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Multifunctional Piezoelectric Energy Harvesting Concepts
by
Steven R. Anton
Dissertation submitted to the Faculty of the
Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
in
Mechanical Engineering
Daniel J. Inman, Chair
Dong S. Ha
Donald J. Leo
Shashank Priya
Henry A. Sodano
April 25, 2011
Blacksburg, Virginia
Keywords: energy harvesting, multifunctional, piezoelectric, unmanned aerial vehicle
Copyright by Steven R. Anton, 2011
Multifunctional Piezoelectric Energy Harvesting Concepts
Steven R. Anton
Virginia Polytechnic Institute and State University, 2011
Advisor: Daniel J. Inman, Ph.D.
A BSTRACT
Energy harvesting technology has the ability to create autonomous, self-powered electronic systems that do not rely on battery power for their operation. The term energy
harvesting describes the process of converting ambient energy surrounding a system into
useful electrical energy through the use of a specific material or transducer. A widely
studied form of energy harvesting involves the conversion of mechanical vibration energy into electrical energy using piezoelectric materials, which exhibit electromechanical
coupling between the electrical and mechanical domains. Typical piezoelectric energy harvesting systems are designed as add-on systems to a host structure located in a vibration
rich environment. The added mass and volume of conventional vibration energy harvesting designs can hinder to the operation of the host system. The work presented in this
dissertation focuses on advancing piezoelectric energy harvesting concepts through the
introduction of multifunctionality in order to alleviate some of the challenges associated
with conventional piezoelectric harvesting designs.
The concept of multifunctional piezoelectric self-charging structures is explored throughout this work. The operational principle behind the concept is first described in which
piezoelectric layers are directly bonded to thin-film battery layers resulting in a single device capable of simultaneously harvesting and storing electrical energy when excited mechanically. Additionally, it is proposed that self-charging structures be embedded into host
structures such that they support structural load during operation. An electromechanical
assumed modes model used to predict the coupled electrical and mechanical response of
a cantilever self-charging structure subjected to harmonic base excitation is described. Experimental evaluation of a prototype self-charging structure is then performed in order to
validate the electromechanical model and to confirm the ability of the device to operate in a
self-charging manner. Detailed strength testing is also performed on the prototype device
in order to assess its strength properties. Static three-point bend testing as well as dynamic harmonic base excitation testing is performed such that the static bending strength
and dynamic strength under vibration excitation is assessed. Three-point bend testing is
also performed on a variety of common piezoelectric materials and results of the testing
provide a basis for the design of self-charging structures for various applications.
Multifunctional vibration energy harvesting in unmanned aerial vehicles (UAVs) is
also investigated as a case study in this dissertation. A flight endurance model recently
developed in the literature is applied to model the effects of adding piezoelectric energy
harvesting to an electric UAV. A remote control foam glider aircraft is chosen as the test
platform for this work and the formulation is used to predict the effects of integrating selfcharging structures into the wing spar of the aircraft. An electromechanical model based
on the assumed modes method is then developed to predict the electrical and mechanical
behavior of a UAV wing spar with embedded piezoelectric and thin-film battery layers. Experimental testing is performed on a representative aluminum wing spar with embedded
self-charging structures in order to validate the electromechanical model. Finally, fabrication of a realistic fiberglass wing spar with integrated piezoelectric and thin-film battery
layers is described. Experimental testing is performed in the laboratory to evaluate the
energy harvesting ability of the spar and to confirm its self-charging operation. Flight testing is also performed where the fiberglass spar is used in the remote control aircraft test
platform and the energy harvesting performance of the device is measured during flight.
All images and figures are property of Steven R. Anton and were captured or created
between June, 2006 and April, 2011.
iii
Dedication
To my wife Tiffany for her unwaivering love and support, and to my daughter Mia for the joy she
brings to my life
And to my parents Michael and Vera for giving me such amazing opportunities throughout life
iv
Acknowledgments
There are countless people that have been a part of my life who have all had a special
influence in guiding me to this point. From teachers and professors to family and friends,
it is impossible to thank every one of these people, but for those of you who have had a
part in this great journey, you know who you are and I am forever grateful.
First and foremost, I would like to extend my deepest thanks to my advisor, Dr. Daniel
J. Inman, for his guidance and support throughout the five years I have spent in graduate school at Virginia Tech. Dr. Inman has been a truly amazing advisor and I could not
imagine a better person to work with. His academic guidance and knowledge is invaluable while his humorous nature has provided for many laughs and great memories. Most
importantly, Dr. Inman has always treated me with respect and made me feel more like a
colleague than a student. He has given me the freedom to explore research topics of my
own interest throughout both my Masters and Ph.D. work. This freedom can be a rarity
among academic advising, and it has allowed me to develop my own research style and
a sense of ownership of my work. Dr. Inman has encouraged me to travel to many research conferences around the US as well as to Scotland, for which I am forever grateful
as I have had the opportunity to meet many great people and see many wonderful places.
He also allowed me to co-teach a senior capstone design course alongside of him, which
was a great learning experience. Throughout graduate school, Dr. Inman has also been
extremely understanding about personal and family issues, always allowing me to take
time to see my family. I consider Dr. Inman to be a great advisor, a great man, and truly a
great friend. Dr. Inman, I look forward to a long and fruitful friendship!
I would like to thank Dr. Dong S. Ha, Dr. Donald J. Leo, Dr. Shashank Priya, and Dr.
Henry A. Sodano for serving on my committee and helping me with several research topics throughout my Ph.D. work. It has been quite beneficial having the support of such
v
knowledgeable professors. I would like to make a special thanks to Henry for introducing
me to the topics of smart materials and energy harvesting back in our Michigan Tech days.
Without your guidance, I would have never found my way to Virginia Tech.
Thanks are certainly due to Ms. Beth Howell, the program manager of the Center for
Intelligent Material Systems and Structures (CIMSS), for all of her help over the past five
years. Anyone working in CIMSS knows that Beth keeps the place running, pun intended!
She has been so helpful with academic, financial, and administrative issues, not to mention
a great person to talk to. I’ll always remember our nice chats about soccer, teaching, and
raising kids.
Several of my fellow CIMSS colleagues have been instrumental in my success during
graduate school. Thanks to Dr. Benjamin L. Grisso, a fellow Michigan Tech Husky, for introducing my wife Tiffany and I to Blacksburg, for your help around the lab and valuable
guidance on graduate school, and for your friendship over the years. Thanks to Dr. Onur
Bilgen for your invaluable help around the lab in understanding much of our equipment
and for your efforts as a lab co-manager. Also, thanks Onur for showing me the wonders
of fiberglasssing and vacuum bagging, which allowed for the fabrication of all of my prototype devices. I would like to thank Justin Farmer, who as a fellow student introduced
me to the topic of piezoelectric vibration energy harvesting when I first arrived at CIMSS
and who now has returned and graciously relieved me of lab management duties, giving
me the chance to finish my Ph.D.! Thanks to Jacob Dodson for your help with anything
math related, my electromechanical modeling would not have been possible without your
help. Also, thanks Jacob for being such a great friend and office mate. Thanks to Na
Kong for your help with the electrical side of piezoelectric energy harvesting. Your help
in the development of various energy harvesting circuits was critical in my success and
resulted in a fruitful collaboration. I would like to thank Amin Karami for your help on
both electromechanical modeling as well as our many nice discussions on searching for
jobs. Thanks to Dr. Stephen A. Sarles (Andy) for your help in many lab related issues, you
are an encyclopedia of knowledge around the lab! Also, thanks Andy for all the advice
you have given me in all academic issues from writing and publishing papers to applying
for jobs. Thanks to Geoff Tizard for your help in configuring the Instron load frame, all of
the static bending tests would not have been possible without your help. Also, thanks to
Robert Briggs and John Coggin for your help in performing the experimental flight testing
vi
in this work.
A special thanks is due to Dr. Pablo A. Tarazaga for introducing me to most of the
equipment around the CIMSS lab and for teaching me the intricacies of experimental vibration testing. Also, for teaching me the ways as a lab manager. Additionally, I would
like to thank Pablo for coaching and inspiring me in both weightlifting and motorcycling.
I have many great memories of early mornings in the gym and beautiful summer evenings
carving the Blue Ridge mountains on our bikes. Thanks Pablo for being such a great friend!
A very special thanks is in order to Dr. Alper Erturk. Alper and I began graduate
school together and we have enjoyed a fruitful collaboration from the beginning. Thanks
Alper for your countless hours of help in coursework, modeling efforts, and experimental
testing. Surly without the help and guidance of Alper, my Ph.D. work would not have
been possible. I feel blessed to have been given the opportunity to collaborate with such
a brilliant and kind individual. I have many fond memories of late night experimental
vibration testing with temperamental laser vibrometers and wonderful philosophical chats
about all walks of life. Alper, it has been amazing working with you and I look forward to
a long friendship.
I would also like to acknowledge a few special friends that I have made along the
way. First, I would like to thank Devon Murphy. Devon and I met during the summer of
2006 as students in the Los Alamos Dynamics Summer School at the Los Alamos National
Laboratory in New Mexico. We had tons of great experiences during our nine weeks at the
lab including hiking to the top of Wheeler Peak, which is the highest point in New Mexico
at 13,161 feet! Throughout our time together in grad school, Devon was an amazing friend,
always making people laugh. We had tons of great talks about research related topic as
well as life in general. Over the years, Devon has become part of my family. Dev, you will
always be my brother!
Also, I would like to thank Corey Pitchford. I became quite close with Corey during
the summer of 2007. . . that is a summer we will never forget! Although our time together
in graduate school was short, I have remained good friends with Corey over the years. We
have made many trips between Blacksburg and Kingsport, TN, from which endless good
memories have been made. Corey is a video game buddy, a golfing buddy, a motorcycling
buddy, and if I could ever convince him, a mountain biking buddy (maybe some day!).
Corey has been very supportive throughout graduate school, and a friend I can always
vii
count on.
I would like to thank Daniel J. Inman II for being a great friend over the years. I met
Daniel when he was in high school and working in the CIMSS lab over the summers.
Our relationship has transformed from me bossing Daniel around the lab and making him
do all the cleaning and organizing us graduate students think we are too good to do, to
Daniel trying to take over my desk in CIMSS now that I am finished! We have become
good friends and have had a lot of fun throughout my time in Blacksburg. Daniel has been
a great lifting and running partner and I owe a lot to him for keeping me motivated (I think
this is a two way street). Also, thanks for allowing me the privilege of kicking your butt in
Halo! OK, I suppose you got me a time or two as well. And thanks for making the CIMSS
office an awesome place to work.
Thanks to Kahlil Detrich, your friendship over the years has been amazing. From
killing me out on the hard climbs at Pandapas Pond on our mountain bikes to enjoying
good scotch and cigars on a summer night, to cooking great food with friends, we have
made many great memories I will never forget. Also, thanks Kahlil for allowing the use of
your house during the final experiments in my Masters work!
I would also like to thank all of my past and present CIMianSS for making CIMSS
such an amazing place to work. The atmosphere in the office and in the lab is one of a
kind. Thanks to Dr. Jamil Renno for your guidance on being a graduate student and for
introducing me to LATEX 2ε . Thanks also to Dr. Vishnu Baba Sundaresan for the help with
LATEX 2ε . Thanks to Austin Creasy for great discussions on raising children. Thanks to Dr.
Armaghan Salehian for your help teaching me graduate vibrations. Thanks to Jason Fox,
Josh Stenzler, Jacob Davidson, Brad Butrym, Matt Castellucci, Mana Afshari, and Carlos
Carvajal for being great friends and for helping me get into trouble during my time at Tech!
A very important thanks is due to my family. To my mother and father for raising
me and always putting their children first, giving me the opportunity to attend college,
supporting me throughout graduate school, and always being there for me to help with
any and all aspects of life. None of this would have been possible without your support
and belief in me. You are amazing people, amazing parents, and I love you both with all
my heart. And thanks to my brother, Chris, for being an inspiration to me and for all of
your advice during graduate school. I am blessed to have an older brother to turn to for
advice on all facets of life.
viii
I would like to thank my beautiful daughter, Mia Jane, for bringing endless joy to my
life. The look in your eyes and the smile on your face at the end of a day make any bad
moments disappear. You have brought a new meaning to life. I love you so much, you
stole my heart with your first breath.
Finally, and most importantly, I would like to thank my amazing wife, Tiffany, for her
unwavering love and support each and every day of my life. You always challenge me
to be the best man I can be. The sacrifices you have made in supporting me throughout
graduate school are immense, and I look forward to paying them back throughout the rest
of our lives. We have accomplished so much since we have been together, from graduating
from college, starting graduate school and your teaching career, to getting married, and to
starting a family. I look forward to every day we will spend together. You are beautiful
inside and out. You are my best friend. I love US.
This work has been supported by the U.S. Air Force Office of Scientific Research MURI
under grant number F9550-06-1-0326 titled “Energy Harvesting and Storage Systems for
Future Air Force Vehicle” monitored by Dr B. L. “Les” Lee. Furthermore, I would like
to thank the Department of Mechanical Engineering at Virginia Tech for their support
through the Pratt Fellowship as well as the Virginia Space Grant Consortium for their
support through the Graduate VSGC Fellowship.
S TEVEN R. A NTON
ix
Contents
Abstract
ii
Dedication
iv
Acknowledgments
v
List of Tables
xv
List of Figures
xvi
Nomenclature
xxii
1
Introduction
1
1.1
Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.1.1
Vibration Energy Harvesting using Piezoelectric Materials . . . . . .
3
1.1.2
Multifunctionality in Energy Harvesting Systems . . . . . . . . . . .
10
1.1.3
Piezoelectric Energy Harvesting in Unmanned Aerial Vehicles . . . .
17
Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
1.2.1
Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
1.2.2
Chapter Summaries . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
1.2
2
Multifunctional Self-Charging Structure Concept and Electromechanical Modeling
2.1
2.2
24
Overview of Piezoelectric Energy Harvesting . . . . . . . . . . . . . . . . . .
25
2.1.1
Introduction to Piezoelectricity . . . . . . . . . . . . . . . . . . . . . .
25
2.1.2
Introduction to Piezoelectric Energy Harvesting . . . . . . . . . . . .
26
Self-Charging Structure Concept . . . . . . . . . . . . . . . . . . . . . . . . .
28
x
2.3
2.4
3
31
2.3.1
Modeling Assumptions and Device Configuration . . . . . . . . . . .
31
2.3.2
Electromechanical Assumed Modes Model . . . . . . . . . . . . . . .
32
2.3.3
Substitution of the Assumed Solution . . . . . . . . . . . . . . . . . .
37
2.3.4
Lagrange Equations with Electromechanical Coupling . . . . . . . .
39
2.3.5
Equivalent Series/Parallel Representation of the Lagrange Equation
41
2.3.6
Solution of the Equivalent Representation of the Lagrange Equations
42
Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
Experimental Testing of Self-Charging Structures
46
3.1
Performance Evaluation of Thin-Film Batteries . . . . . . . . . . . . . . . . .
46
3.1.1
Comparison to Conventional Rechargeable Batteries . . . . . . . . .
47
3.1.2
Battery Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
3.1.3
R
Performance Testing of Thinergy
Batteries . . . . . . . . . . . . . .
50
Device Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
3.2.1
Selection of Piezoelectric and Substrate Materials . . . . . . . . . . .
53
3.2.2
Vacuum Bonding and Electrode Attachment . . . . . . . . . . . . . .
55
3.2
3.3
3.4
4
Electromechanical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Experimental Validation of Electromechanical Model and Self-Charging Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
3.3.1
Electromechanical Model Validation . . . . . . . . . . . . . . . . . . .
58
3.3.2
Self-Charging Structure Concept Validation . . . . . . . . . . . . . . .
65
Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
Strength Analysis of Self-Charging Structures
70
4.1
Static Strength Analysis of Self-Charging Structures . . . . . . . . . . . . . .
70
4.1.1
Strength Calculations for 3-Point Bend Testing . . . . . . . . . . . . .
71
4.1.2
Experimental Testing and Results . . . . . . . . . . . . . . . . . . . .
72
Dynamic Strength Analysis of Self-Charging Structures . . . . . . . . . . . .
76
4.2.1
Strength Calculations for Harmonic Base Excitation Testing . . . . .
76
4.2.2
Experimental Testing and Results . . . . . . . . . . . . . . . . . . . .
78
Static Strength Testing of Various Piezoceramic Materials . . . . . . . . . . .
81
4.3.1
Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
4.3.2
Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
4.2
4.3
xi
4.4
5
4.3.3
Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
4.3.4
Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
4.3.5
Statistical Analysis of Bending Strength Results . . . . . . . . . . . .
90
Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
Performance Modeling of Unmanned Aerial Vehicles with On Board Energy Harvesting
94
5.1
Piezoelectric Energy Harvesting in Unmanned Aerial Vehicles . . . . . . . .
95
5.2
System Level Flight Endurance Model . . . . . . . . . . . . . . . . . . . . . .
96
5.2.1
5.2.2
6
Flight Endurance of an Electric Powered UAV with On Board Energy
Harvesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
Normalized Change in Flight Endurance . . . . . . . . . . . . . . . .
98
5.3
Theoretical Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.4
Experimental Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.4.1
EasyGlider Aircraft Configuration . . . . . . . . . . . . . . . . . . . . 105
5.4.2
Flight Testing Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.4.3
Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.5
Flight Endurance Modeling of Self-Charging Structures in UAVs . . . . . . . 110
5.6
Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Electromechanical Modeling of Multifunctional Energy Harvesting Wing Spar
for Unmanned Aerial Vehicles
6.1
6.2
6.3
113
Electromechanical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.1.1
Modeling Assumptions and Device Configuration . . . . . . . . . . . 114
6.1.2
Energy Formulations for the Electromechanical Spar . . . . . . . . . 114
6.1.3
Substitution of the Assumed Solution . . . . . . . . . . . . . . . . . . 118
6.1.4
Lagrange Equations with Electromechanical Coupling . . . . . . . . 119
6.1.5
Solution of the Equivalent Representation of the Lagrange Equations 120
Experimental Validation of the Assumed Modes Formulation . . . . . . . . 121
6.2.1
Representative Energy Harvesting Wing Spar Configuration . . . . . 121
6.2.2
Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.2.3
Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
xii
7
Energy Harvesting Wing Spar for Electric Unmanned Aerial Vehicle
7.1
Design and Fabrication of Multifunctional Composite Wing Spar . . . . . . 130
7.2
Experimental Evaluation of Energy Harvesting/Energy Storage Performance 132
7.3
7.4
8
129
7.2.1
Frequency Response Measurements . . . . . . . . . . . . . . . . . . . 133
7.2.2
Self-Charging Charge/Discharge Measurements . . . . . . . . . . . . 136
Experimental Flight Testing of Self-Charging Fiberglass Wing Spar . . . . . 138
7.3.1
EasyGlider Aircraft Configuration . . . . . . . . . . . . . . . . . . . . 139
7.3.2
Flight Testing Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Conclusions and Recommendations
143
8.1
Research Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.2
Contributions of the Research . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
8.3
Recommendations and Future Work . . . . . . . . . . . . . . . . . . . . . . . 147
Bibliography
150
Appendices
166
A Piezoelectric Constitutive Equations
166
A.1 Constitutive Equations for Bulk Monolithic Piezoelectric Material . . . . . . 166
A.2 Alternative Representations of the Constitutive Equations . . . . . . . . . . 167
A.3 Reduced Form of the Constitutive Equations for a Thin Beam Operating in
‘31’ Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
B Bending Stiffness and Mass Density Calculations for Composite Cross-Sections 169
B.1 Bending Stiffness Calculation using the Parallel Axis Theorem . . . . . . . . 169
B.2 Self-Charging Structure Sections . . . . . . . . . . . . . . . . . . . . . . . . . 170
B.3 Self-Charging Wing Spar Sections . . . . . . . . . . . . . . . . . . . . . . . . . 172
C Transformation from Relative to Absolute Frame of Reference for Frequency Response Functions
174
D Charge/Discharge Curves for Dynamic Failure Testing
xiii
176
E Load-Deflection Curves for all Piezoelectric Samples Tested
xiv
188
List of Tables
2.1
Equivalent electromechanical coupling and capacitance terms for series and
parallel electrode connections. . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
3.1
Properties of various secondary batteries. . . . . . . . . . . . . . . . . . . . .
