) and phase (θ) becomes,

a = a0
2β 2 (1 + cos kh) 3ω0 2 2
a T
θ= γ+
3α(1 − cos kh) 8α 0 0

(5.26)

Therefore combining equation (5.12) with A = 12 ae−iθ , the frequency ω can be written
as
ω = ω0 +

3 2 a20
θ
= ω0 1 +
T0
8α

112

γ+

2β 2 (1 + cos kh)
3α(1 − cos kh)

(5.27)

5.2.2

Frequency Expression in Displacement Coordinates

The above solution in equation (5.27) is obtained by solving the EOM in strain coordinates
zj = uj+1 − uj , where uj is in the displacement coordinates. We now interprete the solution
in displacement coordinates. Let the solution in u coordinates be uj = De
and uj+1 = De

i(kxj+1 −ωt)

i(kxj −ωt)

+ c.c. = Deikh e

i(kxj −ωt)

zj = D(eikh − 1)e

i(kxj −ωt)

+ c.c.

+ c.c.. Therefore

+ c.c. = Ae

i(kxj −ωt)

+ c.c.

(5.28)

ˆ

Letting D = 21 d0 e−iθ , we get the relationship between the amplitudes in displacement
and strain coordinates as
1
1
ˆ
ˆ
A = d0 e−iθ (eikh − 1) = d0 e−iθ (ρeiφ )
2
2
1
1
ˆ
or
a e−iθ = ρd0 e−i(θ−φ)
2 0
2

(5.29)

where, with help from equation (5.13),

ρ=

(cos kh − 1)2 + sin2 kh =

tan φ =

2(1 − cos kh) =

sin kh
kh
= − cot
= tan
cos kh − 1
2

π kh
+
2
2

ω02
α
(5.30)

a0 = ρd0
Hence the amplitude and phase in strain coordinates are related to the amplitude and
phase in displacement coordinates by
ω2
a20 = 0 d20
α
π kh
θ = θˆ −
+
2
2
113

(5.31)

Therefore the frequency in the displacement coordinate system can be written as
2β 2
ω(k, d0 ) = ω0 1 + 2 γ +
3α

1 + cos kh
1 − cos kh

3ω02 d20
8α2

(5.32)

Also note that, in the above dispersion relationship, ω0 is a function of the wave number
k as in equation (5.13).

5.2.3

Comparison in the Continuum Limit

To check the correlation between the coordinates, we consider a continuum. The displacement at any location x in the continuum is assumed to be of the following form

u(xj ) = De

i(kxj −ωt)

(5.33)

The strain can then be found by taking the partial derivative of the displacement with
respect to position x to get

z(xj , t) =

π i(kx −ωt)
∂u(xj )
i(kxj −ωt)
j
= D(ik)e
= Dkei 2 e
∂xj

(5.34)

To obtain the continuum limit in our MMS solution, we first represent the amplitude and
phase of our solution (strain) in displacement coordinate system and then take the limit as
h → 0. Making use of
sin(kh)
=k
h
h→0
lim

114

(5.35)

and the equations (5.28) - (5.30) the strain can be found as
zj (xj , t)
1 i(kxj −ωt)
= lim A e
h
h→0
h→0 h
1
1
i(kxj −ωt)
i(kxj −ωt)
= lim Dρ eiφ e
= lim D (eikh − 1)e
h
h
h→0
h→0

z(xj , t) = lim

= lim D

2 sin kh
2
h

h→0

(5.36)

π kh i(kx −ωt)
π i(kx −ωt)
j
j
ei( 2 + 2 ) e
= Dkei 2 e

Hence the strains from both the equations (5.34) and (5.36) are identical and confirms the
validity of the coordinate transformations between the strain and displacement coordinates.

5.3
5.3.1

Single Wave Dispersion Analysis in Continuum Limit
Partial Differential Equations for Dispersion Analysis

We will use the Taylor series expansion of u˜j±1 (˜
x) about u˜j (˜
x) as
∂ u˜j
h+
u˜j+1 = u˜j +
∂ x˜
∂ u˜j
u˜j−1 = u˜j −
h+
∂ x˜

∂ 2 u˜j 2
∂ 3 u˜j 3
∂ 4 u˜j 4
h
+
h
+
h + ···
2!∂ x˜2
3!∂ x˜3
4!∂ x˜4
∂ 2 uj 2
∂ 3 u˜j 3
∂ 4 u˜j 4
h
−
h
+
h + ···
2!∂ x˜2
3!∂ x˜3
4!∂ x˜4
∂2u
˜

(5.37)

∂u
˜

j
in equation (5.4), for j = · · · , −2, −1, 0, 1, 2, · · · . Denoting u˜tt =
= u˜¨j , u˜x˜ = ∂ x˜j ,
∂t2

u˜x˜x˜ =

∂2u
˜j
,
∂x
˜2

letting α =

∂nu
˜

· · · , u˜ x˜ . . . x˜ = ∂ x˜nj in equation (5.37), and plugging it into equation (5.4) and

α
˜
mj ,

β

n-times
˜
= mβ , γ
j

= mγ˜ , results in
j

115

u˜tt ≈ α˜
ux˜x˜ h2 + 2β u˜x˜ u˜x˜x˜ h3 +
+
+
+
+
+

α
u˜x˜x˜x˜x˜ + 3γ u˜2x˜ u˜x˜x˜ h4
12

β
β
u˜x˜ u˜x˜x˜x˜x˜ + u˜x˜x˜ u˜x˜x˜x˜ h5
6
3
α
γ
γ
u˜x˜x˜x˜x˜x˜x˜ + u˜2x˜ u˜x˜x˜x˜x˜ + γ u˜x˜ u˜x˜x˜ u˜x˜x˜x˜ + u˜3x˜x˜ h6
360
4
4
β
β
β
h7
u˜ u˜
+ u˜ u˜
+ u˜ u˜
180 x˜ x˜x˜x˜x˜x˜x˜ 60 x˜x˜ x˜x˜x˜x˜x˜ 36 x˜x˜x˜ x˜x˜x˜x˜
α
γ 2
γ
u˜x˜x˜x˜x˜x˜x˜x˜x˜ +
u˜x˜ u˜x˜x˜x˜x˜x˜x˜ + u˜x˜ u˜x˜x˜ u˜x˜x˜x˜x˜x˜
20160
120
20
γ
γ
γ
u˜x˜ u˜x˜x˜x˜ u˜x˜x˜x˜x˜ + u˜x˜x˜ u˜2x˜x˜x˜ + u˜2x˜x˜ u˜x˜x˜x˜x˜ h8
12
12
16
n

(5.38)

n

∂
Nondimensionalizing the above equation by letting x = hx˜ , then ∂∂x˜n = h1n ∂x
n we get

the following partial differential equation (PDE)

u˜tt ≈ α˜
uxx + 2β u˜x u˜xx +
+
+
+
+
+

α
u˜xxxx + 3γ u˜2x u˜xx
12

β
β
u˜x u˜xxxx + u˜xx u˜xxx
6
3
γ
γ
α
u˜xxxxxx + u˜2x u˜xxxx + γ u˜x u˜xx u˜xxx + u˜3xx
360
4
4
β
β
β
u˜x u˜xxxxxx + u˜xx u˜xxxxx + u˜xxx u˜xxxx
180
60
36
α
γ 2
γ
u˜xxxxxxxx +
u˜x u˜xxxxxx + u˜x u˜xx u˜xxxxx
20160
120
20
γ
γ
γ
u˜x u˜xxx u˜xxxx + u˜xx u˜2xxx + u˜2xx u˜xxxx
12
12
16

(5.39)

For small oscillations with the spring deformation near one of the stable equilibria, we
let u˜ = u, u˜tt = utt , u˜x...x = ux...x , etc., in equation (5.39) and cancel
to obtain

116

from both sides,

uxx uxxxx uxxxxxx uxxxxxxxx
+
+
+
2!
4!
6!
8!
β
β
ux uxxxx + uxx uxxx
2βux uxx +
6
3

utt ≈ 2α
+
+
+

β
β
β
ux uxxxxxx + uxx uxxxxx + uxxx uxxxx
180
60
36
2

3γu2x uxx

γ 2
γ
+
ux uxxxx + γux uxx uxxx + u3xx
4
4

(5.40)

γ
γ
γ 2
ux uxxxxxx + ux uxx uxxxxx + ux uxxx uxxxx
120
20
12
γ
γ
+ uxx u2xxx + u2xx uxxxx
12
16
+

For wave dispersion characteristics in the nonlinear chain with both quadratic and cubic
nonlinearities, we will analyze the above partial differential equation by using the multiple
scales perturbation method.

5.3.2

Second-Order Multiple Scales Analysis in Continuum Limit

We analyze the wave equation using the method of multiple scales (MMS). We assume,
u(x, t) = u0 (x, T0 , T1 , T2 ) + u1 (x, T0 , T1 , T2 ) + 2 u2 (x, T0 , T1 , T2 ) + · · ·
d
= D0 + D1 + 2 D2 + · · ·
dt
d2
= D02 + (2D0 D1 ) + 2 D12 + 2D0 D2 + · · ·
dt

(5.41)

∂ . Using equation (5.41) in equation (5.40) and equating like powers of
where Di = ∂T
i

yields the following set of equations
0

: D02 u0 = 2α

u0xx u0xxxx u0xxxxxx u0xxxxxxxx
+
+
+
2!
4!
6!
8!

117

(5.42)

1

2

u1xx u1xxxx u1xxxxxx u1xxxxxxxx
+
+
+
2!
4!
6!
8!
β
β
u u
+ u u
= −2D0 D1 u0 + 2βu0x u0xx +
6 0x 0xxxx 3 0xx 0xxx
β
β
β
+
u0x u0xxxxxx + u0xx u0xxxxx + u0xxx u0xxxx
180
60
36

: D02 u1 − 2α

: D02 u2 − 2α

(5.43)

u2xx u2xxxx u2xxxxxx u2xxxxxxxx
+
+
+
2!
4!
6!
8!

= −2D0 D1 u1 − D12 + 2D0 D2 u0
β
u u
+ u0x u1xxxx
6 1x 0xxxx
β
β
+
u1xx u0xxx + u0xx u1xxx
+
u u
+ u0x u1xxxxxx
3
180 1x 0xxxxxx
β
β
u1xx u0xxxxx + u0xx u1xxxxx +
u
u
+ u0xxx u1xxxx
+
60
36 1xxx 0xxxx
γ
γ 2
u0x u0xxxx + γu0x u0xx u0xxx + u30xx
+ 3γu20x u0xx +
4
4
γ 2
γ
γ
u0x u0xxxxxx + u0x u0xx u0xxxxx + u0x u0xxx u0xxxx
+
120
20
12
γ
γ
+ u0xx u20xxx + u20xx u0xxxx
12
16
+ 2β u1x u0xx + u0x u1xx

+

(5.44)

We seek a traveling wave solution to equation (5.42) in order to solve for zj0 . Let
¯ −i(kx−ω0 T0 )
u0 = y0 + y¯0 = Aei(kx−ω0 T0 ) + Ae
u0 x . . . x = (ik)n y0 + c.c.

(5.45)

n-times

where y0 = Aei(kx−ω0 T0 ) and u0 x . . . x is the nth order partial derivative of u0 with respect
n-times

to x. Using the assumed solution in equation (5.45) in equation (5.42) one can find the
dispersion characteristics of a linear chain as

118

ω02

= 2α

k2 k4 k6 k8
−
+
−
2!
4!
6!
8!

(5.46)

The presence of resonant terms ei(kx−ω0 T0 ) in equation (5.43) will result in unbounded
solutions. For bounded solution u1 , we eliminate the secular terms in equation (5.43). In
doing so, we note that
u0 x . . . x u0 x . . . x =[(−1)m + (−1)n ](ik)m+n y0 y¯0 + (ik)m+n y02 + (−ik)m+n y¯02
m-times

(5.47)

n-times

for m, n = 1, 2, · · · , and has no secular terms i.e. the coefficients of ei(kx−ω0 T0 ) in 1 equation
are zero. Therefore the solvability condition, found by eliminating the secular terms, leads to
D1 A = 0 i.e. amplitude A is independent of slow time-scale T1 . Hence removing the secular
terms we simplify the equation (5.43) as
u1xx u1xxxx u1xxxxxx u1xxxxxxxx
+
+
+
2!
4!
6!
8!
β
β
= 2βu0x u0xx +
u0x u0xxxx + u0xx u0xxx
6
3
β
β
β
+
u0x u0xxxxxx + u0xx u0xxxxx + u0xxx u0xxxx
180
60
36
7
5
(ik)
(ik)
= β 2(ik)3 +
+
A2 e2i(kx−ω0 T0 ) + c.c.
2
20

D02 u1 − 2α

(5.48)

We assume the particular solution to the above equation to be of the form,
u1 = A1 e2i(kx−ω0 T0 ) + c.c.

(5.49)

Plugging this into the 1 equation and balancing the coefficients of e2i(kx−ω0 T0 ) leads to

(−2iω0 )2 − 2α

(2ik)2 (2ik)4 (2ik)6 (2ik)8
+
+
+
2!
4!
6!
8!

