Energy scavenging and wave manipulation in linear and nonlinear dynamical systems

214 Pages·2017·21.21 MB·English

by  Aditya Nanda

Energy scavenging and wave manipulation in
linear and nonlinear dynamical systems
by
Aditya Nanda
June, 2018
A dissertation submitted to the
Faculty of the Graduate School of the
University at Buffalo, State University of New York
in partial fulfillment of the requirements for the
degree of
Doctor of Philosophy
Department of Mechanical and Aerospace Engineering
Acknowledgements
Firstly, I would like to extend my deepest gratitude to my advisor Dr. M. Amin Karami for being the academic
advisor, a wonderful teacher and mentor that he has been to me. Despite being singularly accomplished, he al­
ways has been patient and willing to listen to my ideas/ thoughts. He encouraged me to push hard and push my
mental boundaries when I would have given up and he showed me that even the mightiest of problems can be
tackled by the right combination of creativity and perseverance. He taught me to break up an obstacle or prob­
lem into tinier and simpler bits that are more easily dealt with rather than taking on the monolithic whole. This
PhD has been a valuable opportunity to learn from him not only the trappings of academic research but also the
what-and-how of living a creative, productive and ambitious life. He is the classic leader who leads by example.
I am deeply grateful to him for sharing his teachings, his passion and knowledge with me.
I would also like to extend my heartfelt gratitude to my committee members for agreeing to review my Disser­
tation and for being a part of my PhD. I thank Dr. Puneet Singla for his help with my research and for being a
mentor and advisor when I first came to UB. I thank Dr. Tarunraj Singh for sharing his teachings, his intuition
and wisdom with me. I thank Dr. Mostafa Nouh for many helpful talks and his feedback that helped me with
my metamaterials research.
This dissertation would not have been possible without so many other people. I would like to thank my parents
Debi Prasanna Nanda and Prabasini Nanda for their love and care throughout my life and for making me into
the person I am today. I would like to thank my sister Ayushi Nanda for her companionship and her laughter.
This work is dedicated to them.
I was lucky to have good labmates whose company was conducive to productivity and creativity. Hooman Ansari
(soon to be Dr.) was a pleasure to share the lab with. Dr. Muath Banihani and Antonio Caliber always had a
good sense of humor. I would like to also thank the undergraduates who worked with me on the validation part
for some projects- particularly Shathushan Sivashangaran, Zeshan Muhamed Khetani, Apurv Sharma and oth­
ers.
Among the buffalo gang, I thank Abhishek Venkateswaran for his company, his laughter and his jokes through­
out these past 5 years in Buffalo. I will cherish our long walk-and-talks that were simultaneously about every­
thing and nothing, were humorous (sometimes darkly so) and personal; all at the same time. I thank Alyssa
Wojciechowski for her company, her laughter and her friendship. Special mention to Alex Kline for hosting the
many camping trips where we "Respected Nature" and for appreciating the Borat jokes. Special thanks also
to Guhan Velmurangan, Kushal Bhola, Dr. Poorva Sandlesh, Kundan Goswami, Dr. Prerna Gera, Dr. Adonis
Pitagoras, Adam Ross and others for being awesome friends and the catalyst for many an evening well spent
these past few years.
This work was partially supported by the National Science Foundation (NSF) under award number CMMI-
1649111.
11
Abstract
This dissertation presents results in two broad, inter-related areas. The first three chapters deal with modeling
and analysis of electro-mechanical vibratory systems and harvesting ambient oscillatory energy. The final three
chapters present results for manipulating elastic and sound waves in periodic lattices.
Chapter 1 presents a novel result concerning the rattleback which is an ellipsoidal toy with some curious proper
ties. When placed on a surface, the rattleback has a preferred direction of spin - if spun in the opposite direction,
it will reverse itself. This phenomenon has been termed spin reversal and the effect is attributed to the asym
metry between the inertia and geometry axes of the top . In this work, results are presented demonstrating that
the rattleback, when placed on a vibrating platform, will start to spontaneously spin (in its preferred direction) .
Based on a simple linearization for small angular velocities, we derive an expression for the resonant frequency.
A simple electromagnetic energy harvester is proposed consisting of a magnetized rattleback surrounded by mag
netic coils is proposed and the possible transduced electric power is evaluated.