48
3.2
Geometric and material properties of self-charging structure components. .
55
3.3
Load resistance values used in FRF measurements of self-charging structures along with effective load resistances. . . . . . . . . . . . . . . . . . . . .
60
4.1
Failure loads for three-point bending tests. . . . . . . . . . . . . . . . . . . .
75
4.2
Failure strengths for three-point bending tests. . . . . . . . . . . . . . . . . .
75
4.3
Physical dimensions of various piezoceramic materials tested. . . . . . . . .
83
4.4
Various test parameters for the piezoelectric materials investigated. . . . . .
86
4.5
Bending strength parameters for all materials tested with 95% confidence
interval given in brackets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
5.1
Physical properties of the EasyGlider aircraft. . . . . . . . . . . . . . . . . . . 102
5.2
Physical properties of the piezoelectric devices considered. . . . . . . . . . . 103
6.1
Load resistance values used in FRF measurements of aluminum spar along
with effective load resistances. . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
xv
List of Figures
2.1
Schematic of piezoelectric cantilever harvester configuration subject to harmonic base excitation and its voltage output. . . . . . . . . . . . . . . . . . .
27
2.2
Schematic of self-charging structure. . . . . . . . . . . . . . . . . . . . . . . .
29
2.3
Potential use of self-charging structures: Schematic of small UAV with embedded self-charging structures in wing spar used to provide local power
for low-power sensor node. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4
30
Piezoelectric self-charging structure configurations showing (a) series and
(b) parallel connection of the piezoelectric layers along with the (c) crosssectional views of both composite sections. . . . . . . . . . . . . . . . . . . .
32
3.1
R
R
Photographs of (a) NanoEnergy
and (b) Thinergy
thin-film batteries. . .
47
3.2
Experimental setup used to obtain charge/discharge measurements on thinfilm batteries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
3.3
Transimpedance operational amplifier circuit used for current measurement.
51
3.4
R
Characteristic (a) charge and (b) discharge curves of Thinergy
batteries. .
52
3.5
R
QuickPack
QP10n piezoelectric device. . . . . . . . . . . . . . . . . . . . .
54
3.6
Schematic of vacuum bagging procedure for self-charging structures. . . . .
56
3.7
(a) Vacuum bagging setup; (b) Gast 23 Series vacuum pump. . . . . . . . . .
56
3.8
Complete self-charging structure prototype. . . . . . . . . . . . . . . . . . . .
57
3.9
Experimental setup used to obtain frequency response measurements of
self-charging structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
3.10 Experimental and numerical (a) voltage - to - base acceleration FRFs and (b)
tip velocity - to - base acceleration FRFs of self-charging structure for series
electrode connection case and for various load resistances. . . . . . . . . . .
xvi
61
3.11 Experimental and numerical (a) tip velocity - to - base acceleration FRFs and
(b) voltage - to - base acceleration FRFs of self-charging structure for parallel
electrode connection case and for various load resistances. . . . . . . . . . .
62
3.12 Linear voltage regulator energy harvesting circuit. . . . . . . . . . . . . . . .
63
3.13 Experimental and numerical electrical performance curves of self-charging
structure for the peak voltage output in the (a) series and (b) parallel case,
current output in the (c) series and (d) parallel case, and power output in
the (e) series and (f) parallel case with varying load resistance for the selfcharging structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
3.14 Experimental setup used to obtain charge/discharge measurements of selfcharging structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
3.15 Experimental curves for self-charging structures in (a) charging and (b) discharging under ±1.0 g acceleration at 210.0 Hz. . . . . . . . . . . . . . . . . .
67
4.1
Schematic of three-point bending test. . . . . . . . . . . . . . . . . . . . . . .
71
4.2
Experimental setup used for three-point bend testing including Instron/
MTS 4204 test frame and fixture. . . . . . . . . . . . . . . . . . . . . . . . . .
4.3
72
R
Various specimens after three-point failure testing including (a) Thinergy
R
battery, (b) QuickPack
QP10n piezoelectric, (c) aluminum substrate, (d)
root section of complete self-charging structure, and (e) tip section of complete self-charging structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4
Load-deflection curves for (a) individual layers and (b) complete self-charging
structure sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5
80
Optical microscope image of PMN-PZT sample after dicing showing width
measurement (5X objective lens). . . . . . . . . . . . . . . . . . . . . . . . . .
4.8
78
Charge/discharge capacities measured for (a) power supply and (b) piezoelectric charging for dynamic failure analysis of self-charging structures. . .
4.7
74
Estimates of the maximum dynamic bending stress in the aluminum, piezoceramic, and battery layers. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6
73
84
Optical microscope images of (a) PMN-PZT and (b) PZT-5H samples showing flaw sizes (20X objective lens). . . . . . . . . . . . . . . . . . . . . . . . .
xvii
84
4.9
Diced piezoelectric samples prepared for bending tests including (a) PZT5H, (b) PZT-5A, (c) PZT-8, (d) PZT-4, (e) PMN-PZT, (f) PMN-PT, (g) QP10n,
and (h) QP16n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
4.10 Three-point bend testing experimental setup including (a) Instron 5848 MicroTester frame and (b) fixture. . . . . . . . . . . . . . . . . . . . . . . . . . .
86
4.11 Load-deflection curves for (a) thin samples (PZT-5A, PZT-5H, PMN-PT, PMNPZT, and QP10n) and (b) thick samples (PZT-4 and PZT-8). . . . . . . . . . .
87
4.12 Bending strength values calculated for various piezoelectric materials tested. 88
4.13 Bending strength statistical comparison for all materials tested with error
bars representing 95% confidence interval. . . . . . . . . . . . . . . . . . . .
91
5.1
Multiplex USA EasyGlider remote control aircraft test platform. . . . . . . . 101
5.2
Piezoelectric devices including (a) M8507-P1 Macro-Fiber Composite, and
(b) Piezoelectric Fiber Composite. . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.3
Simulation results for normalized change in flight time of the EasyGlider
aircraft based on the addition of piezoelectric harvesting with varying degrees of multifunctionality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.4
Fiberglass wing spar with (a) surface mounted Macro-Fiber Composite and
Piezoelectric Fiber Composite devices (b) inserted in wing. . . . . . . . . . . 106
5.5
XR5-SE-M-50mv voltage data logger shown (a) installed into cockpit of EasyGlider
and (b) with fiberglass canopy installed. . . . . . . . . . . . . . . . . . . . . . 106
5.6
UA-004-64 accelerometer data logger shown (a) installed on the underside
of the EasyGlider and (b) with fiberglass canopy installed. . . . . . . . . . . 107
5.7
Acceleration measurements recorded during flight in the (a) x-axis, (b) yaxis, and (c) z-axis directions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.8
Voltage measurements recorded during flight for (a) MFC and (b) PFC piezoelectric devices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.9
Updated simulation results for normalized change in flight time of the EasyGlider
aircraft based on the addition of piezoelectric harvesting with varying degrees of multifunctionality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.10 Simulation results for normalized change in flight time of the EasyGlider
aircraft based on the addition of self-charging structures. . . . . . . . . . . . 111
xviii
6.1
Multifunctional piezoelectric energy harvesting wing spar configurations
showing (a) series and (b) parallel connection of the piezoelectric layers
along with the (c) cross-sectional views of both composite sections. . . . . . 115
6.2
Representative aluminum wing spar with embedded self-charging structure. 121
6.3
Experimental setup used to obtain frequency response measurements of the
aluminum wing spar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.4
Experimental and numerical (a) voltage - to - base acceleration FRFs and
(b) tip velocity - to - base acceleration FRFs of the aluminum wing spar for
series electrode connection case for various load resistances. . . . . . . . . . 124
6.5
Experimental and numerical (a) voltage - to - base acceleration FRFs and
(b) tip velocity - to - base acceleration FRFs of the aluminum wing spar for
parallel electrode connection case for various load resistances. . . . . . . . . 125
6.6
Experimental and numerical electrical performance curves of aluminum wing
spar for the peak voltage output in the (a) series and (b) parallel case, current output in the (c) series and (d) parallel case, and power output in the
(e) series and (f) parallel case with varying load resistance. . . . . . . . . . . 127
7.1
Fiberglass wing spar schematic showing foam core with fiberglass layers
and embedded self-charging structures. . . . . . . . . . . . . . . . . . . . . . 130
7.2
Schematic of vacuum bagging procedure for fiberglass wing spar fabrication. 132
7.3
Photographs of fiberglass wing spar curing under vacuum. . . . . . . . . . . 132
7.4
Complete self-charging fiberglass wing spar. . . . . . . . . . . . . . . . . . . 133
7.5
Experimental setup used to obtain frequency response measurements of
fiberglass wing spar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7.6
Frequency response functions of fiberglass wing spar (series electrode connection, 1 MΩ load) showing (a) broad spectrum and (b) detail near the
fundamental resonance frequency. . . . . . . . . . . . . . . . . . . . . . . . . 135
7.7
Experimental setup used to obtain charge/discharge measurements of selfcharging fiberglass wing spar. . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.8
Experimental curves for fiberglass wing spar in (a) charging and (b) discharging under ±0.5 g acceleration at 43.5 Hz. . . . . . . . . . . . . . . . . . 138
7.9
Self-charging fiberglass wing spar inserted into EasyGlider aircraft wings. . 139
xix
7.10 Photograph of EasyGlider aircraft during flight testing of multifunctional
fiberglass wing spar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.11 Flight testing results of fiberglass self-charing spar showing (a) voltage output of piezoelectric device and (b) z-axis acceleration. . . . . . . . . . . . . . 141
B.1 Rigid body with neutral axis x0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
B.2 Cross-sectional views of the two composite sections of the self-charging
structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
B.3 Cross-sectional views of the two composite sections of the self-charging
wing spar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
D.1 Power supply charge and discharge curves for excitation at (a-b) ±0.2 g and
(c-d) ±0.4 g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
D.2 Power supply charge and discharge curves for excitation at (a-b) ±0.6 g, (cd) ±0.8 g, and (e-f) ±1.0 g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
D.3 Power supply charge and discharge curves for excitation at (a-b) ±1.5 g, (cd) ±2.0 g, and (e-f) ±2.5 g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
D.4 Power supply charge and discharge curves for excitation at (a-b) ±3.0 g, (cd) ±3.5 g, and (e-f) ±4.0 g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
D.5 Power supply charge and discharge curves for excitation at (a-b) ±4.5 g, (cd) ±5.0 g, and (e-f) ±5.5 g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
D.6 Power supply charge and discharge curves for excitation at (a-b) ±6.0 g, (cd) ±6.5 g, and (e-f) ±7.0 g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
D.7 Piezoelectric charge and discharge curves for excitation at (a-b) ±0.2 g, (c-d)
±0.4 g, and (e-f) ±0.6 g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
D.8 Piezoelectric charge and discharge curves for excitation at (a-b) ±0.8 g, (c-d)
±1.0 g, and (e-f) ±1.5 g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
D.9 Piezoelectric charge and discharge curves for excitation at (a-b) ±2.0 g, (c-d)
±2.5 g, and (e-f) ±3.0 g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
D.10 Piezoelectric charge and discharge curves for excitation at (a-b) ±3.5 g, (c-d)
±4.0 g, and (e-f) ±4.5 g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
D.11 Piezoelectric charge and discharge curves for excitation at (a-b) ±5.0 g, (c-d)
±5.5 g, and (e-f) ±6.0 g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
xx
D.12 Piezoelectric charge and discharge curves for excitation at (a-b) ±6.5 g and
(c-d) ±7.0 g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
E.1 Load-deflection curves for PZT-5A samples. . . . . . . . . . . . . . . . . . . . 188
E.2 Load-deflection curves for (a) PZT-5H, (b) PZT-4, and (c) PZT-8 samples. . . 189
E.3 Load-deflection curves for (a) PMN-PT, (b) PMN-PZT, and (c) QP10n samples.190
xxi
Nomenclature
A
area
ab
base acceleration
b
width
C
damping matrix
C
battery capacity
CD
coefficient of drag
CL
coefficient of lift
Cp
capacitance
cE
11
elastic modulus at constant electric field
D
electric displacement, also drag force
d31
piezoelectric strain constant
E
electric field
Eb
battery energy
e31
piezoelectric stress constant
eb
battery specific energy
f
forcing function
G
shear modulus
g
acceleration of gravity
H
Heaviside function
h
thickness
I
mass moment of inertia
i
current, also index
Jp
piezoelectric coupling term
xxii
j
unit imaginary number
K
stiffness matrix
L
length, also lift force
l
index
M
mass matrix
M
lumped mass
Mf
failure bending moment
m
Weibull modulus
N
number of modes
P
power
Pf
failure load
Q
electric charge
R
resistance
r
index
S
strain, also wing area
s
crosshead feed rate
sE
11
elastic compliance at constant electric field
T
stress, also kinetic energy, also thrust
t
time
U
potential energy
Vabs
absolute beam velocity
V
volume
V∞
aircraft velocity
v
voltage
W
weight
Wie
internal electrical energy
Wnce
work of non-conservative forces
wabs
absolute displacement
wb
base displacement
wrel
relative displacement
x
spatial coordinate
Y
elastic modulus
xxiii
z
spatial coordinate
α
proportional damping constant
β
proportional damping constant
δ
Dirac delta function
ε
strain rate
εS33
dielectric permittivity at zero strain
η
generalized coordinate
ηb
battery energy extraction efficiency factor
ηp
motor and propeller efficiency factor
θ
electromechanical coupling term
ν
Poisson’s ratio
ρ
mass density
ρ∞
air density
σb
bending strength
σθ
Weibull characteristic strength
φ
admissible function
ψ
aerodynamic constant
ω
frequency
Subscripts
b
battery layer
cr
critical layer
e
epoxy layer
k
Kapton layer
l
lower layer
p
piezoelectric layer
s
structure layer
u
upper layer
Superscripts
xxiv
E
property measured at constant electric field
eq
equivalent representation
p
parallel electrode connection
S
property measured at zero mechanical strain
s
series electrode connection
T
matrix transpose, also property measured at zero mechanical stress
xxv
C HAPTER
1
I NTRODUCTION
W
ITH
recent growth in the development of low-power electronic devices such as mi-
croelectronics and wireless sensor nodes, as well as the global interest in the con-
cept of “green” engineering, the topic of energy harvesting has received much attention
in the past decade. The energy requirements of low-power electronics have steadily decreased with advancements in efficient circuitry such that energy harvesting systems can
be considered feasible solutions in providing power to self-powered systems. Conventional low-power electronics, such as wireless sensor nodes, rely on batteries to provide
power to the device. The use of batteries, however, presents several drawbacks including
the cost of battery replacement as well as limitations imposed by the need of convenient
access to the device for battery replacement purposes. Wireless sensor nodes, for example,
are often used in remote locations or embedded into a structure, therefore, access to the
device can be difficult or impossible. By scavenging ambient energy surrounding an electronic device, energy harvesting solutions have the ability to provide permanent power
sources that do not require periodic replacement. Such systems can operate in an autonomous, self-powered manner, reducing the costs associated with battery replacement,
and can easily be placed in remote locations or embedded into host structures.
Energy harvesting is the process of converting available ambient energy into usable
electrical energy through the use of a particular material or transduction mechanism. Several classes of material exist with various conversion mechanisms that can be used to harvest energy. Some of the common materials include those with photovoltaic coupling to
convert solar energy to electric energy, thermoelectric coupling to convert temperature
gradients into electrical energy, and electromechanical coupling to convert mechanical
vibration energy into electrical energy. Of the various modes of energy harvesting, vibration energy harvesting is the most versatile technique being developed in the litera-
1
ture. Three main mechanisms of vibration-to-electrical energy conversion exist including
electrostatic, electromagnetic, and piezoelectric transduction. Review articles highlighting
work performed on all of the transduction mechanisms are given by Beeby et al. [1] and
Cook-Chennault et al. [2]. Of the three modes of vibration harvesting, piezoelectric energy
harvesting has received the most attention, with three review articles dedicated to recent
research on piezoelectric transduction [3–5].
Piezoelectric vibration harvesting is attractive mainly due to the simplicity of piezoelectric transduction and the relative ease of implementation of piezoelectric systems into
a wide variety of applications as compared to electrostatic or electromagnetic methods
[6, 7]. Unlike electrostatic transduction, which requires the application of an initial voltage
to produce usable energy, piezoelectric material inherently generates a direct voltage when
strained. Additionally, where electromagnetic harvesting systems become increasingly difficult to fabricate at the micro-scale and electrostatic systems are generally restricted to the
micro-scale, piezoelectric materials can easily be fabricated as bulk materials at the macroscale or deposited as thin-films at the micro-scale. Despite these advantages, piezoelectric
harvesting does suffer from the large mass density of piezoceramic materials and, like all
vibration harvesting techniques, the intrusiveness that an added harvesting system has
on a host structure. In an effort to increase the usefulness of various material systems,
researchers have begun to investigate the concept of multifunctionality, which has been
reviewed by Christodoulou and Venables [8] for several different multifunctional material systems. Piezoelectric energy harvesting can benefit from a multifunctional approach
by combining several functions into a single device, such as energy harvesting, energy
storage, and structural load bearing ability, that when combined with a host structure can
provide a more synergistic and efficient overall system. The introduction of multifunctionality into energy harvesting systems holds promise to increase their utility and practicality,
and promote the integration of harvesting systems into many engineering applications.
1.1
Literature Review
The following presents a literature review of the past research performed in the areas of
piezoelectric energy harvesting and multifunctionality in energy harvesting systems. Additionally, previous studies involving energy harvesting in unmanned vehicle systems are
2
also reviewed as multifunctional harvesting in an unmanned aerial vehicle is investigated
as a case study in this dissertation.
1.1.1
Vibration Energy Harvesting using Piezoelectric Materials
Perhaps one of the first suggested applications involving vibration energy harvesting using piezoelectric materials was presented by Wen H. Ko in a 1969 US patent titled “Piezoelectric Energy Converter for Electronic Implants” [9] in which the use of a cantilever
piezoelectric beam with a tip mass is proposed for harvesting energy from heartbeats for
use in powering implanted medical pacemakers. One of the earliest published works in vibration energy harvesting with piezoelectric materials was presented by Taylor and Burns
[10] in 1983 in which they proposed the use of an array of polyvinylidene flouride (PVDF)
piezoelectric polymer film to harvest hydrodynamic energy from ocean waves. Although
no physical system was proposed or built, it was theorized that a 100 MW power plant
utilizing PVDF could deliver power to an onshore grid at a cost of 2.5 cents per kWH.