A1 = βA2 2(ik)3 +

(ik)5 (ik)7
+
2
20
(5.50)

119

Using the equation (5.46) in the above equation (5.51)we get
k5 k7
3k 4 15k 6 63k 8
−
+
A1 = iβA2 2k 3 −
+
4!
6!
8!
2
20

8α

(5.51)

and further simplifying

2iβ
k2 k4
A1 =
1−
+
αk
4
40

k2 k4
1−
+
6
80

−1

A2

(5.52)

The solution to 1 equation is therefore

2iβ
k2 k4
+
u1 =
1−
αk
4
40
k4
k2
Letting A˜1 = 2iβ
αk 1 − 4 + 40

k2 k4
+
1−
6
80
4
2
1 − k6 + k80

−1

−1

A2 e2i(kx−ω0 T0 ) + c.c.

(5.53)

, we can write u1 = A˜1 y02 + c.c., and

u1 x . . . x = (2ik)m A˜1 y02 + c.c.. We will now examine the 2 equation in order to solve for
m-times

A in terms of T2 . Since u0 and u1 are independent of T1 , (−2D0 D1 u1 ) and (D12 u0 ) in
equation (5.44) vanish. For l, m, n = 1, 2, · · · , we also have
u1 x . . . x u1 x . . . x = 2m (ik)m+n A˜1 y03 + (2ik)m (−ik)n A˜1 y02 y¯0 + c.c.
m-times

n-times

(5.54)

u0x . . . x u0 x . . . x u0 x . . . x =
l-times

m-times

n-times

(ik)l+m+n (−1)l + (−1)m + (−1)n y02 y¯0 + (ik)l+m+n y03 + c.c.
Using the above relations in equation (5.44), it is clear that the only terms that contribute
to the secularity i.e. the coefficients of ei(kx−ω0 T0 ) , are the coefficients of y02 y¯0 multiplied by
¯ Plugging u0 and u1 into equation (5.44), we obtain the following solvability condition:
A2 A.

120

2(iω0 )(D2 A) = β 4(ik)3 + (ik)5 +

3
1
1
(ik)7 A˜1 A2 A¯ + γ 3(ik)4 + (ik)6 + (ik)8 A2 A¯
10
2
80
(5.55)

ˆ

ˆ

Letting A = 21 d(T2 )e−iθ(T2 ) , and using the relations A2 A¯ = 18 d3 e−iθ and 2D2 A =
ˆ
∂d , and b = ∂ θˆ in the above equation, we obtain
(d − diθˆ )e−iθ , where d = ∂T
∂T2
2

k 2 k 4 3 −iθˆ
3γk 4
ˆ
1−
+
d e
(iω0 d + ω0 dθˆ )e−iθ =
8
6
80
2
−1
β 2k2
k2 k4
k2 k4
ˆ
+
1−
+
1−
+
d3 e−iθ
α
4
40
6
80

(5.56)

Equating the real and imaginary parts, expanding the square and using the Maclaurin
∞

series expansion (1 − x)−1 =

xn , the solvability condition can be written as
n=0

Im: d = 0
3γk 4
k2 k4
β 2k2
k 2 2k 4
+
+
Re: θˆ =
1−
+
1−
8
6
80
α
3
45

d2
ω0

(5.57)

Solving the equation (5.57), making the integration constant zero without loss of generˆ becomes,
ality, and noting that T2 = 2 T0 , the slowly varying amplitude (d) and phase (θ)
d = d0
k2 k4
β 2k2
k 2 2k 4
3γk 4
1−
+
+
1−
+
θˆ =
8
6
80
α
3
45

a20 2
T0
ω0

(5.58)

ˆ

Therefore combining equation (5.45) with A = 21 de−iθ , the frequency ω can be written
as
ω = ω0 +

2 d2 3γk 4
θˆ
k2 k4
β 2k2
k 2 2k 4
= ω0 1 + 20
1−
+
+
1−
+
T0
8
6
80
α
3
45
ω0

(5.59)

Also note that, in the above dispersion relationship, ω0 is a function of the wave number

121

k. We find that applying the Taylor series expansion of sin2 k and sin2 k2 and taking the first
three terms in the expansion will result in the equation (5.59).
To reduce the error in the dispersion characteristics, one needs to consider more number
of terms, which in turn when goes to infinity results in the trigonometric sines and cosines.
Hence, the frequency in the displacement coordinate system can be accurately represented
as
ω(k, d0 ) = ω0 1 +

2 d2
0
2
ω0

3γ
β2
2
2(1 − cos k) +
sin2 k
8
α

(5.60)

3 2 d20
2β 2
γ(1 − cos k) +
(1 + cos k)
4α
3α

(5.61)

Further simplifying
ω(k, d0 ) = ω0 1 +

Hence our quest to solve for the dispersion characteristics in a continuum with quadratic
and cubic nonlinearities using method of multiple scales in a partial differential equation has
led us to the dispersion relations as shown in equation (5.61), which is an exact match with
the dispersion relation obtained in equation (5.32) of section 5.2 by applying multiple scales
to the discrete chain of mass spring system.

5.4

Wave Dispersion Analysis Using Third-Order Multiple Scales with O(1) DC-Bias

5.4.1

Analysis in Strain Coordinates with O(1) DC Term

Using a second-order perturbation analysis, we found the effect of the quadratic nonlinearity.
However, the true behaviour of a system with quadratic nonlinearity would develop a DC-

122

bias, which we did not take into account during our perturbation analysis. This DC bias is
developed along the chain and is uncovered by the numerical simulations. So to obtain the
dispersion relations, we now take into account the DC-bias which we assume to be in O(1).
We now analyze the system using a third-order MMS analysis, for which we assume,
zj (t) = zj0 (T0 , T1 , T2 , T3 ) + zj1 (T0 , T1 , T2 , T3 )
+ 2 zj2 (T0 , T1 , T2 , T3 ) + 3 zj3 (T0 , T1 , T2 , T3 )
d
= D0 + D1 + 2 D2 + 3 D3
dt
d2
= D02 + (2D0 D1 ) + 2 D12 + 2D0 D2 + 3 (2D0 D3 + 2D1 D2 )
2
dt

(5.62)

∂ . Plugging into equation (5.8) yields
where Di = ∂T
i

D02 zj0 +

D02 zj1 + 2D0 D1 zj0 + 2 D02 zj2 + 2D0 D1 zj1 + D12 + 2D0 D2 zj0

+ 3 D02 zj3 + 2D0 D1 zj2 + D12 + 2D0 D2 zj1 + (2D0 D3 + 2D1 D2 ) zj0
= −α 2 zj0 − zj+10 − zj−10 −
2
2
+ β 2 zj2 − zj+1
− zj−1
0
0
0

α 2 zj1 − zj+11 − zj−11
− 2 α 2 zj2 − zj+12 − zj−12

(5.63)

3
3
− zj−1
+ 2 β 2 zj0 zj1 − zj+10 zj+11 − zj−10 zj−11 + γ 2 zj3 − zj+1
0
0
0

− 3 α 2 zj3 − zj+13 − zj−13 + 2 β 2 zj0 zj2 − zj+10 zj+12 − zj−10 zj−12
2
2
2
2
+ β 2 zj2 − zj+1
− zj−1
+ 3γ 2 zj2 zj1 − zj+1
z
− zj−1
z
1
1
1
0
0 j+11
0 j−11

Equating like powers of ,
0

1

: D02 zj0 + α 2 zj0 − zj+10 − zj−10 = 0

(5.64)

2
2
: D02 zj1 + α 2 zj1 − zj+11 − zj−11 = −2D0 D1 zj0 − β 2 zj2 − zj+1
− zj−1
0
0
0

(5.65)

123

2

: D02 zj2 + α 2 zj2 − zj+12 − zj−12 = −2D0 D1 zj1 − D12 + 2D0 D2 zj0

(5.66)

3
3
− 2 β 2 zj0 zj1 − zj+10 zj+11 − zj−10 zj−11 − γ 2 zj3 − zj+1
− zj−1
0
0
0
3

: D02 zj3 + α 2 zj3 − zj+13 − zj−13 = −2D0 D1 zj2 − D12 + 2D0 D2 zj1
− (2D0 D3 + 2D1 D2 ) zj0 − 2 β 2 zj0 zj2 − zj+10 zj+12 − zj−10 zj−12

(5.67)

2
2
2
2
− β 2 zj2 − zj+1
− zj−1
− 3γ 2 zj2 zj1 − zj+1
z
− zj−1
z
1
1
1
0
0 j+11
0 j−11

We assume a traveling wave solution to solve the 0 equation (5.64). Let the physical
coordinate be x˜. We consider an uniform chain where each mass is separated by a distance h
from its nearest neighbour i.e. x˜j±1 = x˜j ± h. Now to non-dimensionalize we assume x = hx˜ .
Hence xj±1 = xj ± 1. Due to the quadratic nonlinearity we would expect a DC-bias in the
overall spatial configuration when a wave is propagated down the chain. Hence we let
i(k˜
xj −ω0 T0 )

zj0 = A0 (T1 , T2 ) + A(T1 , T2 )e

¯ 1 , T2 )e−i(k˜xj −ω0 T0 )
+ A(T

(5.68)

= A0 (T1 , T2 ) + yj0 + y¯j0
where yj0 = A(T1 , T2 )e

i(kxj −ω0 T0 )

and A0 (T1 , T2 ) is the DC-bias and A(T1 , T2 ) is the

slowly varying amplitude of the propagating wave. Plugging in yj±10 = Ae
i(kxj −ω0 T0 )

e±ik Ae

i(kxj±1 −ω0 T0 )

=

= e±ik yj0 , into equation (5.64) we get
ω02 = 2α(1 − cos k)

(5.69)

The solution to 0 equation can be used in equation (5.65) to find the secular terms. The
secular terms i.e. the DC terms and coefficients of the e
side of the equation (5.65) are

124

i(kxj −ω0 T0 )

terms in the right hand

2D0 D1 zj0 = 2(−iω0 )D1 Ae

i(kxj −ω0 T0 )

+ c.c.
2i(kxj −ω0 T0 )

2
2
2 zj2 − zj+1
− zj−1
= 2 − e2ik − e−2ik A2 e
0
0
0

i(kxj −ω0 T0 )

+ 2A0 A 2 − eik − e−ik e

Hence the solvability condition i.e. coefficients of e

(5.70)

+ c.c.

+ c.c. + (2A20 − A20 − A20 )

i(kxj −ω0 T0 )
,

from equation (5.70) is

2(iω0 )D1 A = 2βA0 A 2 − eik − e−ik

(5.71)

Simplifying we get,
D1 A = −i

βω0
A A
α 0

(5.72)

Removing the secular terms, we rewrite the 1 equation as

D02 zj1 + α 2 zj1 − zj+11 − zj−11 = −β(2 − e2ik − e−2ik )A2 e
=

2i(kxj −ω0 T0 )
−2βA2 (1 − cos 2k)e

2i(kxj −ω0 T0 )

+ c.c.
(5.73)

+ c.c.

Assuming the particular solution to the above equation to be of the form,

zj1 = A1 e

2i(kxj −ω0 T0 )

+ c.c.

(5.74)

and plugging into equation (5.73) and simplifying we get
(−2iω0 )2 + α 2 − e2ik − e−2ik

A1 e

2i(kxj −ω0 T0 )

= −4βA2 sin2 k e

2i(kxj −ω0 T0 )

(5.75)
2i(kxj −ω0 T0 )

Balancing the coefficients of e

A1 =

leads to
βA2 sin2 k
ω02 − α sin2 k
125

(5.76)

Further simplifying equation (5.76) we get
2β(1 + cos k) 2
A
ω02

(5.77)

2β(1 + cos k) 2 2i(kxj −ω0 T0 )
A e
+ c.c.
ω02

(5.78)

A1 =
The solution to 1 equation is therefore
zj1 =

We now examine the 2 equation in order to solve for A0 (T1 , T2 ) and A(T1 , T2 ). Using the
assumed solutions to zj0 and zj1 , various terms in the right hand side of the equation (5.66)
are computed as below:
2D0 D1 zj1 = −i

8β(1 + cos k)
2i(kxj −ω0 T0 )
(D1 A2 )e
+ c.c.
ω0
i(kxj −ω0 T0 )

+ c.c.

i(kxj −ω0 T0 )

+ c.c.

D12 zj0 = D12 A0 + (D12 A)e

2D0 D2 zj0 = −2iω0 (D2 A)e

¯ i(kxj −ω0 T0 ) + A0 e2i(kxj −ω0 T0 ) + Ae3i(kxj −ω0 T0 ) + c.c.
zj0 zj1 = A1 Ae
zj3 = A0 + Ae
0

i(kxj −ω0 T0 )

¯
+ 3(A20 A + A2 A)e

3

¯ i(kxj −ω0 T0 )
+ Ae

i(kxj −ω0 T0 )

¯
= (A30 + 6A0 AA)

2i(kxj −ω0 T0 )

+ 3A0 A2 e

+ A3 e

3i(kxj −ω0 T0 )

+ c.c.
(5.79)

Using the above computations in the 2 equation, we get
D02 zj2 + α 2 zj2 − zj+12 − zj−12
=i

8β(1 + cos k)
2i(kxj −ω0 T0 )
i(kxj −ω0 T0 )
(D1 A2 )e
− (D12 A)e
ω0

+ 2iω0 (D2 A)e

i(kxj −ω0 T0 )

¯ − eik − e−ik )ei(kxj −ω0 T0 )
+ c.c. − D12 A0 − 2βA1 A(2

2i(kxj −ω0 T0 )

3i(kxj −ω0 T0 )

+ A0 (2 − e2ik − e−2ik )e

+ A(2 − e3ik − e−3ik )e

¯ − eik − e−ik )e
− γ 3(A20 A + A2 A)(2

i(kxj −ω0 T0 )

2i(kxj −ω0 T0 )

+ 3A0 A2 (2 − e2ik − e−2ik )e

+ A3 (2 − e3ik − e−3ik )e

+ c.c.