Chapter 2 proposes the use of a piezoelectric curved beam implanted inside the human Carotid artery as a power
source for implanted biomedical devices proximate to the head - neural sensors, neural simulators and pacemak
ers. The curved beam is analytically modeled and based on blood pressure data and arterial distension the dis
placements and induced bending strain are calculated. Based on Gauss's law, the direct piezoelectric effect is used
to calculate the electric power generated across a load resistance. A maximum power of 11.5 µW was harvested
using P ZT - 5H in the Carotid artery.
Chapter 3 presents a methodology for the uncertainty analysis of piezoelectric energy harvesters. Extant methods
in the literature for uncertainty propagation involve approximation and/or assume the parameter stochasticity
to be small and single dimensional. We use the Conjugate Unscented Transform (CUT) method which can ac
curately estimate moments of output variables under multidimensional uncertainty with minimum computation.
The common cantilevered energy harvester consisting of steel substrate with piezoelectric patches and a set of re
pulsive magnets (that serve to make the system nonlinear) is considered. The results show that CUT method is
accurate, computationally efficient and can propagate 4 dimensional uncertainty. Also, we found that a naive
and purely deterministic analysis of energy harvester can grossly overestimate the harvested power. This may
have applications in evaluating worst-case scenarios.
Chapter 4 presents an analysis of elastic waves in a novel metamaterial whose stopbands can be tuned by chang
ing the fold angle. The metamaterial consists of repeating units of panels (modeled as beams) and hinges (mod
eled as torsion springs) so that the panels can fold around the hinges. As such, this model can be used to repre
sent solar arrays, hinged folding structures and origami-inspired structures. The Transfer Matrix (TM) method
is used for the analysis. A comprehensive COMSOL R Finite Element Method (FEM) simulation is conducted
for validation. It is shown that the bandgaps in the structure become wider when the fold angle is increased. Band
width increases of as much as 250 % were observed in the bandgaps when the fold angle is changed from O deg to
90 deg .
Chapter 5 extends the analysis of elastic waves to a nonlinear periodic structure. The linear torsion spring in
Chapter 4 is replaced by a hinge with cubic nonlinearity. No closed form solution for nonlinear lattices exists in
the literature. A harmonic balance formulation is derived and the Galerkin projection is used to obtain a set of
algebraic nonlinear equations that are numerically solved to obtain the dispersion relationship. It is shown that
introducing nonlinearity engenders amplitude-dependent dispersion. Another interesting observation is that only
the leading bandgap edge is affected. Hardening nonlinearity causes the bandgap edges to shift to higher frequen
cies and softening nonlinearity engenders the opposite. The trailing bandgap edge is, however, independent of the
nonlinearity. It is further shown that the wavemode corresponding to the trailing bandgap edge corresponds to the
first modeshape of a simply supported beam. Since the moment at the ends of a simply supported beam are zero,
this explains the independence of the trailing bandgap edge with respect to nonlinearity.
Chapter 6 presents an active wave manipulation strategy for building an acoustic diode using a smart fluid. An
acoustic diode permits sound traveling in only one way. Thus, transmission reciprocity is broken. A smart fluid
like Magnetorheological fluid (MRF) is considered and a time-varying magnetic field is acted upon the fluid. It is
well known in the literature that local sound speed in MRF is dependent on the applied magnetic field. This cre
ates a space-time periodic media that has bandgaps for forward and backward traveling waves in different frequen
cies thereby achieving one-way sound propagation. A dispersion relationship for an unsteady and inhomogeneous
fluid is developed. A set of FEM simulations are used to validate the analysis and reasonably good agreement is
noted between the theoretical and numerical band diagrams. A detailed analysis of the effect of viscosity is con
ducted. It is shown that one-way bandgaps formed at a higher frequency are more susceptible to viscous corrup
tion. A real-world implementation of the device may be possible if the viscosity (µ) is less than 3000 Pa-s
Contents
Acknowledgements ii
Abstract iii
List of Figures vii
List of Tables X
Abbreviations xi
1 Energy Harvesting: Spin resonance of rattleback 1