The following year, Hausler et al. [11] performed what appears to be the first experimental
study using piezoelectric materials to harvest energy. They proposed the use of a PVDF
film as an implantable power source in a biological system. A prototype device was fabricated and implanted in vivo into the ribcage of a dog to harvest energy available from
the relative motion of the ribs during breathing. Experimental testing showed that the
PVDF harvester could generate a peak voltage of 18 V, corresponding to about 17 µW of
power, which was short of the predicted 1 mW goal. Three years later in 1987, Hausler
and Stein [12] published a work in which PVDF film was again proposed for use in harvesting energy from ocean waves. A theoretical system was proposed in which one end of
a PVDF ‘rope’ is attached to the sea floor and the other end to a float located at the surface
of the sea. Relative motion between the float and the sea floor allows the PVDF rope to
be stretched, thus generating electrical energy. It was suggested that the system would
be well suited to provide power to warning or measurement buoys, with 20 kg of PVDF
capable of generating around 100 W of power.
Similar to the work of Hausler et al., Schmidt [13], in 1992, also explored the idea of
harvesting energy using PVDF film, but in windmill applications. As a means of reducing the danger associated with conventional windmills containing large rotating blades,
three independent piezoelectric windmill designs oscillating at high frequency with small
3
amplitude were considered. Experimental testing of each windmill design showed power
levels in the microwatt to milliwatt range. The low power output and high costs of such
piezoelectric wind generators led to the conclusion that significant improvements to the
windmill design and reductions in cost of PVDF film were required in order to produce
practical piezoelectric windmills.
In the mid 1990’s, several researchers returned to the idea of harvesting energy in biological systems, specifically from human body motion. In 1995, Antaki et al. [14] published
the first work on harvesting energy from the human body using piezoelectric material by
embedding piezoceramic stacks into the sole of a shoe to scavenge energy during walking and running. The goal of the work was to investigate energy harvesting solutions for
providing power to electrically powered artificial organs. The design of the shoe harvester
focused on maximum energy transfer to the piezoelectric stacks during the heel-strike and
toe-off portions of gait, while retaining the ergonomics of the shoe. A hydraulic oscillator
was designed to increase the excitation frequency by exciting the stack harmonically during each step to increase the amount of strain induced in the piezoelectrics. Laboratory
simulation experiments on a 1/17 scale harvester yielded peak power outputs ranging
from 150 mW to 2.5 W depending upon body mass and gait.
Several researchers from the MIT Media Lab also investigated piezoelectric harvesting
from human body motion during the same time period. In 1996, Starner [15] published an
article outlining the energy expended during various daily activities including walking,
breathing, and finger and upper limb motion, as well as the energy available from body
heat and blood pressure. It was proposed that the energy scavenged from the body could
be used to power wearable computing devices. Several of the sources produced relatively
low amounts of available energy (<1.0 W) including breating, finger motion, and blood
pressure. Of the higher energy sources including body heat, walking, and upper limb motion, walking was determined to hold the most potential for a practical harvesting system.
Two forms of harvesting during walking were discussed including the integration of a conventional rotary electromagnetic generator into a shoe as well as the concept of embedding
piezoceramic devices into the sole of a shoe. Following the work of Starner [15], Kymissis
[16] further explored the concept of harvesting energy in shoes by fabricating and experimentally testing both piezoelectric and electromagnetic harvester designs. Three designs
were investigated, two piezoelectric systems including a PVDF device installed in the toe
4
area of a shoe and a lead zirconate titanate (PZT) ceramic device installed in the heel, as
well as an electromagnetic generator attached to the outer heel. Although experimental
results show that the electromagnetic generator was capable of harvesting two orders of
magnitude more energy than the piezoelectric designs, the obtrusiveness of the generator limits the practical implementation of the design. The piezoelectric prototypes, on the
other hand, were easily integrated into the shoe and were capable of harvesting around
1-2 mW of energy. Further details of the work on integrating piezoelectric harvesting into
shoes are provided in the Masters Thesis of Shenck [17] of the MIT Media Lab.
Also in the mid 1990’s, work was performed by Umeda et al. [18, 19] to explore the
fundamentals of impact energy conversion using piezoelectric materials. Umeda et al. [18]
first developed an equivalent circuit model to predict the response of a piezoelectric vibrator plate when impacted with a steel ball. Simulation results showed that an optimal load
resistance exists for maximum power transfer from the piezoelectric layers, and that much
of the potential energy from the steel ball is transfered back to the ball when rebounding
off of the vibrator plate, thus decreasing the efficiency of energy conversion. The following year, Umeda et al. [19] performed an experimental study to validate their previously
developed model, and also included rectification and storage of the electrical output of the
piezoelectric vibrator via a bridge rectifier and storage capacitor. Results of the experimental testing showed the ability of the piezoelectric vibrator to charge the storage capacitor
using impact energy, and that for a capacitor pre-charged above 5 V, an efficiency of 35%
was achieved.
In 1999, Goldfarb and Jones [20] performed research to investigate the efficiency of energy conversion using piezoelectric ceramics. Specifically, a piezoelectric stack consisting
of several piezoceramic discs arranged mechanically in series and electrically in parallel
was investigated both analytically and experimentally. Results of the study suggest that
the major challenge in harvesting mechanical energy using piezoceramic material is that a
majority of the mechanical input energy is stored in the piezoelectric and returned back to
the excitation source as mechanical energy. Additionally, results showed that maximum
conversion efficiency occurs at very low excitation frequencies, several orders of magnitude less than the resonance frequency of the stack.
At the start of the 21st century, a surge of research involving piezoelectric energy harvesting occurred. Since the year 2000, hundreds of papers have been published explor5
ing various aspects of vibration energy harvesting using piezoelectric materials including
the development of electromechanical models of piezoelectric harvesters, the efficiency of
various piezoelectric materials and harvester configurations, energy harvesting circuitry,
and various harvesting applications including microelectromechanical systems (MEMS),
self-powered sensors, and biological systems. Several articles highlighting the work in
piezoelectric energy harvesting in the last 10 years are reviewed in the following.
An incessant interest has been placed by the research community on the development
of electromechanically coupled models that can predict the behavior of a piezoelectric energy harvesting system with increasing accuracy. Many of the early modeling efforts utilized a simple model of an electromagnetic harvester developed by Williams and Yates [21]
in 1996. The model was developed to describe electromagnetic harvesting, therefore, its
use is highly inaccurate in modeling piezoelectric transduction. To improve upon the existing models, Roundy and Wright [22] and duToit et al. [23] both presented single-degree-offreedom (SDOF) lumped parameter models of piezoelectric energy harvesters. Although
the lumped-parameter solution showed improvement over the Williams and Yates model
[21], it is limited to describing a single vibration mode and lacks much of the information
necessary to accurately describe the coupled system. In an effort to further refine the piezoelectric harvester models, discrete Rayleigh-Ritz formulations (originally develped by Hagood et al. [24] for piezoelectric actuation) were developed for Euler-Bernoulli cantilever
beam harvesters by Sodano et al. [25] and duToit et al. [23, 26] and later improved upon by
Elvin and Elvin [27]. The Rayleigh-Ritz formulation provides a discretized approximation
(having finite degrees of freedom) of the distributed parameter system and yields more accurate approximations than SDOF models. More recently, Erturk and Inman [28, 29] have
presented the exact electromechanical solution of a cantilever piezoelectric energy harvester under harmonic base excitation based on Euler-Bernoulli beam assumptions. The
exact analytical solution provides the most accurate model of cantilevered piezoelectric
vibration energy harvesting. Additionally, Erturk and Inman [30–32] have also presented
improved approximate distributed parameter modeling of piezoelectric energy harvester
beams that can be used to model nonuniform beams where an exact solution does not
exist.
Significant interest has also been placed on improving the efficiency and practicality of
piezoelectric harvesting through the investigation of various piezoelectric materials as well
6
as physical configurations of vibration energy harvesters. The most common material used
in piezoelectric energy harvesting is lead zirconate titanate (PZT), which is a piezoelectric
ceramic. Although widely used in energy harvesting applications, PZT is extremely brittle, causing limitations in the strain that can be applied to the material without causing
damage. For this reason, researchers have investigated other, more flexible materials for
vibration harvesting. Sodano et al. [33, 34] compared the energy harvesting performance
of monolithic piezoceramic to more flexible piezoelectric composite and fiber-based deR
vices including Macro Fiber Composites (MFCs) and QuickPack
actuators. Lee et al. [35]
and Tylor et al. [36] both explored the use of PVDF film for energy harvesting applications. In general, piezoelectric ceramic devices exhibit large electromechanical coupling
but are susceptible to brittle failure, where more flexible devices have lower coupling but
increased compliance allowing for use in a wider range of potential applications.
Research has also been performed to investigate different physical configurations to
improve the usefulness of energy harvesters. Ng and Lio [37] presented a study in which
they compared the performance of a unimorph cantilever harvester to two bimorph cantilever harvesters, one with the piezoelectric elements connected in series, the other in
parallel. They found that each configuration provides optimal performance under a different excitation level, thus all configurations are useful. Mateu and Moll [38] performed
a similar study in which they compared a unimorph harvester to two bimorphs, one with
a substrate layer, the other without. They also compared a conventional rectangular cantilever configuration to a triangular cantilever and found that the triangular harvester is
capable of obtaining higher strains and providing larger power outputs. Kim et al. [39] developed a piezoelectric “cymbal” harvester design which is capable of withstanding large
loads while evenly distributing stress in the piezoelectric layers. Platt et al. [40] compared
the performance of a piezoceramic stack against a monolithic piezoceramic device of the
same geometry, finding that the two devices gave around the same power output, but the
matched load resistance is much lower and more manageable for the stack configuration.
Studies have also explored the ability to exploit nonlinear phenomenon to create novel energy harvesting configurations. Erturk et al. [41], Barton et al. [42], and Stanton et al. [43]
have all investigated the use of a bistable nonlinear magnetoelastic oscillator for energy
harvesting purposes where permanent magnets provide attractive and repulsive forces
causing nonlinear response in a cantilever beam. The investigations found that nonlinear
7
phenomenon allow for resonance behavior over a broad frequency range, improving upon
the limitations of linear systems that suffer from performance loss when excited away from
resonance.
The development of efficient circuits for energy harvesting applications has captured
the interest of numerous researchers in the field of piezoelectric energy harvesting. A key
element of any energy harvesting system is an electrical circuit that can condition and store
the harvested energy in an efficient manner. Some of the original work on adaptive and
efficient energy harvesting circuits for piezoelectric harvesting was performed by Ottman,
Hofmann, and Lesiuetre [44–46]. Their work involves improving the efficiency of energy
extraction through implementation of a switching DC-DC step-down converter after the
conventional bridge rectifier and smoothing capacitor. They found that the optimal duty
cycle of the switching converter changes significantly with excitation frequency. Lefeuvre et al. [47] later developed the concept of ‘synchronous electric charge extraction,’ a
self-adaptive circuitry in which extraction of the electric charge on a piezoelectric device
is synchronized with the system vibration in order to improve the efficiency of energy
transfer. The circuit contains a diode bridge rectifier and a flyback switching mode DCDC converter. In similar studies, Badel et al. [48], Guyomar et al. [49], and Lefeuvre et
al. [50] developed another synchronous technique called ‘synchronous switch harvesting
on inductor’ (SSHI), which contains an electronic switching device that is triggered on
the maximum and minimum displacements of the piezoelectric device. The switching
device and an inductor in series are placed either in series or in parallel with the piezoelectric before the rectifying diode bridge. More recently, Kong et al. [51] created a nonlinear impedance matching switching circuit that improves upon the circuit developed by
Ottman, Hofmann, and Lesiuetre [44–46] by eliminating some bulky components and providing both step-up and step-down ability by using a buck-boost converter operating in
discontinuous conduction mode. In addition to circuits being used to transfer energy between a piezoelectric harvester and a load, electronic circuits have also been developed to
help optimize energy transfer via frequency tuning. For example, Lallart et al. [52] developed a low-cost self-tuning technique containing both automatic frequency detection and
actuation in which a stiffness tuning technique is employed to alter the natural frequency
of a cantilever piezoelectric harvester to match the source vibration.
With decreases in the energy consumption of electronic devices, researchers are contin8
ually developing novel applications where piezoelectric energy harvesting can be used as a
power source, including MEMS, self-powered sensors, biological systems and unmanned
vehicles. Early work in piezoelectric harvesting for MEMS devices focused on cantilever
designs with similar aspect ratios to conventional macro-scale devices. Lu et al. [53] and
Jeon et al. [54] both fabricated and tested cantilever piezoelectric harvesters on the order
of 1 - 5 mm long and found resonant frequencies in the kHz range. The disadvantage of
these early designs is the impractically high resonance frequency. More recent research
in MEMS harvesting has focused on the development of novel geometries in an effort to
reduce the natural frequency to a practical level. Karami and Inman [55], for example,
have developed a “zig-zag” geometry that is capable of natural frequencies on the order
of hundreds of Hz for a device approximately 1 cm x 1 cm in size.
An attractive application of piezoelectric energy harvesting is in the development of
self-powered sensors that would otherwise depend on batteries as a source of power. Elvin
et al. [56, 57] developed self-powered sensor systems consisting of PVDF film used for both
strain sensing and damage detection purposes as well as energy harvesting to provide
power to the systems. Through experimental testing, it was found that the PVDF generator produced enough energy to allow radio frequency transmission of strain values to a
receiver. Roundy and Wright [22] created a small piezoelectric cantilever generator with
a volume of around 1 cm3 that was capable of generating around 400 µW and powering
a custom radio transmitter. Recently, Zhou et al. [58] developed a self-powered wireless
structural health monitoring (SHM) system. The sensor node uses a cantilever piezoelectric device to harvest vibration energy and a microcontroller along with a surface mounted
piezoelectric device to perform impedance-based SHM. When experimentally tested, the
energy harvesting system was capable of generating around 3 mW of power, sufficient to
run the SHM device once every two minutes.
Several studies have investigated the use of piezoelectric materials for harvesting energy in biological applications. Platt et al. [40, 59] developed an in vivo piezoelectric harvester and sensor to be used in self-powered total knee replacement units. The piezoelectric stacks installed in the unit are capable of sensing important phenomena in the
knee such as joint degradation and misalignment, and also harvesting adequate power
for the sensing circuity to operate and transmit data. Sohn et al. [60] investigated the
use of PVDF film for harvesting energy from fluctuating pressure sources such as blood
9
flow. Experimental testing of a prototype harvester found that when subjected to pressures
similar to human blood pressure, the device was capable of harvesting enough energy to
power a chip that could transmit data, such as DNA information, twice a day. Qi et al.
[61] performed research on a novel concept of integrating piezoelectric ceramic materials
onto stretchable, biocompatible rubber for use in implantable energy harvesting systems.
Unlike PVDF film, the piezoelectric rubber composites contain PZT which exhibits high
coupling, yet the composite retains flexibility. Although still in the concept phase, it is
suggested that this novel technology be used in biological applications as an implantable
power source.
Over the past decade, a vast amount of research has been performed on the topic of
vibration energy harvesting using piezoelectric materials. An overview of some of the
pertinent studies has been presented in this section. For a comprehensive review of the
previous literature on piezoelectric energy harvesting, the reader is referred to the various
review articles published on the topic. Sodano et al. [3] first published a review article
in 2004 highlighting the previous literature on piezoelectric vibration harvesting. In 2006,
Beeby et al. [1] published a review article highlighting previous works on piezoelectric,
electrostatic, and electromagnetic harvesting for micro-scale applications. The next year,
two review articles appeared. Priya [5] provided an overview of vibration harvesting using
piezoelectrics, and Anton and Sodano [4] published a review of piezoelectric energy harvesting from 2003-2006 as an update to the previous work by Sodano et al. [3]. Finally, in
2008, Cook-Chennault et al. [2] provided an overview of various energy harvesting sources
for MEMS applications including solar harvesting, thermal harvesting, and vibration harvesting using electrostatic, electromagnetic, and piezoelectric devices.
1.1.2
Multifunctionality in Energy Harvesting Systems
Conventional works on piezoelectric energy harvesting, such as those described in the
previous section, consider the harvesting device to simply attach to a host structure with
the sole purpose of harvesting energy. Although useful in many applications, this conventional approach for energy harvesting design is not always practical. Host structures
that are mass or volume critical, for example, may be intolerant to the addition of a bulky
energy harvester not originally included in the design of the structure. In order to address
this issue, a multifunctional approach can be considered for vibration energy harvesting
10
systems. The concept of multifunctionality aims to combine several functions into a single
material or device as a means of improving usefulness and optimizing the use of space and
weight. Multifunctional material systems have been of interest to the research community
for the past decade, and several approaches have been explored in the development of
multifunctional structures. Christodoulou and Venables [8] give a review of some of the
earlier efforts in multifunctional structures in which details are given on the development
of structural power material systems, autonomous sensing and actuating material systems,
electromagnetic multifunctional material systems, and survivable, damage-tolerant material systems. The goal of these various technologies is to integrate multiple features into
material systems to enable novel design concepts otherwise not achievable. A second review has been published by Aglietti [62] in which several multifunctional structures for
use in spacecraft and satellite applications have been outlined.
Of the four different classes of multifunctional material systems discussed by Christodoulou and Venables [8], structural power systems, which integrate structural function with
energy storage ability, are of most interest for energy harvesting applications. Some of the
original work on structural power systems involved the development of stiff, lightweight
structures with embedded batteries to store energy for use in spacecraft applications. In
1998, Lyman and Feaver [63] of Boundless Corp. introduced the PowerCoreTM device
which utilizes nickel-metal hydride (NiMH) rechargeable batteries as the core of a honeycomb composite structure for use in satellites. The battery core of the PowerCoreTM device
is fabricated from nickel foam with sintered active electrode materials, and the assembled
core is sandwiched between face sheets to form the composite panel. An effective specific
energy greater than 80 Wh/kg was reported for fabricated samples, however, challenges
involved with the development of large-scale devices were reported. Similar devices have
been reported in the literature in more recent years that make use of more efficient lithium
battery technology for energy storage in the core of honeycomb structures. In 2002, Marcelli et al. [64] of ITN Energy Systems, Inc. described their LiBaCore device which contains
thin-film lithium batteries embedded into the core layers of a conventional aluminum honeycomb composite. Thin-film lithium batteries offer superior performance over the NiMH
batteries used by Lyman and Feaver [63], and it was reported that an optimized design
could yield specific energy on the order of 350 Wh/kg. Fabrication of the LiBaCore devices, however, showed that deposition of thin-film lithium batteries onto an aluminum
11
core proved difficult. Initial studies on structure power honeycomb materials have focused
on the energy storage aspect of the design. More recently, however, Schwingshackl et al.
[65, 66] have investigated the structural properties and dynamic response of honeycomb
structures with integrated batteries. Schwingshackl et al. [65] first analytically explored
ten different honeycomb or corrugated core style composite designs containing embedded
batteries in the core structure and an optimal design was selected. Results of the analytical study and finite element simulations showed that the multifunctional design with
integrated energy storage provided similar dynamic behavior compared to conventional
honeycomb core sandwich panels. In a continuation study, Schwingshackl et al. [66] fabricated and experimentally tested the optimal design found previously [65], and showed
that the multifunctional structure again exhibited similar dynamic response compared to
a conventional design, but with the added functionality of energy storage.