3i(kxj −ω0 T0 )

+ c.c.
(5.80)

126

The secular terms in the above equation are the DC terms and the coefficients of e

i(kxj −ω0 T0 )

Hence the solvability conditions are
D12 A0 = 0
2iω0 (D2 A) − (D12 A)

2βω02
3γω02 2
¯
=
A1 A¯ +
(A0 A + A2 A)
α
α

(5.81)

This results in A0 (T1 , T2 ) = l1 (T2 )T1 + l2 (T2 ). However, to prevent secular behavior, A0
needs to be bounded, which leads to l1 (T2 ) = 0, and hence A0 = l2 (T2 ). Once we eliminate
the secular terms in equation (5.80), we can obtain the particular solution as

zj2 = B1 (T2 )e

2i(kxj −ω0 T0 )

3i(kxj −ω0 T0 )

+ B2 (T2 )e

(5.82)

i(kxj −ω0 T0 )
The secular terms in the 3 equation are the DC terms and the coefficients of e
.

So the secularity contribution from the DC terms from equation (5.67) is

(2D0 D3 + 2D1 D2 )A0 = 0

(5.83)

Since D0 A0 = 0, A0 = l(T2 ) a function of T2 and independent of T0 and T1 , we get
D2 A0 (T2 ) = 0

(5.84)

Hence A0 is a constant and independent of time T0 , T1 , and T2 . From equation (5.72) we
can find
D12 A

β 2 ω02 2
= − 2 A0 A
α

(5.85)

Using equation (5.77) and (5.85) in equation (5.81) results in the reduced solvability
condition
2iω0 (D2 A) =

3γ β 2
− 2
α
α

ω02 A20 A +

3γ
4β 2
+
(1 + cos k) ω02 A2 A¯
2
α
αω0

127

(5.86)

.

Substituting the amplitude with slowly varying amplitude and phase such that, A =
1 a(T )e−ib(T2 )
2
2

and D2 A = 21 (D2 a − ia D2 b)e−ib in the above equation and equating the

real and imaginary parts, the solvability condition can be rewritten as
Im: D2 a = 0
Re: ω0 a D2 b =

3γ β 2
− 2
α
α

a
ω02 A20
2

+

3
3γ
4β 2
2a
(1
+
cos
k)
ω
+
0 8
α
αω02

(5.87)

Using ω0 from equation (5.69), in equation (5.87), we get
Im: D2 a = 0
3γ β 2
− 2
α
α

Re: D2 b =

A20
+
2

3γ
4β 2
+
(1 + cos k)
α
αω02

a2
ω0
8

(5.88)

Solving the equation (5.88), we get
a = a0
b=

3γ β 2
− 2
α
α

A20
+
2

3γ
4β 2
+
(1 + cos k)
α
αω02

a20
ω0 T2 + bc
8

(5.89)

We can neglect integration constant bc without loss of generality. Also we note that T2 =
2T .
0

Hence the slowly varying amplitude (a) and phase (b) becomes,

a = a0
b=

3γ β 2
− 2
α
α

A20
+
2

3γ
4β 2
+
(1 + cos k)
α
αω02

a20
ω0 2 T0
8

(5.90)

Therefore combining equation (5.68) with A = 12 ae−ib , the frequency ω can be written as
b
ω = ω0 +
T0
= ω0 1 +

2

3γ β 2
− 2
α
α

A20
+
2
128

3γ
4β 2
+
(1 + cos k)
α
αω02

a20
8

(5.91)

5.4.2

Transforming Dispersion Relations from Strain into Displacement Coordinates

As we had transformed our coordinates into displacement coordinate in the previous section,
we do the same here to obtain the final dispersion relation in displacement coordinate where
the amplitude and phase in strain coordinates are related to the amplitude and phase in
displacement coordinates by

ω2
a20 = 0 d20
α
π k
φ= +
2 2

(5.92)

Therefore the frequency in the displacement coordinate system can be written as

ω(k, d0 ) = ω0 1 +

2

β2
γ−
3α

3A20
2β 2
+ γ(1 − cos k) +
(1 + cos k)
2α
3α

3d20
4α

(5.93)

One can show the validity and accuracy of the coordinate transformation by considering
a continuum as discussed in previous section. The new dispersion relation has introduced a
correction term to the dispersion obtained without considering any DC-bias. However, the
new correction has a DC-bias even when there is no quadratic term, i.e. by putting β = 0,
in the new dispersion relation, results in an additional γ term than expected without the
DC-bias term. Hence we would now consider introducing the DC bias in z-coordinate to be
of order .

129

5.5

Wave Dispersion Analysis Using Third-Order Multiple Scales with O( ) DC-Bias

5.5.1

Analysis in Strain Coordinates with O( ) DC Term

Using the O(1) DC-bias term in the third-order analysis, we found the effect of the DC-bias
due to quadratic nonlinearity. However, the true behavior of a system with quadratic and
cubic nonlinearities should not develop a DC-bias when the quadratic nonlinearity is absent.
The DC-bias is only developed due to the quadratic nonlinearity. So in the final dispersion
relation, the DC-bias term should only have coefficient of the quadratic nonlinearity. To
uncover the true dispersion relation, we now take into account the DC-bias of O( ) instead
of O(1). We now analyze the same system of equations found using a third-order MMS
analysis, i.e. equations (5.64)- (5.67).

We assume a traveling wave solution to solve the 0 equation (5.64). Let the physical coordinate be x˜. Since we are assuming uniform chain, we consider that each mass is separated
by a a distance h from its nearest neighbour i.e. x˜j±1 = x˜j ± h. Now to non-dimensionalize
we assume x = hx˜ . Hence xj±1 = xj ± 1.
zj0 = A(T1 , T2 , T3 )e

i(k˜
xj −ω0 T0 )

¯ 1 , T2 , T3 )e−i(k˜xj −ω0 T0 )
+ A(T

(5.94)

= yj0 + y¯j0
where yj0 = A(T1 , T2 )e

i(kxj −ω0 T0 )

and A0 (T1 , T2 ) is the DC-bias and A(T1 , T2 ) is the slowly

varying amplitude of the propagating wave. Then plugging in yj±10 = Ae
i(kxj −ω0 T0 )

e±ik Ae

= e±ik yj0 , into equation (5.64) we get

130

i(kxj±1 −ω0 T0 )

=

ω02 = 2α(1 − cos k)

(5.95)

The solution to 0 equation can be used in equation (5.65) to find the secular terms. The
secular terms are the resonant terms w.r.t the e
order analysis, 2D0 D1 zj0 = 2(−iω0 )D1 Ae

i(kxj −ω0 T0 )

i(kxj −ω0 T0 )
+ c.c.

term. Just like in the second-

is the only term contributing to

the secularity, hence resulting in D1 A = 0. Removing the secular terms, we rewrite the 1
equation as
D02 zj1 + α 2 zj1 − zj+11 − zj−11 = −β(2 − e2ik − e−2ik )A2 e
=

2i(kxj −ω0 T0 )
−2βA2 (1 − cos 2k)e

2i(kxj −ω0 T0 )

+ c.c.
(5.96)

+ c.c.

Now we introduce the O( ) DC-bias term i.e. assume the particular solution to the above
equation to be of the form,

2i(kxj −ω0 T0 )

zj1 = B(T1 , T2 , T3 ) + A1 (T1 , T2 , T3 )e

+ c.c.

2i(kxj −ω0 T0 )

and solving equation (5.96), balancing the coefficients of e
to

A1 =

(5.97)

and simplifying leads

4β cos2 k2 2
A
ω02

(5.98)

The solution to 1 equation is therefore

zj1 = B(T1 , T2 , T3 ) +

4β cos2 k2 2
2i(kxj −ω0 T0 )
A (T2 , T3 )e
+ c.c.
ω02

(5.99)

We now examine the 2 equation in order to solve for B(T1 , T2 , T3 ) and A(T2 , T3 ). Using
the assumed solutions to zj0 and zj1 , various terms in the right hand side of the equation (5.66) are computed as below:

131

2D0 D1 zj1 = 0
D12 zj0 = 0
2D0 D2 zj0 = −2iω0 (D2 A)e

i(kxj −ω0 T0 )

i(kxj −ω0 T0 )

¯
zj0 zj1 = (BA + A1 A)e
¯
zj3 = 3A2 Ae
0

i(kxj −ω0 T0 )

(5.100)

+ c.c.
3i(kxj −ω0 T0 )

+ c.c. + AA1 e

+ c.c. + A3 e

3i(kxj −ω0 T0 )

+ c.c.

+ c.c.

Using the above computations, we get the 2 equation as

D02 zj2 + α 2 zj2 − zj+12 − zj−12 = 2iω0 (D2 A)e

i(kxj −ω0 T0 )

+ c.c.

¯ − eik − e−ik )ei(kxj −ω0 T0 )
− 2βA1 (BA + A1 A)(2
3i(kxj −ω0 T0 )

+ AA1 (2 − e3ik − e−3ik )e
− γ 3A20 A(2 − eik − e−ik )e

i(kxj −ω0 T0 )

+ c.c.

+ A3 (2 − e3ik − e−3ik )e

3i(kxj −ω0 T0 )

+ c.c.
(5.101)

The secular terms in the above equation are the DC terms and the coefficients of e

i(kxj −ω0 T0 )

Hence the solvability conditions are
3γω02 2 ¯
2βω02
¯
(BA + A1 A) +
A A
2iω0 (D2 A) =
α
α

(5.102)

Now eliminating the secular terms in 2 equation, the particular solution can be written
as
zj2 = A2 (T2 , T3 )e

3i(kxj −ω0 T0 )

+ c.c.

(5.103)

A2 is independent of time scale T1 because, A2 will only be a function of A which does
not vary with T1 . Hence D1 A2 = 0, and A2 = A2 (T2 , T3 ).
Using the solutions zj0 , zj1 , and zj2 in the 3 equation, we get
132

.

2D0 D1 zj2 = 0
D12 zj1 = D12 B
2D0 D2 zj1 = −4iω0 (D2 A1 )e

i(kxj −ω0 T0 )

i(kxj −ω0 T0 )

2D0 D3 zj0 = −2iω0 (D3 A)e

+ c.c.

+ c.c.
(5.104)

2D1 D2 zj0 = 0
¯ 2e
zj0 zj2 = AA

2i(kxj −ω0 T0 )

+ AA2 e

2i(kxj −ω0 T0 )

zj2 = B 2 + 2BA1 e
1

4i(kxj −ω0 T0 )

¯
+ A21 Ae

+ c.c.

4i(kxj −ω0 T0 )

¯ + A1 A¯2 ) + (BA2 + 2AAA
¯ 1 )e
zj2 zj1 = (AAB
0

+ c.c.

2i(kxj −ω0 T0 )

4i(kxj −ω0 T0 )

+ A2 A1 e

Using the above computations in the 3 equation, we obtain the solvability condition
D12 B = 0 i.e. B = B(T2 , T3 )

(5.105)

2iω0 D3 A = 0
Hence eliminating the secular terms we can assume the particular solution to 3 equation as

2i(kxj −ω0 T0 )

zj3 = A3 e

+ A4 e

4i(kxj −ω0 T0 )

+ c.c.