1.1 EQUATIONS OF MOTION . 3
1.2 SPIN RESONANCE .. . .. . 7
1.3 THEORETICAL ANALYSIS . 10
1.3.1 Linearization . . . . . . 13
1.3.2 Validation of Analysis 19
1.3.2.1 Rattleback - I 19
1.3.2.2 Rattleback - II 22
1.4 PROPOSED APPLICATIONS OF SPIN RESONANCE 24
1.5 CONCLUSIONS AND FUTURE RESEARCH 26
2 Energy Harvesting: Arterial blood pressure 28
2.1 INTRODUCTION ... . . . . . 28
2.2 PRELIMINARY DESIGNS . . . 32
2.3 CURVED BEAM HARVESTER 39
2.3.1 Kinematics of motion .. 40
2.3.2 Governing Equation . . . 43
2.3.3 Validation using MAPDL 48
2.4 HARVESTED POWER AND OPTIMIZATION 50
2.4.1 Harvested power in Carotid artery .... 52
2.4.2 Harvested power in Human Aorta . . . . 54
2.4.3 Variations due to pulse rate, harvester width and optimal resistance 54
2.4.4 Variation due to arterial contractility 56
2.5 CONCLUSIONS .... . . . .. . . . . 58
3 Energy Harvesting: Stochastic analysis 60
3.1 INTRODUCTION .......... . 60
3.2 ENERGY HARVESTING SYSTEMS 64
3.2.1 Linear Energy Harvester . . . . 64
3.2.2 Nonlinear Energy Harvesters . 66
3.3 UNCERTAINTY QUANTIFICATION 68
3.4 RESULTS ... . .... . .... . 73
3.4.1 Numerical Preconditioning .. 74
3.4.2 Linear Harvester . .... . . . 74
3.4.3 Monostable Nonlinear Harvester 76
V
Contents Vl
3.4.3.1 4 dimensional uncertainty in Base frequency, acceleration, initial tip deflection
and magnet gap . . . . . . . . . . . . . . . . . 78
3.4.3.2 Sensitivity Analysis and Design considerations . . . . . . . . . . . . . . . . . . 84
3.4.4 Bistable Nonlinear Harvester . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.4.4.1 3 dimensional uncertainty in base frequency, base acceleration and magnet gap 90
3.4.4.2 Sensitivity Analysis and Design Considerations 92
3.5 CONCLUSIONS .... . . . .. . . . .... . .... . . . 98
4 Metamaterials: Foldable structures and bandgap tuning 100
4.1 Introduction .. . .... . .... . . . .. . . . .... . . 100
4.1.1 Solar arrays with tunable vibration characteristics 102
4.1.2 Flexural waves in Origami structures 103
4.2 Geometry and Transfer Matrix Method 106
4.2.1 Zero fold angle (Straight structure) . . 106
4.2.2 Non-zero Fold angle (Zig-zag structure) 108
4.2.3 FRF for finite periodic structures 111
4.3 Validation using FEM 113
4.4 Analytical Results . . . . . . . . . . . . . 115
4.5 Parametric studies . . . . . . . . . . . . . 120
4.5.1 Implications for deployable metamaterials 130
5 Metamaterials: On nonlinear boundary periodicity 133
5.1 Introduction .. . .... . . 133
5.2 Harmonic balance Method .... . . . .. . . . . 136
5.3 Nonlinear Phononic Crystal ... . . . .. . . . . 143
5.3.1 Comparison with Transfer Matrix method 145
5.3.2 Amplitude-dependent dispersion 148
5.4 Conclusions 153
6 Metamaterials: One-way sound propagation 155
6.1 Introduction .. . .... . .... . . . .. . . 155
6.2 Sound propagation in unsteady and inhomogeneous fluid 158
6.3 Validation using FEM . . . 161
6.4 Real-world Implementation 166
6.5 Conclusions .. . .... . . 173
A AppendixA 175
A.1 Computing 4 dimensional CUTS points 175
Bibliography 178
Vl
List of Figures
1.1 RATTLEBACK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 RATTLEBACK WITH FORCES AND AXES 4
1.3 Plot of (a) spin rate (b) roll and pitch angular velocities with time of a simulated rattleback(conservative
model) rotated in the unstable spin direction with initial w = [0 0 - 5frad/s and u = [.02 .02 -
.9996]T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Plot of steady state spin rate(n) of a simulated rattleback(Conservative model) against frequency
of platform oscillation with initial conditions : u = [.02 .02 .9996]T and w = [0 0 of . . . . . . . . 9
1.5 Plot of steady state spin rate(n) of a simulated rattleback(Dissipative model) against frequency of
platform oscillation with initial conditions : u = [.02 .02 .9996f and w = [0 0 of . . . . . . . . . 10
1.6 Plot of Angular velocities against time of a simulated rattleback with initial w = [0 0 0]T rad/s
and u = [.02 .02 - .9996f. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.7 Plot of the scaled acceleration due to gravity and pitch displacement against time with initial w =
[0 0 0]T rad/s and u = [.02 .02 - .9996]T and !base = 19 Hz (Spin resonant frequency for the
rattleback). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.8 Amplitude spectrum of coupled frequency of a simulated rattleback with initial w = [0 0 of rad/s
and u = [.02 .02 - .9996f. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.9 Amplitude spectrum of uncoupled frequency of a simulated rattleback with initial w = [0 0 ofrad/s
and u = [.02 .02 - .9996f. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.10 Amplitude spectrum of coupled frequency of a simulated rattleback with initial w = [0 0 ofrad/s