In an effort to introduce multifunctional structures into unmanned vehicle systems,
Thomas and Qidwai [67–72] of the Naval Research Laboratory have investigated the concept of the ‘structure-battery.’ In 2002, Thomas and Qidwai [67] first introduced the concept of the multifunctional structure-battery in which polymer-lithium ion battery layers
are used to both store energy and support aerodynamic loads in an unmanned aerial vehicle (UAV) system. Formulations for the change in flight endurance of a UAV with an
integrated structure-battery considering changes in the battery mass, structure mass, and
available energy from the battery were given. Design indices were derived as a metric
for ranking different multifunctional composite designs based on both the system-level
performance of the device as well as the constituent properties and geometry of the components. The design and fabrication of structural batteries consisting of polymer-lithium
ion battery cells, packaging materials, and optional structural additives was outlined. Finally, the Black-Widow flying-wing UAV developed by AeroVironment, Inc. was investigated in this study and it was proposed to replace the original primary (non-rechargeable)
battery with multifunctional structural batteries installed in the wings of the aircraft. Preliminary battery testing and fabrication of embedded batteries was performed, however,
no flight testing occurred. The design indices introduced in [67] were further developed
by Thomas and Qidwai in [68]. In a subsequent study, Thomas and Qidwai [69] investigated the use of structure-batteries in the WASP flying-wing UAV, also developed by
AeroVironment, Inc. Several designs were considered using various structural additives
12
and multiple polymer-lithium ion batteries. Additionally, the shear strength of several
polymer-lithium ion batteries was obtained experimentally as a means of quantifying the
load bearing ability of the batteries. Preliminary analysis and flight testing showed that
the multifunctional design with integrated structure-batteries is capable of outperforming
the conventional non-multifunctional design.
More recently, Thomas and Qidwai [70–72] have investigated the use of structurebatteries in unmanned underwater vehicles (UUVs). Qidwai et al. [70] first described the
concept of integrating multifunctionality into UUVs by replacing existing conventional
batteries with structure-batteries and relocating them into the skin or hull of the vehicle to reduce mass and make additional space available within the vehicle. The design
and fabrication of structure-batteries specifically developed for marine systems, containing lithium-ion batteries embedded within fiber-reinforced polymer layers, was outlined,
and three unique multifunctional designs were produced. Continuing the work, Rohatgi
et al. [71] presented the experimental evaluation of the three fabricated structure-batteries.
The multifunctional composites were tested both mechanically via static three-point bend
testing, and electrically through charge/discharge cycling of the battery layers. Experimental results showed successful integration of the polymer-battery layers into the fiberreinforced structural layers. In their latest work, Qidwai et al. [72] investigated the electrical performance of the various structure-battery designs while under load. Both static
three-point bend testing and hydrostatic loading were considered. Load-deflection measurements during three-point bend testing show that slight thickness changes are observed
and reflected in the load data when charge cycling the composites. Additionally, during
hydrostatic testing, the charge/discharge performance of the composites is found to decrease slightly with increasing load.
The development of robust, load bearing materials that can be directly used as battery component layers (i.e. packaging, anode, cathode, or electrolyte) to provide structural
function integral to the battery itself has also been investigated in the literature. Snyder et
al. [73] explored various solid polymer materials for use as the electrolyte layer in a structural battery. A key component of multifunctional electrolytes is the ability of the electrolyte to exhibit both high ion conductivity and good mechanical integrity. Both electrical
testing of the ion conductivity of the electrolyte materials via electrochemical impedance
spectroscopy, as well as mechanical testing of the stiffness of the electrolytes through com13
pression testing and dynamic mechanical analysis (DMA) were conducted. It was found
that generally, increases in mechanical performance are accompanied by decreases in ion
conductivity, thus an optimal design exists for each specific application. In a continuation study, Snyder et al. [74] studied the use of copolymer electrolytes, as opposed to the
homopolymers investigated previously, with one monomer selected to promote structural
behavior and the other monomer selected to promote ion conduction. Results of the study
showed that the use of a copolymer design exhibits favorable ion conductivity and mechanical strength over an optimized homopolymer design. Snyder et al. [75] have also
examined the usefulness of several commercially available carbon fibers, fabrics, and papers for use as the anode layer in structural batteries. The capacitance as well as the energy
storage ability of each material was experimentally investigated, and it was concluded that
the use of commercially available materials as anodes in power storage devices is viable.
In addition to the work of Snyder et al., Liu et al. [76] have also investigated several different battery components for use in multifunctional structural batteries. Various materials for use as the battery anode, cathode, and electrolyte were considered. A prototype
structural battery was fabricated and electrical charge/discharge measurements as well as
mechanical three-point bending tests were performed. The prototype was shown to simultaneously support load and store energy, however, both the strength and energy density
of the battery showed room for improvement in future designs.
Neudecker et al. [77] have considered another approach to creating structural batteries
in which thin-film batteries are deposited directly onto thin fiber substrates, such as carbon, glass, silicon carbide, or metals, to form what they call PowerFibers. PowerFibers are
made using a rotational shadow mask system that allows sequential vacuum deposition
of each thin-film battery component layer. Several combinations of substrate and thin-film
battery component layers were investigated, and both single fiber PowerFibers as well as
composite PowerFibers composed of multiple individual fibers embedded into an adhesive matrix were fabricated. Experimental charge/discharge testing was performed on
the fabricated devices and results showed excellent cycle life and discharge rate capability. The challenge facing the PowerFiber technology is the relatively low storage capacity
of individual fibers. Lin and Sodano [78–81] have considered a similar approach to creating multifunctional structure power systems in which a novel piezoelectric structural
fiber has been developed. The fiber contains a conventional conductive fiber core which
14
is then coated with piezoelectric material and an outer electrode material. The fiber core
acts as both a structural stiffening layer as well as an inner electrode, and the piezoelectric
layer allows the active fiber to posses sensing, actuation, and energy harvesting ability.
A micromechanics model of the active coated fiber was first developed and validated using a finite element simulation [78]. Next, fabrication of an active fiber containing a silicon
carbide core coated with barium titanate piezoelectric material and an outer electrode composed of silver paint was presented [79]. Electromechanical testing of the fiber was performed using a atomic force microscopy to measure the longitudinal free strain of the fiber
under the application of an electric field, validating the previously developed model [78].
Lin and Sodano [80] then repeated the electromechanical testing of the active fiber in the
presence of an outer polymer lamina layer in order to validate the concept of embedding
the fibers in a matrix to create a multifunctional composite material. Results of the testing
showed good behavior in the multifunctional fiber, thus proving the ability of the fibers
to be embedded into structural composites. Lastly, Lin and Sodano [81] investigated the
energy storage ability of the fibers. The dielectric properties of the active barium titanate
shell were studied in order to create a structural fiber capacitor. Results of experimental
testing showed that the fibers are capable of storing harvested energy.
Multifunctional solar energy harvesting systems have also been investigated in the literature and provide a means of combining structure and energy harvesting capabilities in
a single device. The use of flexible thin-film solar panels permits the integration of solar
harvesting into load bearing structures, and when combined with thin-film battery technology, offers the ability to simultaneously harvest and store energy. In 2000, Clark et al.
[82] of ITN Energy Systems, Inc. explored the combination of thin-film photovoltaics along
with thin-film lithium batteries to create a flexible, lightweight laminate, called the Flexible
Integrated Power Pack (FIPP), that is capable of both harvesting and storing energy. The
FIPP was proposed for use as the skin of a satellite, providing structural function in addition to energy harvesting and storage ability. A prototype FIPP was fabricated and tested
in the laboratory, and test results showed that energy can be successfully transfered from
the thin-film solar panels to the thin-film batteries. Similar research has been presented by
Raffaelle et al. [83–85] in which thin-film solar panels are again combined with thin-film
lithium batteries to create power systems for satellites. An initial prototype was built utilizing conventional polycrystalline solar panel and lithium coin battery technology, and
15
the prototype was launched on the Starshine 3 satellite in 2001. Several other prototype
devices containing thin-film batteries were also fabricated and bench tested and found to
successfully harvest and store energy. More recently, Maung et al. [86] introduced the“cocuring” manufacturing process in which thin-film solar panels are directly integrated onto
carbon-fiber-reinforced epoxy composites to combine harvesting and structural function
in a single device. The process involves simultaneously curing a prepreg epoxy composite
and bonding the solar panel to the composite during the curing process. After fabrication, the electrical performance of a multifunctional solar harvesting device was examined
under cyclic tensile loading, and it was found that significant degradation in solar panel
performance occured above 1.0% strain. Dennler et al. [87] revisited the idea of combining energy harvesting and energy storage and proposed the concept of directly coupling
a flexible solar panel with a conventional lithium-polymer battery. Several novel types of
flexible solar panels were developed and considered for the device, and a unique flexible
interconnection layer was proposed for electrical connection of the solar panel and battery.
The interconnection layer consists of embedded microelectronics on a polymer substrate
to create a flexible power management circuit layer that can be directly integrated with
thin-film solar panels and batteries. A prototype system was built and shown to be capable of providing power to a small temperature sensor. Kim et al. [88] expanded upon the
work of Dennler et al. [87] and fabricated and tested a solar power laminate consisting of
thin-film solar panels and thin-film flexible lithium-based batteries connected with a flexible interconnect circuit. The work focused on the development of the copper nano-ink
piezoelectric inkjet printing method for creating flexible circuits. Once manufactured, the
energy harvesting performance of the multifunctional solar battery composite was tested
under tensile loading and found to operate up to a strain of 0.45%, at which the battery
exhibited failure.
The previous research described above has investigated several aspects of multifunctional structure power systems. The next step in the development of these systems is to
integrate structural function with energy storage and energy generation in a single multifunctional structure. Preliminary work by Dennler et al. [87] and Kim et al. [88] has examined solar energy harvesting multifunctional structures with integrated energy storage.
The goal of the work presented in this dissertation is to design, fabricate, and characterize
a multifunctional piezoelectric vibration-based energy harvesting system with integrated
16
energy storage and structural load bearing ability.
1.1.3
Piezoelectric Energy Harvesting in Unmanned Aerial Vehicles
A wide variety of applications can take advantage of the benefits offered by energy harvesting systems. In this dissertation, a small unmanned aerial vehicle platform is investigated as an example application of multifunctional piezoelectric vibration energy harvesting. Energy harvesting in aircraft has been of interest to the research community for several
decades. The majority of the early work on energy harvesting for small aircraft focuses on
photovoltaic harvesting of solar energy as a means of providing propulsive power for the
aircraft. In 1984, Boucher [89] published a review article highlighting all of the previous solar powered flights at the time, dating back to 1974. A total of seven solar powered flights
were reviewed, most of which were performed using small manned aircraft. In recent
times, a focus has been placed on the development of small unmanned aircraft including
both UAVs and smaller micro air vehicles (MAVs). Pines and Bohorquez [90] discuss the
drive towards reducing the size of unmanned aircraft while highlighting the challenges
facing the development of future MAVs. The topic of solar energy harvesting to provide
power for flight remains a focus of many research efforts on the development of future
unmanned aircraft. Researchers are continually developing novel photovoltaic materials
with increased energy conversion efficiencies and decreased weight that can be used on
small, lightweight aircraft. Several other modes of energy harvesting in UAVs have also
been investigated in the literature. Thomas et al. [91] discuss the potential of harvesting
solar, wind, thermal, and electromagnetic radiation energy on small unmanned vehicles
to supplement the vehicle power supply in order to increase flight endurance. The energy available from each potential source is estimated and the current energy conversion
devices are discussed.
In the past five years, piezoelectric vibration harvesting has gained interest for use in
UAVs as a means of harvesting energy from vibrations persistent during the flight of the
aircraft in order to create local power sources for low-power electronics. Previous research
studies have investigated several aspects of harvesting vibration energy in UAVs using
piezoelectric material. De Marqui Jr. et al. [92] explored the use of piezoelectric harvester
plates on the wings of UAVs for harvesting vibration energy during flight. An electromechanically coupled finite element model was developed for piezoelectric plates that can
17
be used to predict the energy output of harvester plates in UAVs. Both theoretical and
experimental validation was performed on the finite element model against the analytical solution presented by Erturk and Inman [28] and against experimental test results of
a cantilever harvester plate also presented by Erturk and Inman [29]. Additionally, a case
study was presented in which theoretical optimization was performed on a notional UAV
wing spar in which the optimal amount of embedded piezoelectric material is installed
into the spar without exceeding a given mass limitation. Results of this work proved the
effectiveness of the finite element model in predicting the behavior of cantilever piezoelectric harvester plates as well as the ability to determine an optimal amount and optimal
location of piezoelectric material in a UAV wing spar. In a continuation study, De Marqui
Jr. et al. [93] coupled their previously developed finite element model with an unsteady
aerodynamic model and investigated the effects of airflow excitation on the power output of cantilever piezoelectric energy harvester plates which simulate the wing of a UAV.
Various airflow speeds as well as electrode configurations of the piezoelectric (segmented
electrodes vs. continuous electrodes) were analyzed. Results showed that energy harvesting performance is optimized for airflow speeds near the flutter speed, although impractical for actual flight situations, and that segmented electrodes can be useful in eliminating
voltage cancellation experienced under torsional vibrations.
Perhaps one of the more promising uses of piezoelectric energy harvesting in aircraft
is in providing power to low power sensor nodes. Several researchers have investigated
the possibility of utilizing vibration energy harvesting in aircraft to create self-powered
structural health monitoring nodes that are capable of assessing the condition of critical
aircraft components during flight. Durou et al. [94] investigated the combination of thermal and vibration harvesting to power SHM sensors on large-scale aircraft. Simulations
were performed to predict the power output of both harvesting devices as well as the energy storage performance of a supercapacitor used to store the harvested energy. Results
of the simulations suggest that vibration harvesting alone is not suitable to directly power
the SHM node, but that it acts as a valuable supplement to the solar harvesting system
as the output of the piezoelectric harvester is not environmentally dependent. Moss et al.
[95] explored the concept of broadband vibration energy harvesting in aircraft to provide
power to SHM sensor nodes. Considering the variability in excitation characteristics during the flight of an aircraft, it was proposed to develop an energy harvesting device with
18
relatively broadband response. A fairly large vibro-impacting device was created and experimentally tested in which a vibrating mass repeatedly impacts and excites a piezoelectric beam. The mass responds to a range of frequencies, thus the overall device is capable
of harvesting energy in a broadband sense. Results of experimental testing showed that
the harvester responds in the range of 29 - 41 Hz and is capable of harvesting up to 12 mW
of power under 0.5 g excitation. Featherston et al. [96] performed research on vibration
energy harvesting for SHM sensors on both large scale commercial aircraft as well as small
UAVs. A test panel was created using an aluminum plate clamped on all edges in a test
frame and included four surface mounted piezoelectric devices for energy harvesting. The
panel was excited using a shaker with a stinger connection at the center of the plate for various frequencies corresponding to different air speeds, altitudes, and panel locations along
the wing. Results of experimental testing showed that the maximum power harvested by
a single device was 23.5 mW. In order to operate a SHM sensor node requiring 50 mW of
power, it was concluded that approximately 71 cm2 of piezoelectric material is required
under optimal conditions.
Many novel UAVs include a multitude of functions requiring a large amount of technology to be placed into a small platform. The concept of multifunctional piezoelectric
energy harvesting has been investigated in the literature as a means of creating a noninvasive power source that can be used, for example, to power on board self-powered
electronic systems. Apker et al. [97] suggested the use of piezoelectric film on the wings
of a small ornithopter flapping-wing MAV to perform aeroelastic energy harvesting during flight. The piezoelectric material would serve as both a load bearing wing surface as
well as an energy harvesting system, hence its multifunctional use. Although no modeling
efforts were pursued and no physical device was constructed, the work presents a novel
concept in multifunctional harvesting for UAVs. Another novel concept for multifunctional piezoelectric vibration harvesting in UAVs involves the integration of piezoelectric
material into the landing gear of the aircraft. Magoteaux et al. [98] discussed the concept
of using either cantilevered or curved piezoelectric beams as the landing gear for UAVs.
Piezoelectric landing gear can be used not only for conventional takeoff and landing purposes, but to harvest energy and recharge the power source during a mission if the aircraft
is perched on a vibrating structure. In their work, Magoteaux et al. provided a comparison
of the harvesting ability of a traditional piezoelectric cantilever to a curved piezoelectric
19
configuration and found that while having potential to provide larger amounts of power
(≈ 5 mW at ≈ 1.0 g), the resonance frequency of the cantilever configuration is impractically low (≈ 3.5 Hz) compared to the curved configuration (≈ 120 Hz), thus the curved
device, which provides around 3 mW, is best suited for landing gear applications. In a similar study, Erturk et al. [29] investigated a novel L-shaped configuration for piezoelectric
energy harvesting and suggested its use as UAV landing gear. The L-shaped beam mass
structure can be tuned to exhibit a two-to-one internal resonance phenomenon where the
second resonance frequency is roughly two times the first, and has potential for giving
larger power outputs compared to traditional cantilever configurations. Additionally, an
L-shaped device is well suited geometrically for use as landing gear. Erturk et al. compared the energy generation capabilities of their proposed L-shaped device to the curved
harvester proposed by Magoteaux et al. [98] and found that for devices with similar dimensions and mass, the L-shaped device outperforms the curved device.
The integration of vibration energy harvesting systems into unmanned aerial vehicles
is relatively novel and has the potential to provide environmentally independent power
sources for on board sensors of the next generation of aircraft. A multifunctional approach
is paramount as the size of UAVs continues to decrease and the technological demands
steadily increase. The work presented in this dissertation aims to integrate a multifunctional piezoelectric vibration-based energy harvester into a UAV platform to simultaneously harvest and store energy in order to provide power for a low-power sensor node
and to support structural loads during flight.
1.2
Thesis Overview
1.2.1
Research Objectives
It is clear from the previous research highlighted in the Literature Review that piezoelectric
vibration-based energy harvesting is an attractive concept for providing power to various
low-power electronics. The use of energy harvesting provides the ability for electronic devices to become self-powered and no longer rely on batteries which require replacement.
Self-powered devices have the ability to be embedded into a host structure as accessibility is no longer required. Conventional piezoelectric energy harvesting systems, however,
suffer from the large mass density of piezoceramic materials as well as the intrusiveness
20
associated with adding a harvesting system to a host structure. The concept of multifunctionality can be utilized in an effort to overcome the drawbacks of classical piezoelectric
energy harvesting design, which are prominent in applications sensitive to added mass
and volume. The combination of multiple functionalities into a single device can allow
piezoelectric vibration harvesting to become useful in many applications where otherwise
impractical. Previous research has investigated various aspects of multifunctional material
systems including the combination of structural function and energy storage in a single device, and the combination of energy harvesting and energy storage in a single device. The
goal of the research presented in this dissertation is to expand upon the current concept of
multifunctionality in energy harvesting systems and create a novel multifunctional piezoelectric vibration energy harvesting system. The self-charging structure multifunctional concept created in this dissertation combines piezoelectrics with thin-film batteries to create
devices capable of simultaneously harvesting energy, storing the harvested energy, and
supporting structural loads. Multifunctional vibration harvesting in unmanned aerial vehicles is investigated as a case study in which self-charging structures are used in a mass
and volume critical application.