(5.106)

Substituting the amplitude A with slowly varying amplitude and phase such that, A =
1 a(T )e−iθ(T2 )
2
2

and D2 A = 12 (D2 a − ia D2 θ)e−iθ in the solvability condition of 2 equation

and equating the real and imaginary parts, the solvability condition can be rewritten as
Im: D2 a = 0
βω02 aB
4β 2 ω02
3γω02 a3
Re: ω0 a D2 θ =
+
(1 + cos k) +
α
8α
8
αω 2
Simplifying above equation we get
133

(5.107)

Im: D2 a = 0 i.e. a = a0
2β 2
βω0 B
+ γ+
Re: D2 θ =
α
3α

1 + cos k
1 − cos k

3ω0 2
a
8α 0

(5.108)

Assuming B does not vary in time-scale T2 , and solving the above equation (5.108), we get
a = a0
3γ
4β 2
(1 + cos k)
+
α
αω02

β
θ=
B+
α

a20
ω0 T2 + bc
8

(5.109)

Neglecting integration constant bc without loss of generality the slowly varying amplitude
(a) and phase (θ) becomes,
a = a0
β
B+
θ=
α

3γ
4β 2
+
(1 + cos k)
α
αω02

a20
ω0 2 T0
8

(5.110)

Therefore combining equation (5.94) with A = 12 ae−iθ , the frequency ω can be written
as

θ
ω = ω0 +
T0
= ω0 1 +

5.5.2

2

β
B+
α

4β 2
3γ
+
(1 + cos k)
α
αω02

a20
8

(5.111)

Transforming Dispersion Relations from Strain into Displacement Coordinates

As we had transformed our coordinates into displacement coordinate in the previous section,
we do the same here to obtain the final dispersion relation in displacement coordinate where
the amplitude and phase in strain coordinates are related to the amplitude and phase in
displacement coordinates by

134

ω02 2
=
d
α 0
π k
φ= +
2 2

a20

(5.112)

Therefore the frequency in the displacement coordinate system can be written as
ω(k, d0 ) = ω0 1 +

2

2β 2
β
B + γ(1 − cos k) +
(1 + cos k)
α
3α

3d20
4α

(5.113)

The new dispersion relation has introduced a correction term to the dispersion relation
due to the quadratic nonlinearity. The new correction has a DC Bias in the order of

and

vanish from the dispersion relation when the coefficient of quadratic nonlinearity is zero. This
aligns well with our numerical simulation. Though we do not have an analytical expression
for the DC-bias B, we compute it from the numerical simulation and plug it back into the
analytical expression to plot the nonlinear dispersion diagram. For a given frequency ω and
amplitude of oscillation d0 , we simulate a propagating wave in a chain of 1000 mass and
compute the spacial frequency k using FFT and the DC-bias B from the spacial signal at a
particular time.

5.6

Multi-Wave Interactions in an Infinite Nonlinear
Chain Using First-Order Multiple Scales Analysis

5.6.1

Multiple Scales Analysis

We analyze the wave equation using the method of multiple scales (MMS). We consider up
to O( 2 ) error i.e. only time scales of interest are T0 = t and T1 = t, where T0 is the fast
varying and T1 is the slow varying time scale. Expressing our strain variable zj (t) in multiple
135

time scales as in equation (5.9) and plugging into equation (5.8) yields,
D02 zj0 +
−

D02 zj1 + 2D0 D1 zj0 = −α 2 zj0 − zj+10 − zj−10

(5.114)

2
2
α 2 zj1 − zj+11 − zj−11 + β 2 zj2 − zj+1
− zj−1
0
0
0

Equating like powers of ,
0

1

: D02 zj0 + α 2 zj0 − zj+10 − zj−10 = 0

(5.115)

2
2
− zj−1
: D02 zj1 + α 2 zj1 − zj+11 − zj−11 = −2D0 D1 zj0 − β 2 zj2 − zj+1
0
0
0

(5.116)
We assume xj±1 = xj ± h, and letting x0 = 0 without loss of generality, we have xj = jh.
The non-dimensional wavenumber is then µ = kh, where k is the wave number and h is the
spacial distance between the adjacent masses. Assuming a traveling wave solution in spacial
and temporal terms with two harmonics to solve the 0 equation (5.115), we let
zj0 = yjA + yjB = Ae
0
0
i(µA j−ωA T0 )
0
0

where yjA = Ae
0

i(µA j−ωA T0 )
0
0

i(µB j−ωB T0 )
0
0

+ c.c.

i(µB j−ωB T0 )
0
0

+ c.c. are the waves A

+ Be

+ c.c., and yjB = Be
0

(5.117)

and B respectively, and c.c. represents the complex conjugates of all preceding terms. µA0 ,
µB0 are the non-dimensional wave numbers and ωA0 , ωB0 are the corresponding frequencies
of the waves A and B respectively. Then substituting yj±1A = e
0

c.c., and yj±1B = e
0

±iµB
i(µ j−ωB T0 )
0 Be B0
0

+

+ c.c., into equation (5.115) we get

2 = 2α 1 − cos µ
ωA
A0
0

2
ωB
0

±iµA
i(µ j−ωA T0 )
0 Ae A0
0

(5.118)

= 2α 1 − cos µB0

Hence, letting A(T1 ) = 21 a(T1 )e−iθA (T1 ) , and B(T1 ) = 21 b(T1 )e−iθB (T1 ) , the solution to
136

0

equation becomes
1 i(µ j−ωA T0 −θA ) 1 i(µB j−ωB T0 −θB )
0
0
0
zj0 = ae A0
+ be
+ c.c.
2
2

(5.119)

For non-zero amplitudes a and b, the response zj0 in equation (5.119), seems quasiperiodic
with phases (µA0 j −ωA0 T0 −θA ), and (µB0 j −ωB0 T0 −θB ) in continuous drift with difference
θ = (µA0 − µB0 )j − (ωA0 − ωB0 )T0 − (θA − θB ) known as phase drift. Using this solution
of zj0 in equation (5.116) we get
D02 zj1 + α 2 zj1 − zj+11 − zj−11
˜

˜

= iωA0 (a − iaθA )eiθA + iωB0 (b − ibθB )eiθB + c.c.
−
−

β
2
4
a2 e

˜

˜

˜

˜

˜

2iµA 2iθ˜
0e A

2iµB 2iθ˜
0e B

+ b2 e

+ 2abe

i(µB −µA ) i(θ˜ −θ˜ )
0 e B A
0

+ a2 + b2 + 2abe
−

a2 e

˜

a2 e2iθA + b2 e2iθB + 2abei(θA +θB ) + a2 + b2 + 2abei(θB −θA ) + c.c.

−2iµA 2iθ˜
0e A

−2iµB 2iθ˜
0e B

+ b2 e

i(µA +µB ) i(θ˜ +θ˜ )
0 e A B
0

+ c.c.
−i(µA +µB ) i(θ˜ +θ˜ )
0 e A B
0

+ 2abe

−i(µB −µA ) i(θ˜ −θ˜ )
0 e B A
0

+ a2 + b2 + 2abe

˜

+ c.c.
˜

= iωA0 (a − iaθA )eiθA + iωB0 (b − ibθB )eiθB + c.c.
−

β 2
˜
˜
a 1 − cos(2µA0 ) e2iθA + b2 1 − cos(2µB0 ) e2iθB
2
˜

˜

˜

˜

+ 2ab 1 − cos(µB0 + µA0 ) ei(θB +θA ) + 2ab 1 − cos(µB0 − µA0 ) ei(θB −θA ) + c.c.
(5.120)
where, θ˜A = µA0 j − ωA0 T0 − θA , and θ˜B = µB0 j − ωB0 T0 − θB .
The secular terms are the coefficients of e

i(µA j−ωA T0 )
0
0

and e

i(µB j−ωB T0 )
0
0
.

Primary

resonance occurs when
Wave-A :
Wave-B :

iωA (a − iθA a) = 0
iωB (b − iθB b) = 0
137

(5.121)

which simplifies to the following equations
ωA a = 0
ωA aθA = 0

(5.122)

ωB b = 0
ωB bθB = 0
For primary resonance in the above equation, the amplitudes and the phases of both
waves A, and B become independent of the slow time scale T1 hence doesn’t contain any
slow scale dynamics. In that case we need higher order terms to show the effects of the
primary resonances i.e. considering the O( 2 ) terms.
However, we do have interaction between the two waves resulting in super- and subharmonic resonances. Due to the arbitrary naming convention of the A and B waves, the
secular terms arising from super- and sub-harmonics are identical. We let, ωB0 ∼
= 2ωA0 , and
µB 0 ∼
= 2µA0 . Introducing detuning parameters σω , and σµ , we can write,
ωB0 = 2ωA0 + σω

(5.123)

µB0 = 2µA0 + σµ
where σω , and σµ are of order one. σω and σµ are not independent of each other. For
example, given σω and ωA , ωB can be computed from equation (5.123)(a) and µA from
equation (5.118). Then ωB defines µB in equation (5.118). Finally σµ is obtained from
equation (5.123)(b).
i(µA j−ωA T0 )
0
0
)

For wave A, the secular terms (coefficients of e

are

iωA0 (a − iaθA )e−iθA = βab 1 − cos(µB0 − µA0 ) e−i(θB −θA ) ei (σµ j−σω T0 )

138

(5.124)

i(µB j−ωB T0 )
0
0
)

and for wave B, the secular terms (coefficients of e
iωB0 (b − ibθB )e−iθB =

are

β 2
a 1 − cos(2µA0 ) e−i2θA e−i (σµ j−σω T0 )
2

(5.125)

Equating the real and imaginary parts the equations for the slow scale dynamics can be
written as
a =

βab
1 − cos(µB0 − µA0 ) sin φ
ωA0

b =−

βa2
1 − cos(2µA0 ) sin φ
2ωB0

βb
1 − cos(µB0 − µA0 ) cos φ
θA =
ωA0

(5.126)

βa2
θB =
1 − cos(2µA0 ) cos φ
2bωB0
where, ( ) denote the derivative with respect to slowly varying time scale T1 . In above
equation φ is a function of the phases of the waves A and B, the detuning parameters σω
and σµ , and the time scale T1 as
φ = (2θA − θB ) + σµ j − σω T1

(5.127)

φ = (2θA − θB ) − σω
Using equation (5.127) in equation (5.126), the 4-D phase space can be reduced to three as
a =

α1
ab sin φ
ωA0

b =−

α2 2
a sin φ
ωB0

φ = −σω +

(5.128)

2α1 2
α
1
b − 2 a2 cos φ
ωA0
ωB0
b

where α1 = β 1 − cos(µB0 − µA0 ) , α2 = β2 1 − cos(2µA0 ) are functions of nonlinearity
coefficient β and the non-dimensional wave numbers of waves A and B, and are independent

139

of the amplitudes and phases. For non-zero a, b, a , and b dividing the two amplitude
da / db = da , we get
equations and letting dT
db
1 dT1

ωB α b
da
1b
=− 0 1 =−
db
ωA0 α2 a
ra
where r =

α2 ωA
0
α1 ωB
0

=

µA
0
sin2 (µA ) sin
2
0
µA
σ
σ
0
2 sin2
2 + 2 sin µA0 + 2

(5.129)

. Cross multiplying and integrating the

above equation results in

ra2 + b2 = E

(5.130)

where E is the integration constant and can be considered as the total energy of the system.
We find that the slowly varying amplitudes of the waves lie on an ellipse in the amplitude
space for a fixed energy state when both the waves have nonzero amplitudes. The strength of
the quadratic nonlinearity affects the rate at which the energy is exchanged. For different energy states the amplitudes of these waves are defined in different concentric ellipses for given
detuning parameter. In the super- or sub-harmonic resonance condition, the amplitudes of
the waves lie on a fixed ellipse. However, the shape of this ellipse can change depending on
parameters such as the detuning parameter.
When a and b are zero, the phase is either zero or nπ which also represents the steady
state. In this case, the non-zero steady state amplitudes lie as a fixed point on an ellipse
in a-b plane. However when a, b, a , and b all are non-zero, the amplitudes oscillate on an
ellipse in the a-b plane for a particular energy state.
Using equation (5.130) in equation (5.128), the dimension of phase orbits can be reduced
from three to two. This means that we have a 3-D phase space (a, b, φ as the states),
but the motion is constrained/confined to a 2-D surface defined by the ellipses of a, b as in
equation (5.130). Hence the reduced amplitude-phase equations in 2-D are

140

b =−

α1
E − b2 sin φ
ωA0

(5.131)

α
φ = −σω − 1 E − 3b2 cos φ
ωA0 b

In an attempt to solve for the amplitude and phase, we divide the phase and amplitude
equations in (5.131), to get further reduced equation in 1-D
g (b) cos φ g2 (b)
dφ
= 1
−
db
sin φ
sin φ

(5.132)

ωA σ ω
2
0
where g1 (b) = E−3b , and g2 (b) = −
.
b E−b2

α1 E−b2

Letting cos φ = p, and (− sin φ)dφ = dp, we obtain a first order linear differential equation
of the form
dp(b)
+ g1 (b)p(b) = g2 (b)
db

(5.133)

The relationship between the amplitude b and phase φ can then be written as

p(b) = cos φ = e− g1 (b) db

g2 (b)e g1 (b) db db + c1

(5.134)

where c1 is an integration constant, and the integrating factor is e g1 (b) db = b(E − b2 ).
Using the integrating factor, we then get

ωA σω
g2 (b)e g1 (b) db db = − 2α0 b2 . The relation1

ship between the amplitude b and the phase φ is therefore obtained from the solution to our
differential equation as

cos φ =

ωA σω
1
− 0 b 2 + c1
2
2α1
b(E − b )

141

(5.135)

5.6.2

Stability Analysis

For fixed points we will have b = φ = 0 in the 2-D phase space in equation (5.131). We get
the follwing two equations
b20 = E
(5.136)

σω ωA0
√
cos φ0 = ±
2α1 E
For non-zero equilibrium amplitudes, the other fixed point correspond to

sin φ0 = 0
b20
ω

(5.137)

−σ
˜ b0 − E˜ = 0

σ

A ω
where σ
˜ = (±) 3α0 , and E˜ = E3 . The above equation results in the quadratic equation
1

for the fixed point b0 that can be solved to get the equilibrium amplitude as
b0 =

1
σ
˜±
2

σ
˜ 2 + 4E˜

(5.138)

Now for the stability of these steady state amplitude and phases, we first compute the
Jacobian of the equation (5.131), which is



J=


2α1
ωA b0 sin φ0
0
α1
(E
ωA b2
0
0



α
− ω 1 (E
A0

+ 3b20 ) cos φ0

− b20 ) cos φ0 


α1
2 ) sin φ 
(E
−
3b
0
0
ωA b0

(5.139)

0

At the first fixed point from equation (5.136), the Jacobian becomes



 σω tan φ0
J1 = 
 4α
1 cos φ
0
ω
A0

142

0
−σω tan φ0





(5.140)

and the corresponding eigenvalues are
(5.141)

λ1,2 = ±σω tan φ0

Therefore these fixed points are saddle points i.e. unstable. The Jacobian of the other
fixed points for sin φ0 = 0 is



J2 = 


α
− ω 1 (E
A0

0
α1
(E
ω A b2
0 0

+ 3b20 )(±1)


− b20 )(±1) 



0

(5.142)

and the corresponding eigenvalues are
α2
λ2 = − 2 1 2 (E − b20 )(E + 3b20 ) < 0
ω A b0
0

(5.143)

Hence these fixed points are centers i.e. marginally stable fixed points.
Let h(b, φ) = dφ
db as defined in equation (5.132). We observe that h(b, −φ) = −h(b, φ)
and h(b, 2π + φ) = h(b, φ). Therefore, on b-φ space, the phase plot is symmetric about the
b-axis and the phase plot is periodic in φ with period 2π. Within a period of 2π we see that
σω ω

A
√ 0 . The eigenvalues calculated
the fixed points are at φ0 = 0, φ0 = ±π, and cos φ0 = ±
2α1 E

in equations (5.141) and (5.143) at these fixed points prove that the fixed points are either
centers or saddles and that the saddles are connected in the phase plot that results into
situations where there is phase-drift outside the separatrix and weakly phase-locked inside
the separatrix.