and u = [.02 .02 - .9996f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.11 Amplitude spectrum of uncoupled frequency of a simulated rattleback with initial w = [0 0 ofrad/s
and u = [.02 .02 - .9996f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.12 Plot of steady state spin rate of a simulated rattleback against frequency of platform oscillation
with initial w = [0 0 o]T rad/s and u = [.02 .02 - .9996]T. Rattleback parameters in Tab. 1.2 24
1.13 PROTOTYPE RATTLEBACK ENERGY HARVESTER 24
1.14 PROTOTYPE RATTLEBACK VIBRATION SENSOR . . . . . . . . . . . . . . . . . . . . . 26
2.1 (a) Cylindrical harvester and (b) Curved beam harvester (c) First mode-shape of uniform, circu-
lar, curved beam . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2 (a) Curved Beam Harvester (b) Harvester inside artery 40
2.3 Initial and final configurations of harvester placed in artery 41
2.4 Circular curved beam(shown in green) placed in a radially deforming hollow cylinder: (a) FEM
model (b) Tangential bending strain(S0 ) compared with Eq 2.17 . . . . . . . . . . . . . . . . . 43
2.5 Model validation: (a) Screenshot of FEM elements in MAPDL and Comparison of analytical and
FEM results for (b) Average power (c) RMS voltage and (d) Induced bending strains on the out-
ermost surface(S0 ) with R1 = 106ohm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.6 Model validation: (a) Screenshot of FEM elements in MAPDL and Comparison of analytical and
FEM results for (b) Average power (c) RMS voltage and (d) Induced bending strains on the out-
ermost surface(S0 ) with R1 = 106ohm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.7 Arterial radius, r (t) and velocity, r (t) for Carotid artery as a function of time. . . . . . . . . . 51
1 1
2.8 Carotid Artery: Time-averaged harvested power as a function of load resistance for (a) PZT-5H
and (b) MFC. The dotted lines represent power curves which violate the pressure constraint, Eqn.
2.24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.9 Aorta: Time-averaged harvested power as a function of load resistance for PZT-5H in the Aorta.
The dotted line represent power curves which violate the pressure constraint , Eqn. 2.24 54
2.10 Carotid Artery : Harvested mean power with PZT-5H with variation in (a) the harvester width b.
(b) Pulse rate and( c) load resistance Rt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Vll
List of Figures vm
2.11 Harvested mean power in Carotid artery with PZT-5H as function of pulse rate. Load resistance,
Rl = 105 ohm and h = 52.7 µm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
0
2.12 Harvested mean power in Carotid artery with PZT-5H as function of load resistance. h = 52.7 µm 57
0
2.13 Carotid artery: Induced voltage variation with change in arterial radial strain. Material= PZT-
5H. Rl = 109 ohm and h = 52. 7 µm 58
0
3.1 (a) Linear Harvester (b) Nonlinear Harvester 65
3.2 (a)Schematic of specially chosen axes according to CUT methodology in 3-D space (b)Comparison
of number of 8th order quadrature points required for different quadrature schemes, as a function
of dimension of random variable. Adapted from [1] with permission . . . . . . . . . . . . . . . . . 69
3.3 Linear Harvester, mean and min-max bounds for Pwr with (a) base frequency(b) base accelera-
tion. Deterministic trends are in blue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.4 Monostable Harvester, 4 dimensional Gaussian uncertainty: probability density function and Monte
Carlo distribution, with 4th order Moment constraints, (a)p(u) and (b)p(Pwr) . . . . . . . . . . . 81
3.5 Monostable Harvester, 4 dimensional Gaussian uncertainty: Probability density function and Monte
Carlo distribution, with 6th order Moment constraints (a) p(u) and (b)p(Pwr) . . . . . . . . . . . 82
3.6 Monostable Harvester, 4 dimensional Gaussian uncertainty: Joint probability density function and
Monte Carlo distribution, with 4th order Moment constraints, (a)Moments from CUTS and (b)
Moments from GHll . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.7 Monostable Nonlinear Harvester, (a) Time series and (b) Phase portrait for Regime I (in blue,
Base acceleration, F0 = .6 m/s2) and Regime II (in red, base acceleration, F0 = .9 m/s2) . . . 84
3.8 Monostable Nonlinear Harvester, mean and min-max bounds for Pwr: (a) base frequency (b) base
acceleration. Deterministic trends are in blue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.9 Monostable Nonlinear Harvester, mean and min-max bounds for Pwr): (a) initial deflection (b)