1.2.2
Chapter Summaries
An introduction to the topics of piezoelectric vibration energy harvesting and multifunctionality is given in Chapter 1. Motivation behind the development of piezoelectric energy
harvesting systems capable of providing power to low-power electronic devices is given.
A literature review is presented in which previous works in the fields of vibration energy
harvesting, multifunctionality in energy harvesting systems, and energy harvesting in unmanned aerial vehicles are highlighted. A brief history of piezoelectric energy harvesting
is given in which the early works in the literature are reviewed along with some of the
more prominent studies conducted in the 21st century. The concept of multifunctionality
in energy harvesting systems is also reviewed including works on multifunctional structural power systems as well as photovoltaic harvesting and storage systems. Lastly, a
review of the previous work on vibration energy harvesting in unmanned aerial vehicles
is given.
Chapter 2 presents the concept of multifunctional self-charging structures developed in
this dissertation along with an electromechanical model used to predict the behavior of
21
the devices. A brief introduction into piezoelectricity and the concept of piezoelectric energy harvesting is discussed. The design and operational principles behind self-charging
structures are given. An experimentally validated electromechanical assumed modes formulation developed by Erturk and Inman [30–32] is employed to model the cantilever
piezoelectric harvesters subjected to harmonic base excitation. Details of the model are
presented including modeling assumptions, device configuration, and formulations for
both the series and parallel electrode connections of the piezoelectric layers of the bimorph
harvester. The assumed modes model can be used to predict the coupled vibration and
voltage response of the harvester.
Experimental validation of the electromechanical model presented in Chapter 2 as well
as the self-charging structure concept is given in Chapter 3. Performance evaluation of the
thin-film batteries used in self-charging structures is given. Details of the material selection
and fabrication of a prototype self-charging structure are presented. Experimental testing
is performed on the fabricated device in order to validate the model through electromechanical frequency response function measurements. Additionally, charge/discharge experiments are conducted to prove the ability of the structures to simultaneously generate
and store electrical energy in a self-charging manner.
Chapter 4 presents an evaluation of the strength properties of prototype self-charging
structures. Conventional three-point bend testing is first performed to evaluate the static
strength of the fabricated devices and results of the experimental testing are reviewed.
Dynamic strength testing is then performed to evaluate the ability of the harvesters to
support dynamic loading. Considering the variety of piezoelectric materials available for
use in self-charging structures, a separate static strength analysis is then performed on an
assortment of commercially available piezoelectric materials and experimental results are
summarized.
Multifunctional vibration energy harvesting in unmanned aerial vehicles is investigated as a case study in the remaining chapters. Chapter 5 presents the flight endurance
modeling of electric UAVs with on board energy harvesting capability. A flight endurance
model recently presented by Thomas et al. [91] is used to describe changes in the flight
endurance of a UAV based on the power available from added harvesters and the mass of
the added harvesters. Details of the derivation of the model are reviewed. The formulation
is applied to an EasyGlider remote control foam hobbyist airplane, which is selected as the
22
test platform in this work, and theoretical predictions are made to determine the effects
of adding piezoelectric energy harvesting to the aircraft. An experimental case study is
then performed in which data recorded during flight of the EasyGlider aircraft with piezoelectric fiber based devices installed in the wing spar is used to update the simulations. Finally, the flight endurance formulation is used to predict the effects of adding self-charging
structures to the EasyGlider aircraft. Results of the flight endurance modeling are used to
support the concept of multifunctional vibration harvesting in UAVs.
Chapter 6 presents the electromechanical modeling of an energy harvesting wing spar
with embedded self-charging structures. The model is based on the previously described
assumed modes formulation given by Erturk and Inman [30–32] and used in Chapter 2.
Details of the derivation of the model are first given in which both the series and parallel
connection of the piezoelectric layers are considered. Experimental testing is performed on
a representative aluminum wing spar with embedded piezoelectric and thin-film battery
layers in order to validate the model. Modeling simulations are compared to experimentally measured frequency response functions for validation purposes.
Chapter 7 presents the fabrication and testing of a realistic fiberglass UAV wing spar
with embedded self-charging structures for use in the EasyGlider aircraft. Details of the
fabrication of the multifunctional energy harvesting wing spar are given. Experimental
testing in which the cantilever wing spar is subjected to harmonic resonance excitation is
performed to evaluate the energy harvesting performance of the spar and to confirm the
ability of the spar to operate in a self-charging fashion. Finally, flight testing is performed
in which the fiberglass spar is used in the EasyGlider aircraft and the power output of the
multifunctional spar is measured during flight.
The final chapter presents a summary of the results of the research performed in this
dissertation. The major contributions of this work to the research community are described. Lastly, recommendations for future work are presented in which the advances
made by the work in this dissertation can be extended in the future.
23
C HAPTER
2
M ULTIFUNCTIONAL S ELF -C HARGING
S TRUCTURE C ONCEPT AND
E LECTROMECHANICAL M ODELING
T
HIS
chapter introduces the multifunctional self-charging structure concept that is ex-
plored throughout this dissertation. A brief overview of piezoelectricity and piezo-
electric energy harvesting is first given. Multifunctionality in vibration energy harvesting
is then explored through the development of the self-charging structure concept. The concept involves the combination of energy harvesting, energy storage, and load bearing ability in a single composite harvesting device through the direct integration of piezoelectric
material with novel thin-film batteries. The application of an approximate coupled electromechanical model based on the assumed modes method that can be used to predict the
vibration and voltage response of self-charging structures under harmonic base excitation
is discussed. The model is based on Euler-Bernoulli assumptions for thin beams and is
defined for both the series and parallel electrode connections of the bimorph self-charging
structures. Energy formulations are first derived for the potential energy, kinetic energy,
internal electrical energy, and work of non-conservative forces. The assumed series solution consisting of a finite number of trial functions multiplied by generalized coordinates
is then substituted into the energy expressions. The extended Hamilton’s principle is then
applied, which leads to the coupled electromechanical Lagrange equations. Solution of the
Lagrange equations yields the governing mechanical and electrical equations used to describe the system. Finally, the displacement - to - base acceleration and voltage - to - base
acceleration frequency response functions (FRFs) can be defined by assuming harmonic
base excitation imposed on the cantilever self-charging structures.
24
2.1
Overview of Piezoelectric Energy Harvesting
In this section, a brief introduction to piezoelectricity is first given and the most common
form of the piezoelectric constitutive equations is presented. An overview of piezoelectric
energy harvesting is then given in which the typical cantilever harvester configuration is
reviewed.
2.1.1
Introduction to Piezoelectricity
Piezoelectricity was first discovered in 1880 by Pierre and Jacques Curie when they found
that certain crystals, most notably quartz and Rochelle salt, produced a surface charge under a compressive load. This generation of electric charge under mechanical loading is
known as the direct piezoelectric effect. One year later, the opposite (or converse) piezoelectric effect, where an induced voltage will cause mechanical deformation, was mathematically proven by Gabriel Lippman and later experimentally observed by the Curie
brothers. The natural crystals that were initially discovered exhibited weak coupling between the mechanical and electrical domains. More recently however, synthetic piezoelectric materials have been created with increased coupling that enable the use of piezoelectricity in practical applications. The piezoelectric effect exists in several crystalline
materials due to the polarity of the unit cells within the material. This polarity leads to
the production of electric dipoles in the material which give rise to the piezoelectric properties. The application of mechanical strain causes rotation of the dipoles, leading to an
apparent charge flow that can be measured as current by placing electrodes on opposite
faces of a piezoelectric material (direct piezoelectric effect). Similarly, the application of a
voltage across the material will cause rotation of the dipoles which results in an induced
strain in the material (converse piezoelectric effect).
Both the direct and converse piezoelectric effects can be described mathematically through the piezoelectric constitutive equations, given in their most common form as (see Appendix A for a formal treatment of the piezoelectric constitutive equations)
  
 
0
S
sE d T

=
D
d εT E
(2.1)
where S is the mechanical strain, D is the electric displacement, T is the mechanical stress,
E is the electric field, sE is the mechanical compliance (reciprocal of the elastic modulus)
25
measured at constant electric field (denoted by the superscript E), d is the piezoelectric
strain constant, and εT is the dielectric permittivity measured at zero mechanical stress
(denoted by the superscript T ).
The electromechanical coupling of piezoelectric materials allows their ability to act as
both sensors when operating in the direct effect, and as actuators when operating in the
converse effect, thus piezoelectric material is a versatile solution in many applications. A
wide variety of piezoelectric devices are available commercially. The most common piezoelectric material is lead zirconate titanate, a piezoelectric ceramic known as PZT. Piezoelectric ceramics are widely used, however, they are extremely brittle. In an effort to create
flexible piezoelectric material, several piezoelectric fiber-based materials have been developed. These materials contain thin extruded strands of piezoceramic fiber embedded in
an epoxy matrix with flexible surface electrodes. Piezoelectric fiber-based materials have
the advantage of being flexible and can be mounted on curved surfaces, however, their
coupling is typically weaker than that of monolithic ceramics. Additionally, piezoelectric
polymer material, most notably polyvinylidene fluoride or PVDF, provides the highest degree of compliance but the weakest coupling. The wide range of piezoelectric materials
allows for their use in a variety of diverse applications.
2.1.2
Introduction to Piezoelectric Energy Harvesting
Energy harvesting is the process of converting ambient energy surrounding a system into
electrical energy, which is typically used to power an electronic device. Piezoelectric energy harvesting involves the placement of a piezoelectric device in a vibration rich environment to utilize the electromechanical coupling exhibited by piezoelectric materials in
order to create electrical energy. The most common piezoelectric energy harvesting configuration is the cantilever harvester subject to base excitation as shown in Fig. 2.1. Optimal
performance of the energy harvesting system is obtained when the frequency of ambient
vibration closely matches the resonant frequency of the harvester beam. Under this condition, maximum vibration energy is transfered to the harvester and converted to useful
electrical energy. Deviations from resonance typically cause significant decreases in performance, therefore, tuning the system (either passively or actively) is critical. Researchers
have begun to investigate alternative configurations for piezoelectric energy harvesting,
such as nonlinear configurations using magnetoelastic nonlinearity [41–43], in an effort to
26
Voltage Output
Voltage
Piezoelectric
Cantilever Beam
Vibrating
Base
0
Time
Figure 2.1: Schematic of piezoelectric cantilever harvester configuration subject to harmonic base excitation and its voltage output.
create broadband harvester designs as well.
In general, the energy delivered by piezoelectric harvesters is in the form of highvoltage, low-current alternating current (AC) signals, as shown in Fig. 2.1, where the majority of small electronics to be powered require low voltage direct current (DC) power
supplies (piezoelectric harvesting is most often used in powering small electronics such
as sensor nodes). Appropriate electronic circuitry must, therefore, be used to condition
the piezoelectric output to a usable form. Basic conditioning circuitry typically contains a
full bridge rectifier and smoothing capacitor to perform AC/DC conversion as well as a
simple storage medium. Typically, the instantaneous energy produced by a piezoelectric
harvester is inadequate for powering electronics, therefore, a storage medium (such as a
battery or capacitor) is often used to temporarily store the harvested energy before use.
In this case, an operational duty cycle can be defined in which the frequency of operation
of the device to be powered is described based on the amount of harvested power and
the power required to operate the device. The power output of a piezoelectric harvester
depends significantly on the impedance of the circuit or load connected to the piezoelectric device. A matched load exists for each piezoelectric device as well as for each excitation frequency. Several researchers have investigated the design of optimized circuits
to improve the efficiency of energy extraction [44, 47, 52]. Recently, the development of
impedance matching circuitry in which the circuit and storage medium appear as a pure
matched resistive load for maximum power transfer has been investigated as well [51].
The topic of piezoelectric energy harvesting has attracted much attention throughout
the research community in the past decade. Several aspects of piezoelectric energy harvesting have been investigated including the development of novel piezoelectric materi27
als, mechanical configurations to optimize power output, electronic control and harvesting
circuitry, and mathematical models used to predict the response of piezoelectric systems.
Several review articles have been published highlighting the major work in the field [2–5].
2.2
Self-Charging Structure Concept
Conventional piezoelectric energy harvesting systems are designed to be added to a host
structure in order to harvest ambient energy, but often cause undesirable mass loading
effects and consume valuable space. In order to improve the functionality and reduce
the adverse loading effects of traditional piezoelectric harvesting approaches, a multifunctional energy harvesting design is proposed in this dissertation in which a single device
can generate and store electrical energy and also carry structural loads. The proposed selfcharging structures, shown in Fig. 2.2, contain both power generation and energy storage
capabilities in a multilayered, composite platform consisting of active piezoelectric layers
for scavenging energy, thin-film battery layers for storing scavenged energy, and a central
metallic substrate layer. It should be noted that the orientation of the piezoelectric and
thin-film battery layers is arbitrary and that the layers may be interchanged depending on
the application. The operational principle behind the device involves simultaneous generation of electrical energy in the piezoelectric layers when subjected to dynamic loading
causing deformations in the structure, as well as energy storage in the thin-film battery
layers. Energy is transferred directly from the piezoelectric layers through appropriate
conditioning circuitry to the thin-film battery layers, thus a single device is capable of both
generating and storing electrical energy. Additionally, self-charging structures are capable
of carrying loads as structural members when embedded into a host system due to the
stiffness of the composite device. The ability of the device to harvest energy, store energy,
and support structural loads provides true multifunctionality.
The fruition of the self-charging structure concept is mainly attributed to the development of novel thin-film battery technology which allows for the creation of thin, lightweight, and flexible batteries. As discussed in Subsection 2.1.2 most piezoelectric energy harvesters do not produce adequate energy levels for immediate use, thus the energy must
be accumulated for some time prior to use. Conventional energy storage devices, such as
capacitors and traditional rechargeable batteries, are large, bulky devices which add signif-
28
Energy Flow
Piezoelectric Layer
Thin-Film Battery Layer
Harvesting
Circuitry
Substrate Layer
Figure 2.2: Schematic of self-charging structure.
icantly to the overall mass and volume of the harvesting system. Additionally, traditional
storage devices are not suitable for direct integration into the active element of an energy
harvesting system as their mass, volume, and rigidness would hinder the ability of the
device to harvest energy. Mechanical failure is also a concern with conventional storage elements as they may fail under the loads applied to the harvester. Thin-film lithium-based
batteries provide a viable solution for self-charging structures. The batteries are flexible
and have a typical thickness on the order of less than a millimeter, mass of around 0.5
grams, and capacity in the milliamp-hour range. The small capacities of these batteries are
a good match for the low electrical output levels associated with piezoelectric energy harvesting, and their thin, lightweight platform is ideal for direct integration into piezoelectric
harvesters to create compact, multifunctional self-charging structures.
A novel aspect of the self-charging structure concept is that the composite harvester
can be used as a load bearing member in a host structure. The harvester can be embedded
in or used in place of an existing component, thus reducing the total mass added to the
host structure. A potential application that can benefit from the energy harvesting, energy
storage, and load bearing capabilities of self-charging structures, for example, is in powering remote, low-power sensors in UAVs. Such multifunctional composite harvesting devices can be embedded into the wing spar of a UAV, as shown in Fig. 2.3, with the goal of
providing a local power source for remote low-power wireless sensors such as accelerometers, structural health monitoring nodes, or even low-power imaging devices or cameras.
Providing a local power source composed of both harvesting and storage elements is beneficial because it eliminates the need to run wires and tap into the propulsive power supply
of the aircraft, thus reducing mass and complexity while allowing the sensors to operate
29
Self-charging structures embedded in
wing spar (shown with wing removed)
Low-power sensor node being
powered locally by harvesting system
Figure 2.3: Potential use of self-charging structures: Schematic of small UAV with embedded self-charging structures in wing spar used to provide local power for low-power
sensor node.
wirelessly. Additionally, a multifunctional approach in which the composite harvester is
embedded into the wing spar and supports structural loads in the wings is valuable because it can reduce or eliminate the added mass of the harvesting device. The concept of
embedding self-charging structures into the wing spar of a UAV will be investigated as a
case study later in this dissertation.
While the wing spar example described above presents one potential use of self-charging structures, the technology can be used in any low-power application that can benefit
from a multifunctional harvesting solution. Embedded electronic systems and remote selfpowered sensor nodes are examples of applications where the added mass and volume of
conventional harvesting could present design challenges, and where the multifunctional
approach of self-charging structures can be beneficial. Additionally, when coupled with
an appropriate piezoelectric device, the mechanical properties of the self-charging structure can be tailored to the meet the needs of various applications. Piezoelectric devices
range from stiff, brittle monolithic piezoceramic to flexible piezoelectric fiber-based and
polymer-based transducers, allowing a wide range of mechanical properties to be obtained, thus making the self-charging structure a versatile energy harvesting solution.
30
2.3
Electromechanical Modeling
The goal of this section is to develop an electromechanical model that can be used to predict the coupled vibration response and voltage response of the self-charging structures
at steady state. The experimentally validated assumed modes model presented by Erturk
and Inman [30–32] is applied here to the self-charging structures. The assumed modes
method is an approximate distributed parameter modeling technique, closely related to
the more common Rayleigh-Ritz method (in fact both methods yield identical discretized
equations), that uses the extended Hamilton’s principle along with the energy expressions
to derive the equations of motion of a system.
2.3.1
Modeling Assumptions and Device Configuration
Consider the self-charging structure configurations shown in Fig. 2.4. Details of the design of the self-charging structures are described in Chapter 3 of the dissertation. The
energy harvesting beam contains two piezoelectric layers and is symmetric about the xaxis, hence it represents a bimorph configuration. Both series (Fig. 2.4(a)) and parallel
(Fig. 2.4(b)) electrode connections of the piezoelectric devices are considered. The structures are excited under translational base acceleration, ab (t), imposed in the transverse
direction (z-direction) at the clamped end. The cantilever beam is assumed to be sufficiently thin such that Euler-Bernoulli assumptions hold (shear deformation and rotary
inertia effects are negligible). The electrodes covering the piezoelectric surfaces are assumed to have negligible thickness and to be perfectly conductive such that a single elecR
tric potential can be defined across them. QuickPack
piezoelectric devices manufactured
R
by Midé Technology, Corp. [99] are used in the harvester. Each QuickPack
device con-
tains monolithic PZT-5A piezoelectric ceramic bracketed by Kapton, as shown in Fig. 2.4.
The piezoelectric layers are attached to the substrate via high shear strength epoxy (3M
ScotchWeldTM DP460 [100]). At the free end of the beam, two thin-film batteries are also
R
included in the harvester. Thinergy
MEC 101-7SES thin-film batteries manufactured by
R
Infinite Power Solutions, Inc. [101] are used. The batteries are attached to the QuickPack
surfaces using DP460 epoxy as well. All bonding layers are assumed to be perfect and the
thickness of each bonding layer is assumed to be identical.