143

Mass Displacement

0
6N

ï5

6U

ï10
ï15
ï20

0

50

100

150

200
250
Mass Number

300

350

400

Figure 5.3 The dynamics of the wave dispersion in an infinite chain with quadratic and cubic
nonlinearities. The nonlinear chain was simulated by injecting a sinusoidal wave, at the 0th
location, of amplitude d0 = 0.8, and frequency ω = 1.0 with the perturbation parameter
= 0.3. The chain consisting of 400 elements are simulated for 400 seconds. To prevent the
reflection of the waves from the other end, we consider the dynamics at t = 320 seconds.
The wave number is computed through a numerical FFT in space at t = 320 seconds. The
DC-bias, B as in equation (5.113), is computed as B = ∆U
∆N The parameter values for
the nonlinear spring elements are α = 1, β = 1.9, and γ = 1.

5.7
5.7.1

Analytical and Numerical Results
Wave Dispersion in Nonlinear Periodic Chain

The quadratic nonlinearity pose great challenge in characterizing the dispersion of waves
analytically in an infinite chain. However, through several perturbation analysis we have
found an analytical dispersion relation. Though these relations present an elementary understanding of the effect of the quadratic nonlinearities in wave dispersion, there is a need
for further analysis to uncover a more accurate representation of the wave propagation in a
chain with quadratic nonlinear elements.

144

0

Solpe (6U / 6N)

ï0.05
ï0.1
ï0.15
ï0.2
ï0.25
0

0.2

0.4

0.6

0.8
1
1.2
Frequency (t)

1.4

1.6

1.8

2

Figure 5.4 The DC-bais generated in the nonlinear chain with both quadratic and cubic
nonlinearities computed from the numerical simulations. The data points are found by com∆U of the DC-bias in the displacement coordinate as shown in Figure 5.3
puting the slope ∆N
at the frequencies simulated between 0 and 2.0. The parameter values are d0 = 0.8, = 0.3,
α = 1, β = 1.9, and γ = 1.
We find the dispersion relation with with and without the DC-bias correction to the
dispersion relation from the amplitude and phase obtained from the second- and the thirdorder MMS analysis of traveling wave solutions. The dispersion profile as well as the group
and phase velocities with respect to the wave number (k) present an over-all understanding
of the effect of the quadratic nonlinearities in the wave propagation in a nonlinear chain.
Since the third-order perturbation analysis did not reveal an analytical expression for
the DC-bias, B, we compute the DC-bias from the numerical simulation results as seen in
Figure 5.3. From the positions of the masses at a particular time step is used to compute
∆U
the slope zjDC = ∂u
∂x ≈ ∆N . Also, zjDC = B. Hence the DC-bias computed from the

transient dynamics of the wave propagation the quadratic nonlinear chain is B = ∆U
∆N .
Figures 5.3, and 5.4 show the numerically simulated transient dynamics of the wave

145

0.35

FFT Amplitude

0.3
0.25
0.2
0.15
0.1
0.05
0
0

0.5

1

1.5
Frequency (t)

2

2.5

3

Figure 5.5 The FFT in space corresponding to the numerical simulation shown in Figure 5.3
at injected frequency ω = 1.0 and amplitude d0 = 0.8 for the parameters = 0.3, α = 1,
β = 1.9, and γ = 1. In addition to the leading order wavenumber, the single wave creates
super and sub-harmonics.
dispersion in a nonlinear chain with quadratic and cubic nonlinearities. The parameters
used in the quadratic nonlinear chain for the numerical simulations and for the analytical
results are h = 1, d0 = 0.8,

= 0.3, α = 1, β = 1.9 and γ = 1. The numerical simulation

is done such that, a chain of 400 masses connected by nonlinear spring elements is injected
by a sinusoidal wave, at the 0th mass, of the form u0 = d0 sin(ωt) and simulated for 400
seconds. The numerical results in the plots are obtained at time t = 320 seconds to avoid
reflections from the other end of the chain to mimic the wave propagation in an infinite chain.
Figure 5.4 clearly shows the frequency dependence DC-bias in the quadratic chain. Hence
further analysis is needed for an accurate representation of the dispersion characteristics.
Figure 5.5 present the spatial-frequency, i.e. the wavenumber k, by taking the fast Fourier
transform at at t = 320 for an injected wave u0 = 0.8 sin(t). This plot shows the dominant
wave-number as well as various other sub and super harmonic waves in the chain. This

146

Frequency (t)

2

1.5

1

0.5

0
0

0.5

1

1.5
Wave Number (k)

2

2.5

3

Figure 5.6 This is a comparison of the wave dispersion characteristics in linear and nonlinear
chains when a sinusoidal wave of amplitude d0 = 0.8 is propagated down a chain connected
by nonlinear spring elements. Here the x-axis represents the wave number (k) and the y-axis
represents the frequency ω which is a function of both the wave number k and the amplitude
of oscillation d0 . The thick solid (blue) curve corresponds to the quadratic nonlinear case
where α = 1, β = 1.9, and γ = 1. The dashed (red) curve corresponds to the cubic nonlinear
case where α = 1, β = 0, and γ = 1. The thin solid (black) curve corresponds to the
linear case where α = 1, β = 0, and γ = 0. The cross (green +) data points represent
the solution when the DC-bias is taken into consideration, and the triangular (red ∆) data
points correspond to the numerical simulation results.
means that when a single wave of particular frequency is injected in an infinite nonlinear
chain, various sub and super harmonic waves are generated in the chain due to the quadratic
and cubic nonlinearities. The quadratic nonlinearity seem to have developed sub and superharmonic waves at 2k and k/2 spatial-frequencies.
Figure 5.6 present various dispersion trends in an infinite chain with both linear and
nonlinear elements with the inclusion of quadratic and cubic terms predicted by the second
and third-order perturbation analysis. To see a significant trend, the value of

is set to 0.3.

Along with the analytical results, the numerical simulation results validating the theoretical
prediction is presented in the dispersion plot. In the low frequency region, the numerical
147

1.2

Phase Velocity ( t/k )

1.1
1
0.9
0.8
0.7
0

0.5

1

1.5
2
Wave Number (k)

2.5

3

Figure 5.7 The wave speed or the phase velocity, c, shown with respect to the wave number,
k, at an amplitude d0 = 0.8. The thick solid (blue) curve corresponds to the quadratic
nonlinear case where α = 1, β = 1.9, and γ = 1. The dashed (black) curve corresponds
to the cubic nonlinear case where α = 1, β = 0, and γ = 1. The thin solid (black) curve
corresponds to the linear case where α = 1, β = 0, and γ = 0. The cross (green +)
data points represent the analytical solution with the DC-bias obtained from the numerical
simulation results.
simulation results are shown to be closer match to the analytical dispersion result with
DC-bias as seen in equation (5.113) as compared to the analytical result without the DCbias i.e. with B = 0. For the given parameter values, both the cubic and the cubic plus
the quadratic nonlinearities shift the temporal cut-off frequency into a higher level as the
spatial frequency increases as compared to the linear chain. The dispersive waves with
the quadratic nonlinearity develop longer wave lengths compared to the linear chain at a
particular frequency ω.
The phase velocity or the wave speed c and the group velocity vg are functions of the
wave number k and the wave amplitude d0 and can be found from the following relation
between the frequency and wave number based on equation (5.113):

148

Group Velocity (dt/dk)

1.2
1
0.8
0.6
0.4
0.2
0
0

0.5

1

1.5
Wave Number (k)

2

2.5

3

Figure 5.8 The wave number, k, dependent group velocity, vg , at an amplitude d0 = 0.8
shows the wave velocities in the linear and nonlinear chains. TThe thick solid (blue) curve
corresponds to the quadratic nonlinear case where α = 1, β = 1.9, and γ = 1. The dashed
(black) curve corresponds to the cubic nonlinear case where α = 1, β = 0, and γ = 1. The
thin solid (black) curve corresponds to the linear case where α = 1, β = 0, and γ = 0. The
cross (green +) data points represent the analytical solution with the DC-bias obtained from
the numerical simulation results.

ω(k, d0 )
k
dω(k, d0 )
vg (k, d0 ) =
dk
c (k, d0 ) =

(5.144)

Figure 5.7 shows that, at fixed amplitude, the phase velocity is higher for both nonlinear
cases at larger wave numbers as compared to the linear chain. At lower wave number
it is seen that the wave speeds in the cubic chain have smaller deviation from the linear
chain. However, with the introduction of quadratic nonlinearity, the overall wave speed
has increased across all wave numbers in the positive Brillouin zone. Taking DC-bias into
account, we see that the phase velocities for the quadratic nonlinear chain has a deviation
from the predicted results without the DC-bias.
At fixed amplitude d0 = 0.8, the group velocity, shown in Figure 5.8, is found to be higher
149

at lower spatial frequencies in the quadratic nonlinear case as compared to the linear and cubic cases. The cubic nonlinear chain is however seen to have higher group velocities at larger
wave numbers as compared to the linear and quadratic cases. The third-order perturbation
analysis, with the DC-bias show significant deviance from the second-order perturbation results. However, more in-depth analysis is required in order to obtain analytical expressions
for the DC-bias developed
to+ eps
thesigMu))
quadratic
nonlinearity.
x ’ = − sigOm − (1 due
− cos(mu
(E − 3 y2) cos(x)/(y
sqrt(2mu
(1 =− 0.5
cos(mu))))
sigOm = 1
y ’ = − (1 − cos(mu + eps sigMu)) (E − y2) sin(x)/sqrt(2 (1 − cos(mu)))

sigMu = 1.5232

eps = 0.1
E=2

1.5

Amplitude (b) (b)
Amplitude

1

0.5

0

−0.5

−1

−1.5
−5

−4

−3

−2

−1

0

1

Phase (Phi)
Phase
(ϕ)

2

3

4

5

Figure 5.9 The phase plot of the slow scale dynamics corresponding to equation (5.131) of
a multi-wave propagation in an infinite nonlinear chain with weak quadratic nonlinearity.
The x-axis represents the phase φ and the y-axis represents the amplitude b. The figure
corresponds to wave number µ = 0.5, = 0.1, E = 2, and σω = 1, and β = α = 1.