magnet gap. Deterministic trends are in blue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.10 Bistable Harvester,3 Dimensional Gaussian uncertainty: probability density function and Monte
Carlo distribution, with 6th order Moment constraints, (a)p(Pwr) and (b)p(ud) . . . . . . . . . . 92
3.11 Bistable Harvester, 3 Dimensional Gaussian uncertainty: probability density function and Monte
Carlo distribution with 10th order Moment constraints (c) p(Pwr) and (d)p(ud) . . . . . . . . . 93
3.12 Bistable Harvester: Phase portrait and Poincare map of representative samples from (a) Regime I,
(b) Regime II and (c ) Regime III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.13 Bistable Harvester, Spread of samples from Regimes I, II and III over: (a) base frequency and (b)
Base acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.14 Bistable Harvester, mean and min-max bounds for Pwr: (a) base frequency (b) base acceleration
(c ) magnet gap. Deterministic trends are plotted in blue. . . . . . . . . . 97
4.1 Modeling solar array: 4 consecutive cells with panels and torsional springs 102
4.2 A folded sheet showing creases and panels . . . . . . . . . . . . . . . . . 103
4.3 Zig-Zag structure: Configuration at different fold angles - 1¥ and 1¥ 108
1 2
4.4 Schematic showing longitudinal and transverse displacements in neighboring beams . 109
4.5 Displacement FRF for straight structure: from FEM (in solid red), from TMM (Euler bernoulli
(green dashed) and Timoshenko theory (blue dashed). N= 15 cells. Fold angle, 1¥ = 0 . . . . . 114
4.6 Displacement FRF for ziz-zag structure: from FEM (in solid red), from TMM (Euler bernoulli
(green dashed) and Timoshenko theory (blue dashed). N= 15 cells. Fold angle, 1¥ = 60 deg . . 115
4.7 Bandgaps for the straight structure from transfer matrix method (TMM) compared with analyti-
cal solution(solving Equation 4.32. Simulation parameters in Table 4.1 ) . . . . . . . . . . . . 120
4.8 FRF of foldable structure (N = 30 cells) with various fold angles (superimposed over FRF for
1¥=0)(a)1¥=2deg(b)W=8deg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.9 FRF of foldable structure (N = 30 cells) with various fold angles (superimposed over FRF for
w= 0) (a)w = 66 deg (b) w= 90 deg . . . . . . . . . . . . . . . . . . . . . . . 123
4.10 Evolution of bandgaps with fold angle (a) First Bandgap (b) Second Bandgap 124
4.11 Evolution of bandgaps with fold angle (a) Third Bandgap (b) Fourth Bandgap 125
4.12 Effect of fold angle on Band diagrams: Real(R(µ)) and imaginary (8'(µ)) parts ofµ (a) fold angle,
1¥ = 2 deg and (b) fold angle, 1¥ = 60 deg. Only far-field components are plotted . . . . . . . . . 126
4.13 Bandgaps in the periodic structure with variation in torsional stiffness (kt) . . . . . . . . . . . . 127
4.14 Bandgaps in the periodic structure with variation in (a) Young's Modulus (E) (b) Width (b) (c)
Length (L) of the beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.15 Bandgaps in the periodic structure with variation in (a)thickness of beam (b) length of beam . . 129
Vlll
List of Figures lX
5.1 Nonlinear Phononic crystal comprising repeating nonlinear torsion spring and Euler-Bernoulli
beam (a) Isometric view (b) 2 dimensional view . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.2 Comparison of Harmonic balance and Transfer matrix method (TMM) for linear phononic crystal
(a) real part (R(µ)) (b)imaginary part ('s(µ))of propagation constant. Simulation parameters in
Table 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.3 Dispersion relationship in nonlinear torsional PC: (a) Real part of propagation constant R(µ) (b)
Imaginary part of propagation constant 's(µ). f3 = 10-s . . . . . . . . . . . . . . . . . . . . . 149
5.4 Dispersion relationship in nonlinear torsional PC: Real part of propagation constant R(µ) (a)
First bandgap (b) Leading bandgap edge magnified. Hardening nonlinearity, f3 = 10-4 . . . . 150
5.5 Dispersion relationship in nonlinear torsional PC: Real part of propagation constant R(µ) (a)
First bandgap (b) Leading bandgap edge magnified. Hardening nonlinearity, f3 = 10-4 . . . . 151
5.6 Dispersion relationship in nonlinear torsional PC: (a) Real part of propagation constant R(µ) (b)
Imaginary part of propagation constant 's(µ). Softening nonlinearity, f3 = - 10-4 . . . . . . . . . 152
5.7 Normal wavemodes at the leading and trailing bandgap edges. The wavemode at the trailing edge
isoverlayedonsin(Kx/L)) . .. . . . .... . .... . .... . .... . .... . . . .. . . . . 152
6.1 Arrangement used for FEM simulation: Two slender pipes housing a smart fluid (inhomogenous
and time-varying density and sound speed) with sound waves propagating in opposite directions. 162