31
(a)
z
ab (t )
x0
(b)
L1
x
Rl
vs (t )
x
Rl
v p (t )
L
z
ab (t )
x0
L1
L
0  x  L1
(c)
hp
hs
hp
hk
Piezoceramic Layer
he h1
Thin-Film Battery Layer
Substrate Layer
L1  x  L
Kapton Layer
hb
Epoxy Layer
h2
Electrodes
Poling Direction
hb
Figure 2.4: Piezoelectric self-charging structure configurations showing (a) series and (b)
parallel connection of the piezoelectric layers along with the (c) cross-sectional views of
both composite sections.
2.3.2
Electromechanical Assumed Modes Model
The absolute motion of the harvester is a combination of the base motion and the relative
motion of the beam, given by
wabs (x, t) = wb (t) + wrel (x, t)
(2.2)
where wabs is the absolute displacement of the beam, wb is the base displacement and wrel
is the displacement of the beam relative to the moving base.
32
The total potential energy, or strain energy, in the beam is given by


Z
Z
1

U =  Sxx Txx dVs + Sxx Txx dVp 
2
(2.3)
Vp
Vs
where Sxx is the strain, Txx is the stress, subscript s represents structure materials, subscript p represent piezoelectric materials, and the integrations are performed over the volume of the materials. All non-piezoelectric layers (substrate and battery layers) are considered as structure materials.
The only non-zero strain component is given by
Sxx (x, z, t) = −z
∂ 2 wrel (x, t)
∂x2
(2.4)
All materials are assumed to be linear elastic and obey Hooke’s Law. The stress in the
structure layers is given by
(2.5)
Txx (x, z, t) = Ys Sxx
where Ys is the elastic modulus of the structure layer of interest.
The strain energy in the structure layers is then
1
Us =
2
Z
Ys (x)z
2
∂ 2 wrel (x, t)
∂x2
2
dVs
(2.6)
dx
(2.7)
Vs
1
=
2
ZL
Y Is (x)
∂ 2 wrel (x, t)
∂x2
2
0
where I is the area moment of inertia and Y Is (x) is the bending stiffness of the structure
materials, which is a function of x as the beam in non-uniform along the length. The
bending stiffness of the structure layers is given by
Y Is (x) = Y Ic1 H(L1 − x) + Y Ic2 H(x − L1 )
(2.8)
where H(x) is the Heaviside step function, and Y Ic1 and Y Ic2 are the bending stiffnesses
of the structure materials in the first composite section from 0 < x < L1 and in the second
composite section from L1 < x < L, respectively. Calculation of the bending stiffness of
a symmetric composite section about the neutral axis through the use of the parallel axis
theorem is described in Appendix B. Details of the calculation of the bending stiffness of
the self-charging structure composite sections can be found in Appendix B.2.
33
The stress in the piezoelectric layers is (see Appendix A.3)
Txx (x, z, t) = T1 = cE
11 S1 − e31 E3
(2.9)
where cE
11 is the elastic modulus of the piezoelectric measured at constant electric field, S1
is the strain in the x-direction (i.e. S1 = Sxx ), e31 is the piezoelectric stress constant and
E3 is the electric field across the electrodes of the piezoelectric layers (note e31 = d31 /sE
11 ,
E
where d31 is the piezoelectric strain constant and sE
11 = 1/c11 is the elastic compliance of the
piezoelectric layer measured at constant electric field). The electric field can be expressed
in terms of the voltage output of the piezoelectric layers, however, the expressions will differ between the series and parallel electrode connection cases. From this point on, separate
formulations for the series and parallel connections must be given. For the series connection, the voltage across the electrodes of each piezoelectric layer is vs (t)/2, where for the
parallel connection case the voltage of each layer is vp (t). Additionally, e31 has differing
signs depending on the poling direction of the piezoelectric layer. In the series case, the
piezoelectric layers are poled in opposite directions, therefore, e31 has opposite sign. The
instantaneous electric fields, however, are in the same direction (i.e. E3 (t) = −vs (t)/2hp ,
where hp is the thickness of the piezoelectric layer) for both piezoelectric layers. For the
parallel connection, the poling direction is the same for both layers so e31 has the same sign,
however, the instantaneous electric fields are in opposite directions with E3 (t) = −vp (t)/hp
in the top layer and E3 (t) = vp (t)/hp in the bottom layer.
The total strain energy in the piezoelectric layers is the sum of the strain energy in the
top and bottom layers, giving
#
2
2
Z "
2 w (x, t)
1
v
(t)
∂
∂
w
(x,
t)
s
rel
rel
Ups =
z 2 cE
− ze31
dVpu
11
2
∂x2
2hp
∂x2
Vpu
1
+
2
#
2
2
Z "
2 w (x, t)
∂
∂
w
(x,
t)
v
(t)
s
rel
rel
z 2 cE
+ ze31
dVpl
11
∂x2
2hp
∂x2
(2.10)
Vpl
Upp
1
=
2
Z "
z 2 cE
11
∂ 2 wrel (x, t)
∂x2
2
vp (t) ∂ 2 wrel (x, t)
− ze31
hp
∂x2
#
dVpu
Vpu
1
+
2
#
2
2
Z "
vp (t) ∂ 2 wrel (x, t)
∂ wrel (x, t)
2 E
+ ze31
dVpl
z c11
∂x2
hp
∂x2
(2.11)
Vpl
where superscript s and p stand for series and parallel connection of the piezoelectric layers and subscript u and l represent the upper and lower piezoelectric layers. Recalling
34
that the upper and lower piezoelectric layers are identical and that the beam is symmetric
about the x-axis, the total strain energy in the piezoelectric layers can be simplified and
added to the potential energy of the structure to give the following total potential energy
expressions
1
Us =
2
2
2
ZL (
∂ wrel (x, t)
Y Is (x)
∂x2
(2.12)
0
+ 2 cE
11 Ip
1
U =
2
p
∂ 2 wrel (x, t)
∂x2
2
∂ 2 wrel (x, t)
− Jps vs (t)
∂x2
!)
dx
2
2
ZL (
∂ wrel (x, t)
Y Is (x)
∂x2
(2.13)
0
+ 2 cE
11 Ip
∂2w
rel (x, t)
∂x2
2
− Jpp vp (t)
∂2w
rel (x, t)
∂x2
!)
dx
where cE
11 Ip is the bending stiffness of the piezoelectric layer (again, found using the parallel axis theorem as shown in Appendix B), and the piezoelectric coupling terms are
ZZ
e31
s
Jp =
z dy dz
(2.14)
2hp
p
ZZ
e31
z dy dz
(2.15)
Jpp =
hp
p
where the integrals are evaluated over the domain of the piezoelectric layers.
The total kinetic energy of the beam is


2
2
Z
Z
1
∂wabs (x, t)
∂wabs (x, t)

Tbeam =  ρs
dVs + ρp
dVp 
2
∂(t)
∂(t)
(2.16)
Vp
Vs
which can be rewritten as
Tbeam
1
=
2
ZL
(ρAs (x) + ρAp (x))
(2.17)
0
"
×
∂wb (t)
∂t
2
+2
∂wb (t) ∂wrel (x, t)
+
∂t
∂t
∂wrel (x, t)
∂t
2 #
dx
where ρAs (x) and ρAp (x) are the mass density functions of the structure and piezoelectric
layers given by
ρAs (x) = ρAc1 H(L1 − x) + ρAc2 H(x − L1 )
(2.18)
ρAp (x) = ρAp
(2.19)
35
where ρAc1 and ρAc2 are the mass densities of the structure material in the first composite
section from 0 < x < L1 and in the second composite section from L1 < x < L, respectively, and ρAp is the mass density of the piezoelectric layers. The mass densities in each
section can be found by simply summing the mass density of each material layer in that
section. The mass density functions for the self-charging structure composite sections are
given in Appendix B.2.
The internal electrical energy in the piezoelectric layers is given by
Z
1
Wie =
E3 D3 dVp
2
(2.20)
Vp
The electric displacement, D3 , in the piezoelectric layers is (see Appendix A.3)
D3 = e31 S1 + εS33 E3
(2.21)
where εS33 is the dielectric permittivity measured at zero strain (denoted by the superscript
S). The signs of the electric fields, E3 , and the piezoelectric stress constants, e31 , again depend on the poling direction and electrode configuration, and follow the same relationship
given when formulating the stress in each piezoelectric layer (Eq. (2.10) and Eq. (2.11)). The
total internal electrical energy in the piezoelectric layers is the sum of the electrical energy
in the top and bottom layers, which gives
Z 2
1
vs (t)
∂ 2 wrel (x, t)
s
S vs (t)
Wie =
e31 z
+ ε33
dVpu
2
2hp
∂x2
4h2p
Vpu
1
+
2
Z Vpl
Wiep
1
=
2
Z "
Vpu
1
+
2
2
vs (t)
∂ 2 wrel (x, t)
S vs (t)
−
e31 z
+
ε
dVpl
33
2hp
∂x2
4h2p
v 2 (t)
vp (t)
∂ 2 wrel (x, t)
S p
e31 z
+
ε
33
hp
∂x2
h2p
Z "
Vpl
(2.22)
#
vp (t)
∂ 2 wrel (x, t) S vp2 (t)
−
e31 z
ε33 2
hp
∂x2
hp
dVpu
(2.23)
#
dVpl
Due to the symmetry of the identical piezoelectric layers, the internal electric energy
can be simplified to
Wies
1
=
2
ZL
2Jps vs (t)
∂ 2 wrel (x, t)
1
dx + Cp vs2 (t)
2
∂x
4
(2.24)
∂ 2 wrel (x, t)
dx + Cp vp2 (t)
∂x2
(2.25)
0
Wiep
1
=
2
ZL
2Jpp vp (t)
0
36
where the internal capacitance of a piezoelectric layer, Cp , is given by
Cp = εS33
Ap
hp
(2.26)
where Ap is the electrode area.
The effects of base excitation are considered in the kinetic energy term and mechanical
damping is to be introduced later in the form of proportional damping, therefore, the only
non-conservative work is due to the piezoelectric charge output, giving
(2.27)
Wnce = Q(t)v(t)
where Q(t) is the electric charge output of the piezoelectric layers.
2.3.3
Substitution of the Assumed Solution
The assumed modes method involves the discretization of the energy expressions by substitution of an assumed series solution for the unknown relative beam displacement composed of kinematically admissible functions (or trial functions), φr (x), multiplied by generalized (or modal) coordinates, ηr (t), of the form
wrel (x, t) =
N
X
(2.28)
φr (x)ηr (t)
r=1
The admissible functions must satisfy the geometric boundary conditions. A simple admissible function satisfying the essential boundary conditions of a clamped-free thin beam
is [102]
φr (x) = 1 − cos
(2r − 1)πx
2L
(2.29)
Substitution of the assumed solution, Eq. (2.28), into the potential energy expressions,
Eq. (2.12) and Eq. (2.13), gives
ZL (
N
N
X
X
1
00
00
s
U =
Y Is (x)
φr ηr (t)
φl ηl (t)
2
r=1
0
+ 2 cE
11 Ip
N
X
00
φr ηr (t)
r=1
1
U =
2
p
l=1
N
X
00
φl ηl (t) − Jps vs (t)
(2.30)
N
X
!)
00
φr ηr (t)
dx
r=1
l=1
ZL (
N
N
X
X
00
00
Y Is (x)
φr ηr (t)
φl ηl (t)
r=1
0
+ 2 cE
11 Ip
N
X
r=1
00
φr ηr (t)
l=1
N
X
00
φl ηl (t) − Jpp vp (t)
(2.31)
N
X
r=1
l=1
37
!)
00
φr ηr (t)
dx
where prime represents ordinary differentiation with respect to the spatial variable, x.
The total kinetic energy can be written in a similar manner as
1
T =
2
ZL
(ρAs (x) + ρAp (x))
0
"
×
(2.32)
∂wb (t)
∂t
2
#
N
N
N
X
X
∂wb (t) X
φr (x)η˙r (t) +
φr (x)η˙r (t)
+2
φl (x)η̇l (t) dx
∂t
r=1
r=1
l=1
where an overdot represents ordinary differentiation with respect to the temporal variable,
t.
Similarly, the internal electrical energy becomes
Wies
1
=
2
ZL
2Jps vs (t)
r=1
0
Wiep
1
=
2
N
X
ZL
2Jpp vp (t)
1
00
φr (x)ηr (t) dx + Cp vs2 (t)
4
N
X
00
φr (x)ηr (t) dx + Cp vp2 (t)
(2.33)
(2.34)
r=1
0
The potential, kinetic, and internal electrical energies can be written as
N
N
1 XX
(ηr (t)ηl (t)krl − 2ηr (t)vs (t)θrs )
U =
2
s
Up =
T =
1
2
r=1 l=1
N X
N
X
1
2
(ηr (t)ηl (t)krl − 2ηr (t)vp (t)θrp )
(2.35)
(2.36)
r=1 l=1
N
N
XX
(η˙r (t)η̇l (t)mrl + 2η˙r (t)pr )
r=1 l=1
ZL
1
+
2
(ρAs (x) + ρAp (x))
∂wb (t)
∂t
(2.37)
2
dx
0
Wies
N 1X
1
s
2
=
2ηr (t)vs (t)θr + Cp vs (t)
2
2
Wiep =
r=1
N
X
1
2
2ηr (t)vp (t)θrp + 2Cp vp2 (t)
(2.38)
(2.39)
r=1
where
ZL
krl =
00
00
Y Is (x) + 2cE
11 Ip φr (x)φl (x) dx
(2.40)
0
θrs
ZL
=
00
Jps φr (x) dx
0
38
(2.41)
θrp
ZL
=
00
Jpp φr (x) dx
(2.42)
0
ZL
mrl =
(ρAs (x) + ρAp (x)) φr (x)φl (x) dx
(2.43)
∂wb (t)
dx
∂t
(2.44)
0
ZL
pr =
(ρAs (x) + ρAp (x)) φr (x)
0
2.3.4
Lagrange Equations with Electromechanical Coupling
The extended Hamilton’s principle for electromechanical systems is [103]
Zt2
(δT − δU + δWie + δWnc ) dt = 0
(2.45)
t1
where δT , δU and δWie are the first variations of the kinetic energy, potential energy and
internal electrical energy, and δWnc is the virtual work of all non-conservative forces. Based
on the extended Hamilton’s principle, the electromechanical Lagrange equations are
∂U
∂Wie
∂T
d ∂T
+
−
=0
(2.46)
−
dt ∂ η̇i
∂ηi ∂ηi
∂ηi
∂U
∂Wie
d ∂T
∂T
+
−
=Q
(2.47)
−
dt ∂ v̇i
∂vi ∂vi
∂vi
where Q is the electric charge output resulting from the non-conservative electrical work,
Wnce .
The second term in the first of the Lagrange equations is zero, leaving the following
non-zero terms
N
N
∂T
1 XX
=
∂ η̇i
2
1
=
2
∂U s
1
=
∂ηi
2
1
=
2
r=1 l=1
N X
N
X
[(δri η̇l + δli η̇r ) mrl + 2δri pr ] =
r=1 l=1
N
N XX
r=1 l=1
N X
N
X
∂ η̇r
∂ η̇l
∂ η̇r
η̇l +
η̇r mrl + 2
pr
∂ η̇i
∂ η̇i
∂ η̇i
N
X
(2.48)
(mil η̇l + pi )
l=1
∂ηr
∂ηl
∂ηr
ηl +
ηr krl − 2
vs θrs
∂ηi
∂ηi
∂ηi
[(δri ηl + δli ηr ) krl −
r=1 l=1
2δri vs θrs ]
=
N
X
l=1
39
(2.49)
(kil ηl −
vs θis )
N
N
∂U p
1 XX
=
∂ηi
2
1
=
2
r=1 l=1
N X
N
X
∂ηr
∂ηr
∂ηl
p
ηl +
ηr krl − 2
v p θr
∂ηi
∂ηi
∂ηi
[(δri ηl + δli ηr ) krl − 2δri vp θrp ] =
r=1 l=1
∂Wies
∂ηi
∂Wiep
∂ηi
N
X
(2.50)
(kil ηl − vp θip )
l=1
N 1X
∂ηr
s
=
2
v s θr =
2
∂ηi
r=1
N 1X
∂ηr
=
2
vp θrp =
2
∂ηi
r=1
N
1X
2
1
2
r=1
N
X
(2δri vs θrs ) = vs θis
(2.51)
(2δri vp θrp ) = vp θip
(2.52)
r=1
The temporal derivative of the kinetic energy term gives
d
dt
where
∂pi
=
∂t
∂T
∂ η̇i
N X
∂pi
=
mil η̈l (t) +
∂t
(2.53)
r=1
ZL
(ρAs (x) + ρAp (x)) φi (x)
∂ 2 wb (t)
dx
∂t2
(2.54)
0
Using these results, the first set of Lagrange equations can be written as
N
X
(mil η̈l + kil ηl − 2θis vs − fi ) = 0
(series connection)
(2.55)
(mil η̈l + kil ηl − 2θip vp − fi ) = 0
(parallel connection)
(2.56)
l=1
N
X
l=1
where fi is the forcing due to base excitation given by
∂pi
=−
fi = −
∂t
ZL
(ρAs (x) + ρAp (x)) φi (x)
∂ 2 wb (t)
dx
∂t2
(2.57)
0
The non-zero terms in the second of the Lagrange equations are
N
∂U s
1X
=
(−2ηr θrs )
∂v
2
(2.58)
r=1
∂U p
∂v
=
N
1X
2
(−2ηr θrp )
(2.59)
r=1
N
N
1X
1
1X
∂Wies
=
(2ηr θrs + Cp vs ) = Cp vs +
(2ηr θrs )
∂v
2
2
2
r=1
∂Wiep
∂v
=
N
1X
2
(2ηr θrp + 4Cp vp ) = 2Cp vp +
r=1
40
1
2
r=1
N
X
r=1
(2ηr θrp )
(2.60)
(2.61)
which gives the second set of Lagrange equations as
N
X
1
Cp v s + Q +
(2ηr θrs ) = 0
2
2Cp vp + Q +
r=1
N
X
(2ηr θrp ) = 0
(series connection)
(2.62)
(parallel connection)
(2.63)
r=1
Taking the temporal derivative of Eq. (2.62) and Eq. (2.63) gives
N
X
1
Cp v̇s + Q̇ +
(2η̇r θrs ) = 0
2
2Cp v̇p + Q̇ +
r=1
N
X
(2η̇r θrp ) = 0
(series connection)
(2.64)
(parallel connection)
(2.65)
r=1
where the time rate of change of the electrical charge output, Q̇, is equal to the electrical
current passing through the load resistor, given by
Q̇ =
v
Rl
(2.66)
Therefore, the second set of Lagrange equations becomes
N
1
vs X
Cp v̇s +
+
(2η̇r θrs ) = 0
2
Rl
2Cp v̇p +
2.3.5
vp
+
Rl
r=1
N
X
(2η̇r θrp ) = 0
(series connection)
(2.67)
(parallel connection)
(2.68)
r=1
Equivalent Series/Parallel Representation of the Lagrange Equation
At this point it is convenient to introduce an equivalent representation of the electromechanical Lagrange equations for the series and parallel cases, as suggested by Erturk and
Inman [32], in which a single formulation with modified electromechanical coupling and
capacitance terms is introduced. Observe that the only differences in the coupled Lagrange
equations between the series and parallel electrode connection cases involve the electromechanical coupling terms (i.e. θi ) and the capacitance terms. Considering these differences,
one can define the following equivalent Lagrange equations
N
X
(mil η̈l + kil ηl − θieq v − fi ) = 0
(2.69)
l=1
N
Cpeq v̇ +
X
v
(η̇r θreq ) = 0
+
Rl
r=1
41
(2.70)
where the equivalent electromechanical coupling, θeq , and capacitance, Cpeq , are selected
from Table 2.1 depending on whether a series or parallel solution is desired.