5.7.2

Wave-Wave Interactions in Periodic Nonlinear Chain

We can confer from this analysis and from the phase plot in Figure 5.9 that near the centers
the waves behave like a standing wave with a change of shape of the peak amplitudes due
150

Waveïwave interactions phase plot
Amplitude (b)

1.5
1
0.5
0
ï0.5
ï1
ï1.5
ï6

ï4

ï2

0

2

4

6

Phase (Phi)

Waveïwave interactions in traveling waveform
Displacement

2
1
0
ï1
ï2
0

10

20

30

40

50

60

Location in medium

Figure 5.10 This plot shows the phase plot corresponding to equation (5.131) and a snapshot of the time evolution of wave propagation in a chain. The blue dots on the top figure
represents the fixed points and the green lines represent the phase trajectories for initial
conditions other than the fixed points. The time evolution snap-shot on the bottom plot
correspond to a point on the thick blue-green curve on the top figure. The figures correspond
to wave number µA = 0.5, = 0.1, E = 2, and β = α = 1, and σω = 1.
to the weak phase-locking and away from the center i.e. outside the separatrix one would
expect a shape-shifting behavior of the traveling wave due to the drifting phase.
For both the plots all the parameters are fixed except for the wave number µ. Parameters
used here are σω = 1, β = 1, α = 1,

= 0.1, and E = 2. Keeping these parameters fixed

and varying the wave number we notice that for µ = 0.5, the centers at φ = 0 for positive b
disappears giving rise to only phase drift situations.
In both the plots in Figure 5.9, the phase diagram is symmetric about the φ axis. For
151

Amplitude (a)

2

1.5

1

0.5

0

0

20

40

60

80

100

120

140

160

180

200

120

140

160

180

200

Time (t)
Amplitude (b)

2

1.5

1

0.5

0

0

20

40

60

80

100

Time (t)
Figure 5.11 Time evolution of slowly varying amplitudes. The initial conditions are b0 =
0.1, φ0 = 0, and the parameter values for this particular plot are µA = 0.5, E = 2, =
0.1, α = 1, β = 1, σω = 1.
µ = 2, there exists two centers for φ = 0 and another two centers for φ = π. In between these
centers we observe the connecting saddles. The phase drifts are seen outside the separatrix .
The equations (5.128) to (5.131) and the solutions in equation (5.135), shows that the
strength of the quadratic nonlinearity affects the rate at which the energy is exchanged.
For different energy states the amplitudes of these waves are defined in different concentric
ellipses for given detuning parameter. Given the detuning parameter, the frequency of one
of the waves, and the strength of the nonlinearity, one can find the corresponding frequency
and wave numbers of the other sub- or the super-harmonic wave.
From the two-dimensional phase plot in Figure 5.9, for a particular energy state (E = 2),
we observe that there exist two centers for both φ = 0 and φ = π. In between these centers
152

Phase (eA)

0
ï0.5
ï1
ï1.5
ï2
ï2.5
ï3
0

20

40

60

80

100

120

140

160

180

200

120

140

160

180

200

120

140

160

180

200

Time (t)
Phase (eB)

8

6

4

2

Relative Phase (q)

0

0

20

40

60

80

100

Time (t)

0
ï5
ï10
ï15
ï20
ï25
ï30
ï35
0

20

40

60

80

100

Time (t)
Figure 5.12 Time evolution of slowly varying phases. The initial conditions are b0 = 0.1, φ0 =
0, and the parameter values for this particular plot are µA = 0.5, E = 2, = 0.1, α = 1, β =
1, σω = 1.
there are connecting saddles. The phase drifts are seen outside the separatrix. Near the
centers the waves have nearly fixed shape, but fluctuating due to the weak phase-locking.
Outside the separatrix one would expect a shape-shifting behavior of the traveling wave due
to the drifting phase.
The interactions (i.e. a, b, φ) are periodic or whirling, such that the combined propagating
waves are quasi-periodic with both phase-drifts and weakly phase-locking behavior.

153

Amplitude (a)

2

1.5

1

0.5

0

0

20

40

60

80

0

Amplitude (b)

100

120

140

160

180

200

120

140

160

180

200

Time (t)

ï0.5

ï1

ï1.5

ï2
0

20

40

60

80

100

Time (t)
Figure 5.13 Time evolution of slowly varying amplitudes. The initial conditions are b0 =
−0.1, φ0 = 0, and the parameter values for this particular plot are µA = 0.5, E = 2, =
0.1, α = 1, β = 1, σω = 1.

5.8
5.8.1

Summary
Effects of Quadratic Nonlinearity in Single Wave Dispersion

In this chapter we presented an extensive perturbation analysis for single wave propagation
in one dimensional periodic chain with quadratic and cubic nonlinearities. Due to the development of a DC-bias, the quadratic nonlinearity pose a challenge while solving for wave
dispersion characteristics in the chain. Through several different approaches using second
and third-order multiple scales analysis we have developed a method to quantify the DC-bias
through analytical solutions and numerical simulation results. We performed three different
models for the wave analysis. First, we solved for traveling waves with no DC-bias, which
154

Phase (eA)

0

ï1

ï2

ï3

ï4

0

20

40

60

80

0

Phase (eB)

100

120

140

160

180

200

120

140

160

180

200

120

140

160

180

200

Time (t)

ï5
ï10
ï15
ï20
ï25
ï30

Relative Phase (q)

0

20

40

60

80

100

Time (t)
1
0.5
0
ï0.5
ï1
0

20

40

60

80

100

Time (t)
Figure 5.14 Time evolution of slowly varying phases. The initial conditions are b0 =
−0.1, φ0 = 0, and the parameter values for this particular plot are µA = 0.5, E = 2, =
0.1, α = 1, β = 1, σω = 1.
resulted in the dispersion characteristics and the quadratic effects. Though the major contribution of the quadratic nonlinearity was discovered in this method, we found deviation in
results as compared to the numerical simulation.
Through the numerical simulation, we found the DC-bias developed in the spatial coordinate when the wave propagates down the chain. This developed a slight deviation in the
analytical dispersion relations. To resolve this and include the DC-bias effect on the wave
dispersion, we first introduce a DC term in the first order correction in the multiple scales
analysis. However, this resulted in a erroneous dispersion relation since the analytical result
still showed DC-bias effect in the absence of the quadratic nonlinearity.
155

Finally we introduced the DC-bias in order of , that resulted in a more accurate representation of the dispersion characteristics that takes the DC-bias from the numerical simulation
into account. We compute the DC term from the numerical simulation and use that in
the analytical solutions. Though the frequency corrections obtain from the third-order perturbation analysis account for the DC-bias, the results are not complete. The numerical
expression for the DC-bias is required in order to find an accurate dispersion characteristics
in the quadratic chain. Even higher-order perturbation analysis might be necessary to obtain
an analytical expression for the frequency dependent DC-bias.

5.8.2

Effects of Quadratic Nonlinearity in Wave-Wave Interactions

Here we have presented a detailed study of interactions of two traveling waves in an infinite
nonlinear periodic chain with weak quadratic nonlinear elements. With first-order multiplescales analysis, we solved for slowly varying amplitude and phase relations as well as the
fixed points and the stability of the wave propagation. This analysis uncovered the energy
exchange between the two propagating waves through the chain with drifting phase.
We observed that when two waves are injected in an infinite chain with weak quadratic
nonlinear elements, they create phase drift and weak phase locking while the energy exchange
between the two waves form an ellipse in amplitude space in 2:1 resonance condition. At
steady state in the other hand, the traveling wave form has fixed shape without any phase
drift. There are centers for both φ = 0 and φ = nπ. In between these centers there are
connecting saddles. The phase drifts are seen outside the separatrix. Near the centers the
waves have nearly fixed shape, but fluctuating due to the weak phase-locking. Outside the

156

separatrix we observed shape-shifting behavior of the traveling wave due to the drifting
phase. Depending on the initial conditions, the interaction of the two waves will either form
fixed shaped traveling wave-fronts or wave-fronts with continuous phase-drift.
In case of three-wave interactions when the frequencies and the wave numbers are in a
1:2:4 ratio, we would see that the amplitudes of these three waves lie on a three-dimensional
ellipsoid corresponding to a particular energy state. With second-order perturbation analysis,
which in difficult to solve analytically at this point, would reveal the effects of both the
quadratic and cubic nonlinearities.

157

Chapter 6
Conclusions and Future Work

6.1
6.1.1

Summary and Conclusion
Bifurcations and Transient Chaos in Twinkling Oscillators

A complete analytical solution for the equilibria of the symmetric SDOF and 2-DOF twinkling oscillators has been presented. Two unique bifurcation types are discovered for the
2-DOF twinkling oscillator. The spring mass twinkler consists of nonlinear springs with cubic spring forces, and a quasistatic pull at the end spring. We observed complicated highly
degenerate bifurcations: star bifurcations [71, 73] and an eclipse bifurcation [72, 74]. We
have shown the degree of degeneracy in terms of codimension for the symmetric system and
different symmetry breaking cases for both the star and eclipse bifurcations, which unfolds
into saddle-node bifurcations with a general perturbation.
Many different equilibrium configurations can coexist for a simple 2-DOF twinkler. Equilibria of this type of twinkler may have differing energy states. The global picture for the
energy provides insight into the optimum harvestable energy in the pull parameter region.
Predicting the final energy of the system requires analysis of transients, which are influenced
by the structure of both stable and unstable equilibria. This study provides insight into the
dynamics of simple 2-DOF twinkling oscillators, and hence is a building block for further
analysis of more complicated oscillators.

158

We have shown the chaotic nature of the system with fractal basins of the energy levels
in the twinkler in the snap-through regime. With parametric variation, we show regions in
the parametric where it is easy to determine the associated energy level, as well as regions
where it is difficult to predict the associated energy state of the system. In other words in the
parametric space, we have shown the coexistence of non-chaotic, chaotic regions, that gives
us better understanding of the system for various initial conditions and different masses.
The transient chaos study through fractals confirms our suspicion about the chaoticnature and the rich dynamics of the snap-through structures. However, as science progress
and new and better analytical methods are developed, one would be able to find regions
in the parametric space to avoid chaos in an experimental study. In the other hand, the
chaotic, however stable, nature of the simple 2-DOF twinkling oscillator can be used to our
advantage in maximizing power output of a snap-through energy harvester, or minimizing
the vibration isolation and absorption time for industrial use.

6.1.2

Experimental Nonlinear Energy Harvesting

We have built a novel experiment for converting low-frequency input into high-frequency
oscillations and harvest energy by taking advantage of the high-frequency oscillations. The
nonlinear springs consisted of attracting magnets and linear springs. The experimental energy harvesting has been done for both the SDOF and 2-DOF oscillators. We have developed
two mathematical models to compare the analytical results into the experimental results.
The second model has shown to be a more accurate representation of the experimental set-up.
In conclusion, we have shown experimentally that with a 0.1 Hz input displacement
oscillation, results in an frequency up-conversion to 2.5 Hz. This is in agreement with the
numerical simulation results. There are also other super-harmonics present in the solution
159

at frequencies 5 Hz, 7.5 Hz, and 10 Hz in the FFT of the voltage output. These super
harmonics are possibly the results of the unsymmetric spring forces. This system can be
studied using perturbation analysis for analytical study of the energy harvester. This is an
interesting problem to be investigated in the future.

6.1.3

Traveling Waves in Nonlinear Oscillators

A detailed study of the perturbation analysis for wave dispersion in one-dimensional, discrete,
nonlinear periodic structures has been presented. The unit cells are considered featuring a
sequence of masses connected springs that has both the quadratic and cubic nonlinearities
along with the linear stiffness. The traveling wave behavior in the chain is studied for lowamplitude oscillations. We adapted an approximate closed-form, second-order dispersion
relations that capture the effects of quadratic and cubic nonlinearities on harmonic wave
propagation [89], which in turn highlights the effects of the quadratic nonlinearity. These
relationships document amplitude-dependent behavior to include tunable dispersion curves
and cutoff frequencies, which shift with wave amplitude.
The quadratic nonlinearities is seen to have effected the dispersion in the mid-range
spatial frequencies on the dispersion characteristics. A comparison between the quadratic
and the cubic cases show that at a fixed wave amplitude, the quadratic terms lead to much
higher phase and group velocities for lower wave numbers. At higher wave numbers, however,
the group velocities at a fixed amplitude for the quadratic case are smaller as compared to
the cubic case. With spatial frequency fixed, both the wave speed and the group velocity
increase with increase in wave amplitude for the quadratic and cubic case. This analysis
uncovered the effects of quadratic and cubic nonlinearities in the dispersion relationships as
well as wave amplitude-dependent frequency, wave speed, and group velocities.
160

This example demonstrate the manner in which nonlinearities in periodic systems may
be exploited to achieve amplitude-dependent dispersion properties for the design of tunable
acoustic devices. Follow-on work could include the study of stronger wave amplitudes in
snap-through twinkling chains, in which multiple sub- and super-harmonic frequencies can
be expected. Additionally, the same techniques presented could be used to study the effect
of nonlinearity on wave propagation in two- and three-dimensional periodic lattices.

6.2
6.2.1

Ongoing Work
Exploratory Study of Wave Propagation in Snap-Through
Periodic Chains

So far we have discussed the wave propagation behavior in one-dimensional chains with
weakly nonlinear elements of quadratic and cubic nonlinearities. These wave propagations
have been studied with these nonlinear elements when the oscillations of each element is
restricted to be about one stable equilibrium, i.e. small amplitude oscillations.
In this chapter, we wish to look into the wave propagation behavior in a chain of mass
and spring elements where each spring element exhibits snap-through. Though the research
on wave propagation in periodic chains has gained significant momentum in the past few
years, the wave propagation behavior in nonlinear-snap-through chains are yet to be studied. Various authors have studied wave dispersion in infinite chains with weakly nonlinear
elements [80, 89, 125, 126], solitons in a quadratic nonlinear chain [135, 136], and strong nonlinearity with a single equilibrium [137,138]. However, one needs more sophisticated analyses
to model mathematically the wave behaviors in periodic chains with snap-through elements.