6.2 Fourier transform of pressure excitation(P(!1)) used to induce sound waves as shown in Fig. 6.1.
!1 is non-dimensional frequency (Equation 6.12.) . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.3 Case I: Band diagrams for smart fluid with constant sound speed (c = c0) and constant density
(p = p0 ). Simulation parameters in Table 6.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.4 Case II: Band diagrams for spatial ONLY modulation of sound speed and density (Km= 3 rad/s
and Wm = 0). Also, Cm = 0.2co, Pm = 0.2po. Simulation parameters in Table 6.1. . . . . . . . . . 165
6.5 Case III: Band diagrams for spatial AND temporal modulation of sound speed and density (Km=
3 rad/sand Wm= 0.15Kmco)- Also, Cm= 0.2co , Pm= 0.2po. Simulation parameters in Table 6.1. 165
6.6 Schematic of MRF with Linear induction motor mechanism for one-way sound . . . . . . . . . . 168
6.7 Low frequency bandgaps: Acoustic coefficient (Cv defined in Equation 6.14) plotted against fre-
quency for different viscosities (a)µ = 50 Pa-s (b) µ = 5 x 104 P a-s (c) µ = 5 x 105 Pa-s.
Simulation parameters in Table 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.8 High frequency bandgaps: Acoustic coefficient (Cv defined in Equation 6.14) plotted against fre-
quency for different viscosities (a)µ = 50 P a-s (b) µ = 5 x 104 Pa-s (c) µ = 5 x 105 P a-s. All
simulation parameters in Table 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.9 Effect of viscosity on one-way bandgaps: Bandgap attenuation (20log(f30 )) plotted against fluid
viscosity (µ) . Simulation parameters for low frequency bandgaps are in Table 6.1. Simulation pa-
rameters for high frequency bandgaps are in Table 6.2 . . . . . . . . . . . . . . . . . . . . . . . . 173
lX
List of Tables
1.1 SIMULATION PARAMETERS . .. . . . .. . .... . . 7
1.2 SIMULATION PARAMETERS FOR RATTLEBACK -II 22
2.1 Parameters and dimensions of harvester used for simulations 49
2.2 Peak power and critical thickness( her) for various arteries and materials 52
3.1 Descriptions of coefficients. See Equation 3.1 and Equation 3.2 . .... 66
3.2 Descriptions of Parameters 67
3.3 Numerical values of the coefficients used for simulations of linear harvester. See Equation 3.1 75
3.4 Numerical values of the coefficients used for simulations of monostable harvester. See Equation 3.2
77
3.5 Convergence of Monte Carlo runs in capturing fourth order moments. Reference = GHll 79
3.6 Mean square errors in Moment integrals from GH5, GH7, MCS0000 and CUTS. Reference =
GHll . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.7 Numerical values of the coefficients used for simulations of bistable harvester. See Equation 3.2 90
3.8 Convergence of Monte Carlo runs in capturing moments up to fourth order. Reference = GH13 90
3.9 Mean square errors in Moments from different order Gauss-Hermite and CUTS. Reference =
GH13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.1 Geometry and material parameters the structure 113
5.1 Geometry and material parameters the structure 145
6.1 Geometry and material parameters used in simulation 164
6.2 Geometry and material parameters used in high frequency simulation 170
A.1 htp 176
A.2 htp 176
X