Table 2.1: Equivalent electromechanical coupling and capacitance terms for series and
parallel electrode connections (from Erturk and Inman [32]).
Series Connection
θieq
Cpeq
RL
Parallel Connection
00
Jp φi (x) dx
2
0
RL
00
Jp φi (x) dx
0
1
2 Cp
2Cp
where the piezoelectric coupling term, Jp , is given by
ZZ
e31
Jp =
z dy dz
hp
(2.71)
p
2.3.6
Solution of the Equivalent Representation of the Lagrange Equations
Rewriting the equivalent Lagrange equations given by Eq. (2.69) and Eq. (2.70) in matrix
form and introducing proportional damping in the first equation gives
[M] η̈ + [C] η̇ + [K] η − Θeq v = f
v
+ Θeq η̇ = 0
Cpeq v̇ +
Rl
(2.72)
(2.73)
where the mass, stiffness and damping matrices ([M], [K], and [C]) are N × N , the generalized coordinates, η, the forcing vector, f , and the electromechanical coupling vector, Θeq ,
are N × 1, and the damping matrix is given by
[C] = α [M] + β [K]
(2.74)
where α and β are constants of proportionality.
Assuming harmonic base excitation of the form wb (t) = wb ejωt , which leads to a base
acceleration of ab ejωt (where ab = −ω 2 wb ), the forcing vector becomes
f = Fejωt
(2.75)
The components of the forcing vector are given by
ZL
Fi = −ab
(ρAs (x) + ρAp (x)) φi (x) dx
0
42
(2.76)
The assumed harmonic forcing results in a harmonic solution for the generalized coordinates and the voltage of the form
η = ηejωt
(2.77)
v = V ejωt
(2.78)
Substitution of the assumed solutions into Eq. (2.72) gives
− ω 2 [M] ηejωt + jω [C] ηejωt + [K] ηejωt − Θeq V ejωt = Fejωt
(2.79)
The exponential function is never equal to zero for all positive time, which leads to
−ω 2 [M] + jω [C] + [K] η − Θeq V = F
(2.80)
Similarly, substitution of the assumed solutions, Eq. (2.77) and Eq. (2.78), into Eq. (2.73)
gives
jωCpeq V ejωt +
V ejωt
+ jωΘeq T ηejωt = 0
Rl
where superscript T stands for transpose. Again this can be written as
1
eq
jωCp +
V + jωΘeq T η = 0
Rl
(2.81)
(2.82)
Equation (2.82) yields
V =
−jωΘeq T η
jωCpeq + R1l
(2.83)
Substituting Eq. (2.83) into Eq. (2.80) gives
jωΘeq Θeq T
−ω 2 [M] + jω [C] + [K] +
jωCpeq + R1l
!
η=F
(2.84)
Recalling our assumed solution of η = ηejωt , Eq. (2.84) can be rewritten as
η(t) =
!−1
1 −1 eq eq T
eq
Θ Θ
Fejωt
−ω [M] + jω [C] + [K] + jω jωCp +
Rl
2
(2.85)
Substitution of Eq. (2.85) back into Eq. (2.83) and recalling our assumed solution of
v = V ejωt gives
v(t) = −jω
jωCpeq
1
+
Rl
−1
Θeq T −ω 2 [M] + jω [C] + [K]
!−1
1 −1 eq eq T
eq
+jω jωCp +
Θ Θ
Fejωt
Rl
43
(2.86)
Substitution of Eq. (2.85) into the assumed series solution for the relative displacement
given by Eq. (2.28) gives the following expression for the relative displacement
wrel (x, t) = ΦT (x) −ω 2 [M] + jω [C] + [K]
!−1
1 −1 eq eq T
eq
Θ Θ
Fejωt
+jω jωCp +
Rl
(2.87)
Finally, the following electromechanical frequency response functions (FRFs) can be
defined for the relative displacement and voltage per base acceleration, as described by
Erturk and Inman [30–32]
wrel (x, t)
= ΦT (x) −ω 2 [M] + jω [C] + [K]
ab ejωt
!−1
1 −1 eq eq T
eq
+ jω jωCp +
Θ Θ
F̃
Rl
1 −1 eq T
v(t)
eq
−ω 2 [M] + jω [C] + [K]
=
−jω
jωC
+
Θ
p
ab ejωt
Rl
!−1
−1
1
+jω jωCpeq +
Θeq Θeq T
F̃
Rl
(2.88)
(2.89)
where the components of the forcing vector are given by
ZL
F̃i = −
(ρAs (x) + ρAp (x)) φi (x) dx
(2.90)
0
Equation (2.88) and Eq. (2.89) can be used to predict the coupled vibration and voltage
response of the self-charging structures when subjected to harmonic base excitation.
2.4
Chapter Summary
An overview of piezoelectric energy harvesting and an introduction to the concept of multifunctional self-charging structures is presented. A brief introduction to both piezoelectricity and piezoelectric energy harvesting is first given. The self-charging structure concept is then discussed. Finally, the application of an approximate coupled electromechanical model used to predict the response of self-charging structures is presented.
The multifunctional self-charging structure concept presented in this dissertation involves the realization of several functionalities in a single energy harvesting device. Throu44
gh the combination of piezoceramic devices and novel thin-film batteries, a single composite energy harvester can simultaneously harvest energy in the piezoelectric layers and store
the harvested energy in the battery layers. Additionally, if such a composite structure is
embedded into a host system, it can be used to help support structural load. Combining
energy harvesting, energy storage, and load bearing ability in a single device provides a
multifunctional solution to vibration energy harvesting.
An approximate electromechanical model based on the assumed modes method is presented to describe the response of the multilayer, nonuniform self-charging structures introduced here. The assumed modes model is based on Euler-Bernoulli beam assumptions
and utilizes the extended Hamilton’s principle from which the electromechanical Lagrange
equations arise. Both series and parallel electrode connections of the two piezoelectric
layers in the symmetric bimorph energy harvesters are considered. Derivation of the electromechanical Lagrange equations for each electrode configuration proves the formulation
to be similar for each configuration and leads to an equivalent representation that is used
for the remainder of the modeling. Displacement - to - base acceleration and voltage - to base acceleration FRFs are derived by assuming harmonic base excitation. The FRFs can be
used to predict the coupled vibration and voltage response of the self-charging structures.
45
C HAPTER
3
E XPERIMENTAL T ESTING OF
S ELF -C HARGING S TRUCTURES
T
HE self-charging structure concept and electromechanical assumed modes model were
described in Chapter 2. This chapter presents experimental investigation of self-
charging structure prototypes. A detailed investigation of two types of thin-film batteries
is first given and the optimal battery type is selected. Fabrication of the prototype selfcharging structures is next discussed including selection of the substrate and piezoelectric layers along with details on vacuum bonding. Finally, experimental testing results
are given for both the validation of the electromechanical assumed modes model through
frequency response measurements as well as the confirmation of the self-charging ability
through charge/discharge measurements.
3.1
Performance Evaluation of Thin-Film Batteries
R
NanoEnergy
thin-film batteries manufactured by Front Edge Technology, Inc. [104] and
R
Thinergy
thin-film batteries produced by Infinite Power Solutions, Inc. [101] are both
investigated. Both batteries, shown in Fig. 3.1, are flexible lithium-based secondary (i.e.
rechargeable) cells that utilize all solid-state components. Common to both types of thinfilm batteries, the active elements include lithium-metal anodes, lithium cobalt dioxide
(LiCoO2 ) cathodes, and lithium phosphorous oxynitride (LiPON) electrolyte layers. The
R
R
most significant difference between NanoEnergy
and Thinergy
batteries lies in their
construction and sealant layers, which will be discussed later in this section. The ability to
produce extremely thin (less than 200 microns) and flexible batteries can be attributed to
the use of the solid-state LiPON electrolyte as opposed to liquid or gel electrolytes found in
46
(a)
(b)
R
R
Figure 3.1: Photographs of (a) NanoEnergy
and (b) Thinergy
thin-film batteries.
conventional rechargeable batteries. LiPON, developed at Oak Ridge National Laboratory
[105, 106], exhibits a high lithium ion mobility, lending to its performance as an electrolyte,
and a low electron mobility, allowing for low self-discharge rates.
3.1.1
Comparison to Conventional Rechargeable Batteries
Compared to traditional rechargeable batteries, thin-film batteries offer a clear advantage
in form factor. Table 3.1 presents both physical and electrical properties of various types of
R
R
secondary batteries, in which the mass and volume of both NanoEnergy
and Thinergy
batteries are shown to be 1-2 orders of magnitude less than conventional batteries. Thinfilm batteries are also flexible, where conventional batteries are rigid. Additionally, thinfilm battery technology offers superior cycle life (on the order of 1,000 - 10,000 cycles)
compared to conventional rechargeable designs (typically limited to 100 - 1,000 cycles).
The main drawback to thin-film battery technology lies in the low storage capacity of the
cells. The limited storage ability in turn causes the energy density and specific energy of
the batteries to suffer. Table 3.1 also provides a comparison of both the energy density
and specific energy of the various secondary batteries, in which it can be seen that the
R
R
NanoEnergy
cells compare reasonably well to the traditional batteries. The Thinergy
cells, however, exhibit a much lower energy density and specific energy, which is mainly
due to the fact that the packaging material contributes significantly to the mass and volume
of the batteries. Although their small capacity restricts their use to low energy applications,
the flexibility, slimness, and superior cycle life of thin-film batteries allows them to be used
in applications where previously impractical, such as direct integration into composite
structures, thus creating countless new possibilities for energy storage systems.
Another important difference between conventional batteries and thin-film lithium-
47
Table 3.1: Properties of various secondary batteries.
Voltage
(V)
Capacity
(mAh)
Mass
(g)
Volume
(cm3 )
Specific
Energy
(Wh/kg)
Energy
Density
(Wh/l)
Energizer NH15-2450
NiMH AA
1.2
2450
30.00
8.34
98.00
352.52
Energizer NH22-175
NiMH 9V
8.4
175
42.00
21.52
35.00
68.31
Varta V15H NiMH
button type
1.2
15
1.30
0.32
13.85
60.00
Samsung AB463446FZ
Li-ion cell phone
3.7
800
17.90
8.36
165.36
354.07
AA Portable Power
Corp. PL-383562-C2
Li-Polymer single cell
3.7
850
18.00
7.26
174.72
152.73
Front Edge Technology,
R
Inc. NanoEnergy
Lithium thin-film
4.2
4
0.45
0.11
76.36
152.73
Infinite Power
Solutions, Inc.
R
Thinergy
MEC101-7
Lithium thin-film
4.0
0.7
0.22
0.11
6.22
25.45
Battery
based batteries is the internal resistance of the cells. The thin-film batteries under investigation have typical internal resistances on the order of 50 - 200 Ω. This is extremely high
compared to the internal resistance of most conventional alkaline, nickel-metal hydride,
and lithium-ion secondary batteries, which are on the order of 0.1 - 1 Ω. As current flows
through a battery, there is a voltage drop across the internal resistance of the battery equal
to
vdrop = i × Rint
(3.1)
where vdrop is the voltage drop, i is the current flowing through the battery, and Rint is the
internal resistance of the battery. This voltage drop decreases the terminal voltage as well
as the efficiency of charging and discharging. A large internal resistance is detrimental
to battery performance because it causes a significant voltage drop under loads drawing
a high amount of current. Due to the large internal resistance of thin-film batteries, it is
difficult to source high current levels while maintaining rated cell voltage. Additionally,
48
R
R
the manufacturers of the Thinergy
and NanoEnergy
batteries suggest that they not be
discharged below 2.1 V and 3 V, respectively, to prevent damage to the cells. High currents,
therefore, can only be sourced from the thin-film batteries for a short period of time before
reaching the cut-off voltage.
3.1.2
Battery Selection
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R
Both NanoEnergy
and Thinergy
cells are considered for use in self-charging structures.
As stated previously, the primary difference between the cells lies in the packaging material and encapsulation method.
R
NanoEnergy
thin-film batteries are built by encasing the active elements between a
R
top and bottom mica substrate and sealing the substrate layers with a Surlyn
(DuPont)
sealant layer around the perimeter of the active elements. Electrical leads are given in the
form of 100 µm thick metal foil tabs. Typical battery dimensions are 28.96 mm x 25.40 mm
with a thickness of 150 µm and a mass of around 0.220 grams. The nominal voltage of the
cells is 4.2 V and their capacities are around 4 mAh. The manufacturer publishes that the
batteries have more than 1000 charge/discharge cycles at 100% depth of discharge, can be
charged to 70% of rated capacity in just 2 minutes, and can be discharged at rates up to
10C [104]. (Note that battery charge and discharge currents are given in terms of their rated
capacity, C. A rate of 10C for a battery with a capacity of 4 mAh, for example, corresponds
to a current of 40 mA.)
R
Thinergy
batteries are assembled using a proprietary encapsulation method, how-
ever, it can be observed that they utilize a metal foil substrate for the top and bottom outer
layers, which act as electrodes, and a sealant layer between the electrode layers to prevent
electrical shorting. The nominal operating voltage of the batteries is 4.0 V and the nominal capacity of the cells used here is 0.7 mAh. Typical dimensions are 25.40 mm x 25.40
mm with a thickness of 170 µm and a mass of about 0.450 grams. Infinite Power Solutions
R
claims that the Thinergy
cells can exhibit 10,000 cycles at 100% depth of charge before
deteriorating to 80% of the initial capacity at a C/2 discharge rate. They also state that the
batteries can be charged to 90% of rated capacity in 10 minutes and can be discharged at
rates up to 40C [101].
Differences in the packaging and sealant layers of the thin-film batteries primarily affect the handling and ease of use of the cells. Preliminary testing of both types of thin-film
49
R
batteries has revealed that the metal foil substrate of the Thinergy
batteries facilitates
R
convenient electrode application as opposed to the metal foil tabs of the NanoEnergy
cells, which are fragile and difficult to use. Additionally, the metal foil substrate also apR
pears more robust compared to the mica substrate used on the NanoEnergy
cells, which
can peel away from the sealant under shear loading. Previous research has reported simiR
lar observations in regards to the fragility of the NanoEnergy
cells [107, 108]. Although
R
R
the capacity of the NanoEnergy
batteries is superior to that of the Thinergy
cells (see
R
Table 3.1), Thinergy
batteries are selected for use in self-charging structures due to their
increased robustness.
3.1.3
R
Performance Testing of Thinergy
Batteries
R
Prior to combining the Thinergy
thin-film batteries with piezoelectric devices to create
self-charging structures, the performance of the batteries is evaluated experimentally. The
experimental setup used for battery charge/discharge measurements is shown in Fig. 3.2.
The batteries are placed in a fixture which allows convenient and temporary electrode
connections. The batteries are charged using an HP (now Agilent Technologies, Inc. [109])
6825A power supply/amplifier (not shown), which provides constant-voltage charging,
and discharged through standard carbon film resistors. During charging and discharging, the battery voltage as well as the current flowing in/out of the battery are monitored
and recorded using a National Instruments Corp. [110] NI 9215 4-channel analog voltage input card with 16-bit precision running a custom LabVIEW code written to perform
the data acquisition task. In order to measure the current flowing through the battery, a
transimpedance operational amplifier circuit (powered by two HP 6218A power supplies)
utilizing an Analog Devices, Inc. [111] OP177 ultra precision op-amp, shown schematically
in Fig. 3.3, is used to convert the current into a voltage that is measurable with the NI 9215
card. The circuit utilizes the large impedance of the op-amp to force the input current to
flow through the set resistor, Rset , which can then be measured as a voltage at the output terminal of the op-amp. A portion of the input energy is dissipated in the resistor,
however, the op-amp compensates for this dissipated energy by supplying an appropriate
amount of energy back to the input to hold the non-inverting input terminal at ground.
The set resistor, Rset , must be chosen appropriately based on the expected current to yield
a voltage output in a range compatible with the data acquisition system.
50
(a) Dual power
supplies
(b) Transimpedance
op-amp circuit
(c) Thin-film battery
test fixture
(d) NI 9215 data
acquisition card and
computer
(a)
(b)
(c)
(d)
Figure 3.2: Experimental setup used to obtain charge/discharge measurements on thinfilm batteries.
The current measured during charge/discharge testing can be used to quantify the
amount of energy flowing through the battery. Batteries capacities are rated in milliamphours (mAh), which describes charge over time. The capacity achieved during charging
and discharging can be calculated by performing numerical integration of the current measurement over time as follows [107, 108]:
Z
C=
(3.2)
i dt
where C is the battery capacity, i is the measured current, and t is time. Typical voltage
R
and current measurements during charging and discharging of the Thinergy
batteries
are shown in Fig. 3.4(a) and Fig. 3.4(b), respectively. Charging is performed by supplying
4.1 V of potential to the battery (per manufacturer specification) using the power supply
until only about 35 µA of current is sourced by the battery, at which time the battery is
considered fully charged. Discharging is performed by applying a resistive load of 2749
Ω across the battery terminals in order to draw roughly 2C (1.4 mA) of current until a
Rset
I in
OP 177
Vout   I in Rset
+
Figure 3.3: Transimpedance operational amplifier circuit used for current measurement.
51
(a)
8
4.1
Voltage
3
Current
4
2
2
Charge Capacity:
0.702 mAh
Voltage (V)
Current (mA)
6
1
0
0
1000
Time (sec)
0
1500
3
4.1
3.8
Voltage
Current (mA)
2
Load
Disconnected
Current
3
2
1
Voltage (V)
(b)
500
Discharge Capacity:
0.704 mAh
1
0
0
500
1000
Time (sec)
1500
2000
0
R
Figure 3.4: Characteristic (a) charge and (b) discharge curves of Thinergy
batteries.
voltage of 3.0 V (the cutoff voltage recommended by the manufacturer) is reached. A rate
of 2C is chosen arbitrarily in order to provide a reasonable time to discharge the battery
(approximately 30 minutes). The batteries are capable of achieving discharge rates up to
40C, however, such large discharge rates can degrade the performance of the batteries
over time. The charge and discharge characteristics displayed in Fig. 3.4 are typical for
rechargeable batteries. In charging, the voltage is held constant and an initial current spike
is observed, after which the charge current decreases and levels out at a lower value. The
discharge curve shows relatively constant voltage and current until the battery is nearly
fully discharged, at which point the voltage and current drop rapidly. For this particular
52
battery, an initial voltage drop during discharging of 0.061 V (from 4.096 V to 4.035 V) and
a current of 1.467 mA are observed. The corresponding internal resistance of this battery
can be calculated using Eq. (3.1) as 41.58 Ω, which is slightly below the manufacturer’s
specification of 50 Ω. Carrying out the capacity calculation given in Eq. (3.2), the capacity
in charging is calculated as 0.702 mAh, and in discharging as 0.704 mA, which correlate
well with the manufacturer’s specification of 0.7 mAh. It is expected that these capacities
be reasonably close to one another, which is the case, and in both charging and discharging,
R
the full 0.7 mAh capacity can be obtained. Overall, the Thinergy
battery performs as
expected, showing reasonable charge/discharge characteristics and good charge cycling
ability.