161

The dynamics of snap-through oscillations have been characterized for their stability and
chaotic nature in a simple 2-DOF twinkling oscillator in the Chapter 3. In Chapter 2, we
have shown a method to compute the amplitudes and frequency of a truncated Fourier series
approximation to periodic solutions of an undamped snap-through Duffing oscillator. Here,
we are motivated by these studies to explore the wave propagation phenomenon in a snapthrough chain. Combining these studies with wave analysis in discrete weakly nonlinear
chains in Chapter 5, we extend the analysis to characterize the dispersion of a periodic
solitary wave in a periodic strongly nonlinear chain consisting of snap-through elements.
From here on we refer to these waves as snap-through waves.

6.2.2

Snap-Through Solitary Wave Equation Formulation

We consider an infinite periodic chain connected by nonlinear snap-through elements each
with two stable equilibria in the potential well. For simplicity, we consider these springs to
be symmetric. I.e. the potential well is symmetric about the y-axis as shown in Figure 6.1.
The main idea of this study is to explore the wave dynamics when each of these elements
exhibit intra-well dynamics in the phase space. As described in Chapter 5, the undamped
equation of motion (EOM) of the j th mass in a one-dimensional nonlinear chain is

d2 u˜j
uj+1 − u˜j ) − (˜
uj − u˜j−1 ) + γ (˜
uj+1 − u˜j )3 − (˜
uj − u˜j−1 )3
m 2 = α (˜
dτ

(6.1)

where u˜j = u˜j (τ ), j = . . . − 2, −1, 0, 1, 2, . . ., is the displacement of the mass element in the
j th location in the chain. For snap-through, each spring element has a characteristic spring
force with two stable equilibria and the corresponding double-well potential shown in the
Figure 6.1.
162

1.0

Force f

j

0.5
0.0
0.5
1.0
1.5

1.0

0.5

0.0

0.5

1.0

1.5

0.5

1.0

1.5

j

Potential V

j

1.0
0.5
0.0
0.5
1.0
1.5

1.0

0.5

0.0
j

Figure 6.1 The force and the potential energy of each snap through spring elements in the
strain coordinates zj where f (zj ) = −αzj + γzj3 , where zj = uj+1 − uj . Dashed (black),
Dot-dashed (blue) and solid (red) curves correspond to (α = 1, γ = 0), (α = 0, γ = 1), and
(α = 1, γ = 1) respectively.
Using nondimensionalization similar to the nondimensionalization shown in Chapter 2,
for negative stiffness snap-through elements, i.e. with α < 0 and γ > 0 in the above equation,
we substitute τ =

m/(−α) t, and u˜j =

(−α)/γ uj , to obtain

d2 uj
= − (uj+1 − uj ) − (uj − uj−1 ) + (uj+1 − uj )3 − (uj − uj−1 )3
2
dt

(6.2)

Assuming a traveling waveform as the solution to the above equation we let
˜
uj = f (˜
xj − c˜t) = f (φ)

163

(6.3)

Noting that xj±1 = xj ± h, we have
uj±1 = f (˜
xj ± h − c˜t) = f (φ˜ ± h)

(6.4)

˜ we get
Applying a Taylor series expansion about φ,
2 2 ˜
3 3 ˜
4 4 ˜
˜
˜ + h ∂f (φ) + h ∂ f (φ) + h ∂ f (φ) + h ∂ f (φ) + · · ·
uj+1 = f (φ˜ + h) = f (φ)
2! ∂ φ˜2
3! ∂ φ˜3
4! ∂ φ˜4
∂ φ˜
2 2 ˜
3 3 ˜
4 4 ˜
˜
˜ − h ∂f (φ) + h ∂ f (φ) − h ∂ f (φ) + h ∂ f (φ) − · · ·
uj−1 = f (φ˜ − h) = f (φ)
2! ∂ φ˜2
3! ∂ φ˜3
4! ∂ φ˜4
∂ φ˜

(6.5)

which is valid for small h. This implies wavelengths that are large relative to the mass
spacing. That is, the approximation is good for “small” wave numbers. Substituting the
above Taylor series expansion into equation (6.2), we get
c˜2 fφ˜φ˜ = −h2 fφ˜φ˜ −

where fφ˜φ···
˜ φ˜ =

˜
∂ n f (φ)
.
n
˜
∂φ

h4
f
+ 3h4 f 2˜ fφ˜φ˜ + O(h6 )
φ
12 φ˜φ˜φ˜φ˜

(6.6)

Now let x˜ = hx, x being the nondimensional spatial coordinate,

n
∂ n , for n = 1, 2, . . .. Hence the
which leads to φ˜ = hφ, c˜ = hc, φ = x − ct, and ∂˜n = h1n ∂φ
n
∂φ

new equation becomes
(1 + c2 )fφφ +

1
f
− 3fφ2 fφφ = 0
12 φφφφ

(6.7)

(1 + c2 )fφφ +

1
f
− ((fφ )3 )φ = 0
12 φφφφ

(6.8)

and further reduces to

which is valid for large wavelengths. Integrating equation (6.8) once with respect to φ results
in
(1 + c2 )fφ +

1
f
− (fφ )3 = constant
12 φφφ

164

(6.9)

In the limiting condition fφ , fφφ , fφφ → 0 as φ → ±∞. Hence the constant vanishes. Letting
g(φ) = fφ , we get
g + 12(1 + c2 )g − 12g 3 = 0

(6.10)

d .
where () = dφ

Using scaling factors similar to those seen in Chapter 2, we substitute φ =
and g =

1
12(1+c2 )

ζ

(1 + c2 ) w into equation (6.10) to obtain the Duffing equation in w(ζ):
w + w − w3 = 0

(6.11)

1.0

Force f

0.5
0.0
0.5
1.0
1.5

1.0

0.5

0.0

0.5

1.0

1.5

1.5

1.0

0.5

0.0

0.5

1.0

1.5

1.0

Potential V

0.5
0.0
0.5
1.0

Figure 6.2 The force and the potential energy in w-space for f (w) = αw − γw3 . Dashed
(black), Dot-dashed (blue) and solid (red) curves correspond to (α = 1, γ = 0), (α = 0,
γ = 1), and (α = 1, γ = 1) respectively.
This scaled equation represents a traveling wave in the continuum limit, and therefore is
165

good for small wave numbers. The solution to this equation is discussed in the next section.

6.2.3

Analytical Wave Solutions

6.2.3.1

Solutions of Snapping Waves Using Harmonic Balance Method

The equation above is of the form of a simple Duffing equation with positive linear and
negative cubic nonlinear term. The force equivalent term in equation (6.11) is f (w) = w−w3
2
4
and the potential is V (w) = w2 − w4 , and is plotted versus w in Figure 6.2. For a bounded

solution we must have −1 < w < 1.
The Duffing equation can be solved using the harmonic balance method as discussed
in Chapter 2. Since f (−w) = −f (w) and w(ζ + π) = −w, the harmonic balance solution
contains only odd harmonics. The harmonic solution to equation (6.11), with up to three
terms, based at initial conditions w(0) = w0 , w (0) = 0, has the form
w(ζ) ∼
= A1 (w0 ) cos(ωζ) + A3 (w0 ) cos(3ωζ) + A5 (w0 ) cos(5ωζ)
where ζ =

(6.12)

12(1 + c2 )(x − ct). Substituting equation (6.12) into equation (6.11), and

balancing harmonics, we get
3
1 − ω 2 − A21 1 + k52 + k32 (2 + k5 ) + k3 (1 + 2k5 ) = 0
4
1
1 − 9ω 2 k3 − A21 1 + 3k33 + 3k5 + 6k3 (1 + k5 + k52 ) = 0
4
3
1 − 25ω 2 k5 − A21 k3 + k32 (1 + 2k5 ) + k5 (2 + k52 ) = 0
4

(6.13)

Furthermore, the harmonics relate to w0 as
A1 + A3 + A5 = w0

166

(6.14)

Amplitude Ratios k3 , k5

0.05
k5

0.00

k3

0.05
0.10
0.15
0.0

0.2

0.4
0.6
Initial Condition w0

0.8

1.0

Nonlinear Frequency Ω

1.0
0.8
0.6
0.4
0.2
0.0
0.0

0.2

0.4
0.6
Initial Condition w0

0.8

1.0

Figure 6.3 Solution of the Duffing equation (6.11) for frequency ω and amplitude ratios with
respect to initial condition w0
In equation (6.13), k3 and k5 , defined as
A3
= k3 ,
A1

A5
= k5
A1

(6.15)

are the amplitude ratios of the third harmonic to that of the first harmonic, and the fifth
harmonic to that of the first harmonic. Then, we use the same algorithm that we used in
Chapter 2 to solve for amplitudes and frequencies, to get

167

A21 =
ω2

=

32k3
3
+ 51k3 − 3k5 − 6k3 k5 + 54k32 k5 + 27k33 k5 + 48k3 k52
−1 − 3k3 + 3k32 + 3k33 − 3k5 − 6k3 k5 + 6k32 k5 + 3k33 k5
−1 + 21k3 + 27k32 + 51k33 − 3k5 − 6k3 k5 + 54k32 k5 + 27k33 k5 + 48k3 k52
−1 + 21k3 + 27k32

(6.16)

Plugging the expressions from equation (6.16) into the third equation in equation-set (6.13)
we solve for k3 in terms of k5 . An assumed value of k5 will give us a possible value of k3 ,
and this in-turn can be used to get values of A21 and ω 2 . So plugging the values of A1 , k3
and k5 back into A1 (1 + k3 + k5 ) = w0 from equations (6.14) and (6.15), we find the value of
w0 . In this way, choosing k5 instead of the value of w0 as the given parameter value makes
the computation more efficient. The main difference in the solutions here as compared to
the solutions in Chapter 2 is that the initial condition is constrained as −1 < w0 < 1.

6.2.3.2

Dispersive Solitons and their Space-Time Modulation

The amplitudes of the harmonics are dependent on the initial condition w0 at t = 0, where
f
w= √ φ

1+c2

x0

. Also since φ = (x − ct), then at t = 0, φ0 = x0 , hence ζ0 = φ0

12(1 + c2 ) =

12(1 + c2 ).
The approximate solution fφ is
3

fφ (φ) =

1 + c2

A2i−1 (w0 ) cos (2i − 1)[ω(w0 )] 12(1 + c2 )φ

(6.17)

i=1

Integrating the above equation we get

3

1
A2i−1 (w0 )
f (φ) = √
sin (2i − 1)[ω(w0 )] 12(1 + c2 )φ + k1
12 ω(w0 ) i=1 2i − 1

(6.18)

We ignore the constant term without loss of generality. Since ωζ = [ω(w0 )] 12(1 + c2 )(x−
168

Numerical

Ζ

0.5

Amplitude

1.0

0.0

3 term HB

0.5
1.0

w0 0.9714, Ω w0

0

10

0.4536, Ζ

12 1

20
30
Space Time Variation Ζ

c2 x ct

40

Numerical

0.6

50
3 term HB

0.4
'Ζ

0.2
0.0
0.2
0.4
0.6

Ζ

w0 0.9714

1.0

0.5

0.0
Ζ

12 1

0.5

c2 x ct

1.0

Figure 6.4 The top figure shows the harmonic balance solution of the Duffing equation (6.11)
based on equation (6.12) when the initial condition w0 is very close to the separatrix. The
bottom figure is the phase portrait of the numerical solution compared to that of the threeterm harmonic balance solution, where w (ζ) = ∂w
∂ζ . These solutions are valid for any wave
speed, c, since the equation 6.11 is independent of the wave speed c.
ct) is in the form of Kx − Ωt, then K = [ω(w0 )] 12(1 + c2 ), and Ω = [ω(w0 )] c 12(1 + c2 ).
We find the dispersion relation by writing the wave speed, c, in terms of wave number K,
i.e. c =

K2
12[ω(w0 )]2

− 1. Hence the dispersion relation is
Ω(K) = K

K2
−1
12[ω(w0 )]2

(6.19)

In the snap-through chain, for given w0 , one can find the values of A1 (w0 ), A3 (w0 ),
A5 (w0 ), and ω(w0 ) from Tables B.1 and B.2 in Appendix B.
169

Since, w0 = w(ζ0 ) =

w(x0

12(1 + c2 )), for a particular value of ω(w0 ) from the tables, one can obtain a partic-

ular dispersion relation specific to w0 . Note that, for an analytical solution, we must have
√
K > 12 [ω(w0 )] for real wave speed, c.

6.2.3.3

Comparison with a Linear Chain

In a linear chain with α > 0, we let τ =

mt
α

and u˜j = uj , x˜ = hx where x is the nondi-

mensional spatial coordinate, which leads to φ˜ = hφ, c˜ = hc, φ = x − ct in equation (6.2),
to obtain

d2 uj
= (uj+1 − uj ) − (uj − uj−1 )
dt2

(6.20)

Letting uj = fl (xj − ct), and gl = fl , the equations (6.10) and (6.11) in case of a linear
chain become
gl + 12(1 − c2 )gl = 0

(6.21)

wl + wl = 0

(6.22)

and

where wl = gl , and φ =

ζl
12(1−c2 )

. Here we note that the wave speed c must be between 0

and 1. We also note that equations (6.21) and (6.22) are in the continuum limit, and therefore
are valid for wavelengths that are large relative to the mass spacing h, or equivalently, for
small wave numbers.
The solution of equation (6.22) is
12(1 − c2 )φ

gl (φ) = wl (0) cos

(6.23)

and therefore
fl (φ) =

wl (0)
12(1 − c2 )

sin

170

12(1 − c2 )φ

(6.24)

Using the technique similar to that of last section, the dispersion relation for the linear
chain would then be
Ωl (Kl ) = Kl

1−

Kl2
12

(6.25)

which is valid for small wave numbers, Kl .
√
The wave number Kl in expression (6.25) has a limit of 12. However, the frequency
√
Ωl = c 12(1 − c2 ) also has a limit of Ωlmax = 3 corresponding to wave speed c = √1 . So,
2
√
the new limit on the wave number, Kl , corresponding to Ωlmax , is Klmax = 6. Obviously
the minimum values in the linear chian are Ωlmin = 0 and Klmin = 0 corresponding to wave
speed c = 0.

6.2.4

Summary and Questions

Further in-depth analysis is needed in order to find various constrains on the wavenumber
and frequency limits. Also it is important to connect the analysis to the physical system and
check for ambiguity and whether or not the snap-through criterion of each element is satisfied.
In a discrete chain, the wave length is constrained by λmin = 2, i.e. the shortest wave has
nodes on the consecutive mass elements that are separated by unit distances. Hence, the
maximum wave number allowable is Kmax = λ 2π = π. However, this wavelength constraint
min
is applicable at a steady state wave-propagation in a linear or nonlinear chain consisting of
elements that has only one steady-state equilibrium. Since in the snap-through chain each
element has two stable equilibria and can snap-through to either equilibrium, the minimum
allowable wavelength can be much smaller and hence allowing higher wave number solutions
to exist. This should be studied more carefully by investigating the spring characteristics
in detail such as its undeformed natural length, the allowable compression of the nonlinear
bi-stable spring etc..
171

x1

x2

xn

x3

y

(a)

S

fm
S

h

N
N

Spring Force f s

0.3

2fmx

fm

0.2
0.1
0.0
0.1
0.2
0.3
2

(b)

1
0
1
Spring Deformation s

2

(c)

Figure 6.5 (a) A possible arrangement of masses with magnets and linear spring to produce
a chain of bistable elements. The masses consist of magnets which are arranged in a way
that they repel each other. Each mass consist of 3 magnets (filled circles), two on the right
(sleeve) and one on the left (tongue) except for the first and the last mass. Here y is the
quasistatic pull distance of the end spring. (b) The repelling forces fm between the magnets
on the tongue and the sleeves is shown. S and N represent the south and the north pole of
the magnets respectively, h is the separation distance between the two sleeves, and 2fmx is
the resultant magnetic force in the direction of motion of the tongue and sleeves. (c) Total
nonlinear magnetic spring force (solid curve) that can be achieved from combining the forces
due to the magnets (dashed curve) and the linear spring (dotted line).

6.3

Future Works

In this dissertation we have looked at various aspects of the snap-through nonlinearities and
it’s usage in energy harvesting, wave propagation and system dynamics analysis. Several
potential future research topics emerged during our analytical, numerical, and experimental
investigations of the snap-through oscillators. This section is dedicated for advanced research
ideas. Some of these ideas are currently being investigated by the author of this dissertation.

172

6.3.1

Bandgaps in Weakly Nonlinear Chains with Quadratic and
Cubic Nonlinearities

Wave propagation analysis using perturbation methods revealed the dispersion and wave
speed as well as phase drifting and phase locking behavior of the wave-wave interaction in
the weakly nonlinear chain with both quadratic and cubic nonlinearity. Similar methods can
be used to uncover bandgaps in the nonlinear chain with quadratic and cubic nonlinearities.
In this dissertation we have used monatomic chain i.e.!with one type of mass and one type
of nonlinear springs. However, we expect bandgap behavior in a nonlinear periodic chain
consisting of wither two different mass elements or two different spring elements. These
can be studied for weakly nonlinear chain using similar perturbation techniques used in this
thesis.

6.3.2

Large Amplitude Waves in Snap-Through Twinkling Chains
and its Experimental Investigation

We have developed an analytical model for wave propagation behavior in a nonlinear periodic chain where each element has snap-through characteristics. Using harmonic balance
approximation, we have shown the dispersive behavior of the solitary waves in a snap-through
chain. It would be interesting to build experiments to study wave propagation behavior in
a snap-through chain. To show the qualitative behavior it is possible to use finite number
of elements in an experimental investigation. Several arrangements possible for this type of
experiments. For example, one can use magnets and linear springs in combination, as shown
in Figure 6.5 to form the bistable snap-through elements. Another configuration can be using
only linear springs in such a fashion that it forms a snap-through elements [139]. Bandgap
173

behavior can be studied with periodic chains consisting of two different mass elements, or
two different nonlinear spring elements.

6.3.3

Inverse System Identification and Perturbation Analysis of
the Experimental TEG

When designing experimental snap-through oscillators, it is important to know the rough
estimation of the parameters to be used. The design of an analytical framework before hand
and performing numerical simulations can help achieve desired output. This type of inverse
system identification techniques can help designing an experimental snap-through oscillator
for bifurcations study and energy harvesting purposes.
From the experimental twinkling energy harvesting we observed higher order super harmonics. We suspect this is due to the nonuniform nonlinear spring elements. For example,
the linear spring stiffnesses used in the SDOF experiment are not identical. An analytical
perturbation analysis can reveal and confirm this phenomenon in the TEGs.

6.3.4

Wave Propagation in Two–Dimensional Nonlinear Snap–
Through Periodic Structures

Two-dimensional periodic structures are of particular interest for their applications in design
of metamaterials and bandgap structures. Because of the chaotic nature of the snap-through
structures, researchers have have difficulty in analyzing snap-through periodic structures.
However, the snap-through wave analysis in Chapter 6.2.1 can be used as a building block
and extended to two-dimensional periodic snap-through structures to study wave propagation
and band-gap behavior.

174

APPENDICES

175

Appendix A
Conditions for a General Quadratic
Curve to be an Ellipse
From [140], the conditions for a general quadratic curve

av12 + 2bv1 u2 + cv22 + 2dv1 + 2f v2 + g = 0

(A.1)

to be an ellipse are ∆ = 0, G > 0, and ∆
I <0
a b d
a b
where ∆ =

b c f

, and I = a + c.

, G=
b c

d f g
The center (v10 , v20 ) of the ellipse is given by

ad − bf
v10 = 2
b − ac

af − bd
v20 = 2
b − ac

(A.2)

and the semi axis lengths are

r1 =

r2 =

2 af 2 + cd2 + gb2 − 2bdf − acg
b2 − ac

(a − c)2 + 4b2 − (a + c)

2 af 2 + cd2 + gb2 − 2bdf − acg
b2 − ac

−

(a − c)2 + 4b2 − (a + c)

176

(A.3)

In our case in Section 3.4.3 of Chapter 3, d = f = 0 results in ∆ = (ac − b2 )g, G = (ac − b2 )
and I = (a + c). We also have a = c = 1, b = − 21 , and g = −1, which gives us ∆ = − 49 = 0,
9
0 0
G = 43 > 0, I = 2, and ∆
I = − 8 < 0. Hence we have an ellipse with center at (v1 , v2 ) = (0, 0)
√
√
and semi-axis lengths r1 = 2, and r2 = √2 .
3

177

Appendix B
Harmonic Balance Solutions of the
Snap–Through Wave Equation
Table B.1 Harmonic balance solutions look-up table for snap-through waves corresponding
to equation 6.11 for solutions closer to the separatrix.
k5
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.010
0.011
0.012
0.013
0.014
0.015
0.016
0.017
0.018
0.019
0.020
0.021
0.022
0.023
0.024
0.025
0.026
0.027
0.028
0.029
0.030

k3
-0.053291
-0.061274
-0.068249
-0.074508
-0.080225
-0.085513
-0.090449
-0.095091
-0.099483
-0.103656
-0.107639
-0.111452
-0.115113
-0.118638
-0.122038
-0.125324
-0.128506
-0.131592
-0.134588
-0.137501
-0.140336
-0.143099
-0.145793
-0.148423
-0.150992
-0.153504
-0.155961
-0.158366

A1
0.91034
0.94259
0.96702
0.98658
1.00282
1.01668
1.02873
1.03938
1.04890
1.05749
1.06532
1.07251
1.07914
1.08529
1.09102
1.09639
1.10143
1.10618
1.11066
1.11491
1.11895
1.12279
1.12646
1.12996
1.13331
1.13652
1.13960
1.14256

A3
-0.048513
-0.057756
-0.065998
-0.073508
-0.080452
-0.086939
-0.093048
-0.098836
-0.104347
-0.109616
-0.114670
-0.119533
-0.124223
-0.128756
-0.133146
-0.137404
-0.141540
-0.145563
-0.149482
-0.153301
-0.157029
-0.160670
-0.164230
-0.167712
-0.171120
-0.174460
-0.177733
-0.180942

178

A5
0.0027310
0.0037703
0.0048350
0.0059194
0.0070198
0.0081334
0.0092586
0.0103938
0.0115379
0.0126899
0.0138492
0.0150151
0.0161871
0.0173646
0.0185474
0.0197350
0.0209271
0.0221235
0.0233239
0.0245281
0.0257359
0.0269470
0.0281614
0.0293789
0.0305993
0.0318224
0.0330483
0.0342767

w0
0.864559
0.888600
0.905852
0.918987
0.929391
0.937874
0.944943
0.950935
0.956087
0.960568
0.964503
0.967990
0.971102
0.973898
0.976424
0.978719
0.980814
0.982736
0.984505
0.986140
0.987657
0.989069
0.990388
0.991624
0.992784
0.993878
0.994911
0.995890

ω(w0 )
0.638932
0.608088
0.583456
0.562906
0.545251
0.529756
0.515938
0.503458
0.492071
0.481594
0.471885
0.462835
0.454355
0.446373
0.438830
0.431677
0.424872
0.418379
0.412170
0.406217
0.400498
0.394994
0.389686
0.384559
0.379600
0.374796
0.370136
0.365610

Table B.2 Harmonic balance solutions look-up table for snap-through waves corresponding
to equation 6.11 close to the steady-state equilibrium.
k5
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0.0007
0.0008
0.0009
0.0010
0.0011
0.0012
0.0013
0.0014
0.0015
0.0016
0.0017
0.0018
0.0019
0.0020
0.0021
0.0022
0.0023
0.0024
0.0025
0.0026
0.0027
0.0028
0.0029
0.0030

k3
-0.0099501
-0.0140425
-0.0171712
-0.0198010
-0.0221121
-0.0241967
-0.0261098
-0.0278870
-0.0295533
-0.0311266
-0.0326207
-0.0340460
-0.0354112
-0.0367229
-0.0379868
-0.0392076
-0.0403894
-0.0415355
-0.0426488
-0.0437319
-0.0447871
-0.0458163
-0.0468212
-0.0478035
-0.0487645
-0.0497056
-0.0506278
-0.0515321
-0.0524196
-0.0532911

A1
0.513693
0.590138
0.637095
0.671129
0.697812
0.719731
0.738308
0.754410
0.768603
0.781281
0.792727
0.803152
0.812717
0.821548
0.829745
0.837390
0.844549
0.851278
0.857624
0.863625
0.869316
0.874726
0.879880
0.884799
0.889504
0.894011
0.898335
0.902491
0.906489
0.910341

A3
-0.0051113
-0.0082870
-0.0109397
-0.0132890
-0.0154301
-0.0174151
-0.0192771
-0.0210383
-0.0227147
-0.0243186
-0.0258593
-0.0273441
-0.0287792
-0.0301696
-0.0315193
-0.0328321
-0.0341108
-0.0353582
-0.0365766
-0.0377680
-0.0389342
-0.0400767
-0.0411970
-0.0422965
-0.0433763
-0.0444373
-0.0454807
-0.0465072
-0.0475178
-0.0485131

179

A5
0.00005137
0.00011803
0.00019113
0.00026845
0.00034891
0.00043184
0.00051682
0.00060353
0.00069174
0.00078128
0.00087200
0.00096378
0.00105653
0.00115017
0.00124462
0.00133982
0.00143573
0.00153230
0.00162949
0.00172725
0.00182556
0.00192440
0.00202372
0.00212352
0.00222376
0.00232443
0.00242551
0.00252697
0.00262882
0.00273102

w0
0.508633
0.581969
0.626347
0.658109
0.682731
0.702748
0.719548
0.733975
0.746580
0.757744
0.767740
0.776772
0.784994
0.792528
0.799470
0.805897
0.811874
0.817452
0.822677
0.827585
0.832208
0.836574
0.840706
0.844626
0.848351
0.851898
0.855280
0.858510
0.861600
0.864559

ω(w0 )
0.896672
0.861609
0.837038
0.817691
0.801574
0.787685
0.775440
0.764465
0.754506
0.745379
0.736947
0.729107
0.721776
0.714890
0.708394
0.702246
0.696407
0.690847
0.685540
0.680462
0.675594
0.670918
0.666420
0.662085
0.657903
0.653862
0.649952
0.646166
0.642495
0.638932

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