3.2
Device Fabrication
Assembly of the self-charging structures involves several steps including the selection of
piezoelectric and substrate materials, bonding the battery, piezoelectric, and substrate layers, and connecting leads to the electrodes of the thin-film batteries and piezoelectric devices. These steps are outlined in the following sections.
3.2.1
Selection of Piezoelectric and Substrate Materials
A commercially available piezoelectric material will be used as the active energy harvesting element in self-charging structures. Several companies produce piezoelectric materials
that can be considered. Piezoelectric materials to be used in bending can be categorized
into three classes including monolithic piezoceramic, piezoceramic fiber-based, and piezoelectric polymer film devices. Piezoelectric polymer films such as polyvinylidene fluoride
(PVDF) are extremely compliant with an elastic modulus around 3 GPa, and typically exhibit low electromechanical coupling. The power output of piezoelectric polymer films is
typically orders of magnitude less than monolithic and fiber based piezoelectric materials, therefore, they will not be considered for use in self-charging structures. Piezoelectric
fiber-based devices such as the Macro Fiber Composite (MFC) (Smart Material Corp. [112])
or the Piezoelectric Fiber Composite (PFC) (Advanced Cerametrics, Inc. [113]) consist of
monolithic piezoceramic fibers embedded in a polymer matrix with interdigitated electrodes. The structure of such devices allows flexibility and moderate energy generation
53
capabilities. Monolithic piezoceramic materials exhibit high energy generation abilities
compared with fiber and polymer based devices, however, they are brittle in nature and
susceptible to failure under loading. As a compromise between high energy generation
R
and strength under dynamic loading, QuickPack
piezoelectric devices manufactured by
R
Midé Technology Corp. [99] are selected for use in self-charging structures. QuickPack
devices contain monolithic piezoceramic (PZT-5A) active elements bracketed by Kapton
R
(DuPont) layers to protect the active element and provide robustness. QuickPack
QP10n
piezoelectric devices, shown in Fig. 3.5, are considered in this work.
Several substrate materials can be considered for use in self-charging structures. Typical piezoelectric bimorph energy harvesters contain a thin, relatively flexible substrate,
such as brass or aluminum, such that the stiffness of the substrate does not dominate the
overall structural stiffness in order to allow adequate vibration energy to be induced in the
piezoelectric elements. The substrate layer material selected for use in the self-charging
structures is 1100-O aluminum alloy. Alternative substrates, however, can be used to alter
the characteristics of the self-charging structures to fit the design parameters of a given
application.
Important physical parameters of the various components used to construct the selfR
charging structures are given in Table 3.2. As stated previously, the QuickPack
devices
consist of a central monolithic piezoceramic (PZT-5A) layer bracketed by 0.0635 mm thick
Kapton layers that include embedded electrodes. Dimensions of both the overall device
and the active element are given in the table. The piezoelectric material properties are
taken from the manufacturer’s datasheet [114] (QP10n utilizes 3195HD piezoelectric material from CTS Corp.), where the battery properties are found from experimental tests and
measurements. The mass density of the battery is found by simply measuring its volume
and mass, where the elastic modulus is found by fitting a distributed parameter bending
R
Figure 3.5: QuickPack
QP10n piezoelectric device.
54
Table 3.2: Geometric and material properties of self-charging structure components.
Aluminum
Substrate
Property
R
R
Quickpack
Quickpack
R
Thinergy
QP10n
QP10n
MEC
Overall
Active
101-7SES
Device
Element
Length (mm)
63.500
50.80
45.97
25.40
Width (mm)
25.400
25.40
20.574
25.40
Thickness (mm)
0.152
0.381
0.254
0.178
69
-
67
55
2730
-
7800
4000
-
-
-190
-
-
-
14.60
-
Elastic Modulus (GPa)
(kg/m3 )
Mass Density
Piezoelectric Strain
Constant, d31 (pC/N)
Dielectric Permittivity
Constant, S33 (pF/m)
beam model to experimental FRF data obtained with a battery mounted in a cantilever configuration and subjected to base excitation. For modeling purposes, the elastic moduli of
the Kapton layers and epoxy layers (not listed in the table) are taken as 3.7 GPa and 3 GPa,
respectively, and the mass densities are taken as 1233 kg/m3 and 1000 kg/m3 , respectively
[115].
3.2.2
Vacuum Bonding and Electrode Attachment
Fabrication of the self-charging structures is performed by separately bonding each layer
using a vacuum bagging procedure, shown schematically in Fig. 3.6, to achieve thin, uniform bonding layers. The vacuum bonding setup includes a commercial vacuum pump,
vacuum gauge, vacuum bag surrounding both the workpiece and work surface sealed
with vacuum bagging tape, and several material layers surrounding the workpiece. Peel
ply layers are placed on both the top and bottom surfaces of the workpiece which prevent
excess epoxy from bonding to the work surface or other material layers. Additionally, a
bleeder/breather layer is placed on top of the workpiece above the peel ply layer to absorb any excess epoxy and to allow even vacuum over the entire workpiece. Photographs
of the vacuum bagging setup are shown in Fig. 3.7. When fabricating the self-charging
structures, approximately 20 in-Hg of vacuum is achieved using a Gast Manufacturing,
Inc. [116] 23 Series 0523-101 vacuum pump, shown in Fig. 3.7(b). 3M [100] ScotchWeld
55
TM
Vacuum Gauge
15
20
10
25
5
0
30
in Hg
Vacuum Pump
Workpiece
Work Surface
Breather/Bleeder
Cloth
Peel Ply
Vacuum Bag
Tape
Figure 3.6: Schematic of vacuum bagging procedure for self-charging structures.
DP460 two part epoxy is chosen for the bonding layer due to its high shear strength (27.58
MPa when bonded to aluminum) and high volume resistivity (2.4 x 1014 ohm-cm). Bonding is achieved by applying a thin layer of epoxy between two component layers, placing
the device in vacuum, and allowing it to cure for 6 hours. After curing, any excess epoxy
is removed from the edges of the device and the process is repeated until the self-charging
structure is complete. The average epoxy thickness between device layers is measured as
0.0205 mm. The thin-film batteries are selected as the outermost layers to facilitate attachment of electrical leads. They are placed towards the free end of the device to reduce the
induced strain in the batteries in order to help prevent electrical or mechanical failure.
With all of the self-charging structure layers bonded, the final step in fabrication involves attaching electrical leads to both the piezoceramic and battery layers. The Quick(a)
(b)
GAST
PUMP
PIC
Hose to
vacuum pump
Vacuum bag
Self-charging
structure
Figure 3.7: (a) Vacuum bagging setup; (b) Gast 23 Series vacuum pump.
56
R
Pack
devices contain an electrical connector (Fig. 3.5), however, it is removed to reduce
the length and mass of the piezoceramic layer. With the connector removed, a small area
of the flat electrodes is exposed by removing the Kapton coating with a razor blade. 28gauge insulated and stranded wire is then soldered to the exposed electrodes to create an
R
electrical connection. The entire outer faces of the Thinergy
batteries serve as electrodes,
and there is a slight overlap on one of the edges of the battery such that both positive and
negative electrodes are accessible from a single surface. Electrical leads are attached to
the batteries by directly soldering 28-gauge wire to the electrode surfaces. A very small
amount of solder is used to prevent shorting of the device when attaching the lead to
the overlapping electrode, therefore, an additional epoxy coating is placed over the elecR
trode connections to provide mechanical strength as well as electrical insulation. Loctite
[117] 3381 UV curable epoxy is used to coat the connection points and is cured in about
3 minutes using LED UV light. A photograph of a complete self-charging structure with
electrical leads can be seen in Fig. 3.8.
3.3
Experimental Validation of Electromechanical Model and SelfCharging Concept
Experiments are performed on the fabricated self-charging structure shown in Fig. 3.8 in
order to verify the electromechanical model described in Chapter 2 and to confirm the ability of the device to simultaneously harvest and store electrical energy. The performance of
the self-charging structure is evaluated by mounting the device in a cantilever fashion and
subjecting it to base excitations while monitoring the mechanical and electrical response,
Figure 3.8: Complete self-charging structure prototype.
57
including the energy transfer between the piezoceramic layers and the battery layers. The
following sections describe the results of the experimental characterization.
3.3.1
Electromechanical Model Validation
In order to validate the electromechanical assumed modes model and to determine the
resonant frequency and optimal load resistance of the clamped device, experiments are
conducted to obtain the frequency response functions of the self-charging structure for a
set of resistive electrical loads (ranging from 100 Ω to 1 M Ω). The experimental setup
used for validation of the electromechanical model is shown in Fig. 3.9. The self-charging
structure is clamped to a small TMC Solution [118] TJ-2 electromagnetic shaker (powered
by an HP [109] 6825A power supply/amplifier operating as a fixed gain amplifier) with an
overhang length of 44.2 mm. A DSP Technology, Inc. (now Spectral Dynamics, Inc. [119])
SigLab 20-42 data acquisition system with four input channels and two output channels is
used for all FRF measurements. SigLab is used to generate a low amplitude chirp signal
that is used to drive the shaker. Five averages are used for each FRF measurement. The
input acceleration, which is used as the reference channel for the FRF measurements, is
measured using a PCB Piezotronics, Inc. [120] U352C67 accelerometer attached to the base
of the clamp with wax. A PCB Piezotronics, Inc. 482A16 charge coupler is used to condition the accelerometer signal prior to measurement with the SigLab acquisition system.
The tip velocity is measured using a Polytec, Inc. [121] OFV-303 laser Doppler vibrometer
and OFV-3000 controller by placing a small piece of retroreflective tape at the tip of the cantilever structure, and the voltage output of the self-charging structure is measured directly
with the SigLab data acquisition system. Two frequency response functions are, therefore,
measured; the tip velocity - to - base acceleration FRF and the voltage - to - base acceleration FRF. It is worth noting that the laser vibrometer measures the absolute tip velocity of
the beam as opposed to the relative tip displacement that is predicted by Eq. (2.88), however, Eq. (2.88) leads to the following expression for the absolute tip velocity FRF, as shown
by Erturk and Inman [29] (see Appendix C for details of the transformation).
Vabs (L, t)
1
=
+ jωΦT (L) −ω 2 [M] + jω [C] + [K]
jωt
ab e
jω
!−1
−1
1
+ jω jωCpeq +
F̃
Θeq Θeq T
Rl
58
(3.3)
(a)
(b)
(f)
(c)
(d)
(g)
(h)
(e)
(e)
(a) Fixed gain amplifier
(b) Accelerometer coupler
(c) Laser vibrometer controller
(d) Laser vibrometer head
(e) Electromagnetic shaker
(f) Accelerometer
(g) Clamp
(h) Self-charging structure
(i) SigLab data acquisition system
(i)
Figure 3.9: Experimental setup used to obtain frequency response measurements of selfcharging structures.
Electromechanical frequency response functions are measured for 15 different load resistance values (standard carbon film resistors are used), as listed in Table 3.3. The input
channels of the SigLab data acquisition system have an impedance of 1 MΩ which acts in
parallel with the load resistors placed across the piezoelectric layers. The effective load
that is seen by the self-charging structure is, therefore, affected by the impedance of the
measurement system. Table 3.3 also lists the effective load seen by the piezoelectric layers
by considering the 1 MΩ input channel impedance.
Experimental voltage - to - base acceleration and tip velocity - to - base acceleration
FRFs measured for both the series and parallel connection of the electrodes are shown in
Fig. 3.10 and Fig. 3.11, respectively (where the base acceleration is given in terms of the
acceleration of gravity, g = 9.81 m/s2 ). In order to verify the electromechanical assumed
modes formulation for the self-charging structures, the voltage and tip velocity FRFs are
59
Table 3.3: Load resistance values used in FRF measurements of self-charging structures
along with effective load resistances.
Load Resistor
Values (Ω)
Effective
Resistance Seen
by Piezoelectric
Layers (Ω)
100
1200
6700
10000
11800
14700
17600
20000
22000
33000
47000
100000
330000
470000
1000000
99.99
1198.56
6655.41
9900.99
11662.38
14487.04
17295.60
19607.84
21526.42
31945.79
44890.16
90909.09
248120.30
319727.89
500000.00
predicted using Eq. (2.89) and Eq. (3.3), respectively, and plotted over the experimental
results in Fig. 3.10 and Fig. 3.11. Thirty modes are used in the assumed modes formulation
(N = 30) to ensure convergence of the fundamental natural frequency using the admissible functions given by Eq. (2.29). From the results, it is clear that the model accurately
predicts the coupled electrical and mechanical response of the structure. As the load resistance increases from 100 Ω (near the short-circuit condition) to 1 MΩ (near the open-circuit
condition), the experimentally measured fundamental natural frequency shifts from 203.9
Hz to 211.6 Hz for the series connection case and from 204.2 Hz to 211.7 Hz for the parallel
connection case. These frequencies are predicted by the assumed modes model as 204.0 Hz
and 211.6 Hz, respectively (the same for both series and parallel connections). The model
predictions of the magnitude of the FRFs are also well matched. The experimental voltage
FRFs show a maximum peak voltage output (obtained for the largest load resistance) of
31.61 V/g for the series connection and 18.65 V/g for the parallel connection. It should be
60
noted, however, that these measurements are frequency response based linear estimates
obtained from low-amplitude chirp excitation and they are not necessarily accurate for
large amplitude excitations with nonlinear response characteristics. The maximum voltage output is predicted by the model as 27.30 V/g for the series case and 15.54 V/g for the
parallel case. The maximum tip velocity for the short-circuit condition is measured experimentally as 0.408 (m/s)/g and 0.401 (m/s)/g for the series and parallel cases, respectively,
and is predicted by the model as 0.458 (m/s)/g and 0.446 (m/s)/g. For the open-circuit
condition, maximum tip velocities of 0.392 (m/s)/g and 0.460 (m/s)/g are measured for
(a)
Experiment
Model
2
|Voltage FRF| (V/g)
10
Increasing Rl
0
10
-2
10
180
|Tip Velocity FRF| ((m/s)/g)
(b)
190
200
210
220
Frequency (Hz)
230
240
250
0.5
Experiment
Model
Experiment
(circuit connected)
0.4
0.3
0.2
0.1
Increasing Rl
0
190
200
210
220
230
Frequency (Hz)
240
250
Figure 3.10: Experimental and numerical (a) voltage - to - base acceleration FRFs and (b)
tip velocity - to - base acceleration FRFs of self-charging structure for series electrode connection case and for various load resistances.
61
(a)
Experiment
Model
2
|Voltage FRF| (V/g)
10
Increasing Rl
0
10
-2
10
180
190
200
210
220
Frequency (Hz)
230
240
250
|Tip Velocity FRF| ((m/s)/g)
(b) 0.5
Experiment
Model
0.4
0.3
0.2
0.1
Increasing Rl
0
180
190
200
210
220
Frequency (Hz)
230
240
250
Figure 3.11: Experimental and numerical (a) tip velocity - to - base acceleration FRFs and
(b) voltage - to - base acceleration FRFs of self-charging structure for parallel electrode
connection case and for various load resistances.
the series and parallel cases, respectively, and the predicted values are 0.387 (m/s)/g and
0.440 (m/s)/g.
After the preliminary analysis for the resistive load case, the piezoceramic and thin-film
battery layers are connected to the input and output of a simple linear voltage regulator
energy harvesting circuit consisting of a full bridge diode rectifier, smoothing capacitance,
and a Texas Instruments, Inc. [122] TPS71501 adjustable output voltage regulator, shown
schematically in Fig. 3.12. The electrical boundary conditions of the piezoceramic layers
then become more sophisticated. The tip velocity FRF is measured for the series electrode
62
connection case (which will be used in the following section for the charge/discharge measurements), and is plotted in Fig. 3.10(b). It appears from the figure that the case with the
largest resistive load (effectively 500 kΩ, which is close to open-circuit conditions) most
closely represents the vibration response of the self-charging structure when connected to
the circuit. The resonant frequency in this configuration is found to be around 210.0 Hz.
Based on the voltage FRFs given in Fig. 3.10(a) and Fig. 3.11(a), several electrical performance curves can be extracted to better describe the electrical behavior of the system.
The variation of the peak voltage, current, and electrical power with load resistance can be
determined at both the short-circuit and open-circuit resonance frequencies, which are of
particular importance for energy harvesting purposes as they provide bounds on the fundamental resonance frequency of the system under examination. For any load resistance,
the fundamental natural frequency will lie between the short-circuit and open-circuit resonance frequencies. The variation of the peak voltage with load resistance is presented in
Fig. 3.13(a) and Fig. 3.13(b) for the series and parallel electrode connections, respectively.
The model predictions are in good agreement with the experimental data. The voltage
initially follows a linearly increasing trend with increasing load resistance, but approaches
a horizontal asymptote at large load resistance values. The peak voltage output values
predicted by the model for the largest load resistance at the short-circuit and open-circuit
resonance frequencies are 10.26 V/g and 27.30 V/g, respectively for the series case and 5.21
V/g and 15.54 V/g, respectively for the parallel case.
Figure 3.13(c) and Fig. 3.13(d) show the variation of the peak current output with load
TPS71501
Voltage Regulator
+
C1 100 F
PZT
C 2 0.22 F
Bandgap
Reference
-
C 3 0.22 F
+
Vo
-
R1
2.24 M
R2
950 k
Figure 3.12: Linear voltage regulator energy harvesting circuit.
63
(a)
2
(b)
10
1
10
Voltage (V/g)
Voltage (V/g)
1
0
10
Experiment (211.6 Hz)
Experiment (203.9 Hz)
Model (211.6 Hz)
Model (204.0 Hz)
-1
10
-2
10
2
10
4
10
Load Resistance ()
10
0
10
Experiment (211.7 Hz)
Experiment (204.2 Hz)
Model (211.6 Hz)
Model (204.0 Hz)
-1
10
-2
10
6
2
10
10
(c)
4
10
Load Resistance ()
6
10
(d)
0
0
Current (mA/ g)
10
-1
10
Experiment (211.6 Hz)
Experiment (203.9 Hz)
Model (211.6 Hz)
Model (204.0 Hz)
-2
10
2
10
Power (mW/g2)
(e)
-1
10
10
6
10
(f)
0
10
-1
10
Experiment (211.6 Hz)
Experiment (203.9 Hz)
Model (211.6 Hz)
Model (204.0 Hz)
-2
10
-3
2
4
10
Load Resistance ()
2
10
1
10
Experiment (211.7 Hz)
Experiment (204.2 Hz)
Model (211.6 Hz)
Model (204.0 Hz)
-2
4
10
Load Resistance ()
10
10
10
Power (mW/g2)
Current (mA/ g)
2
10
4
10
Load Resistance ()
1
10
0
10
-1
10
Experiment (211.7 Hz)
Experiment (204.2 Hz)
Model (211.6 Hz)
Model (204.0 Hz)
-2
10
-3
10
6
10
6
10
2
10
4
10
Load Resistance ()
6
10
Figure 3.13: Experimental and numerical electrical performance curves of self-charging
structure for the peak voltage output in the (a) series and (b) parallel case, current output
in the (c) series and (d) parallel case, and power output in the (e) series and (f) parallel case
with varying load resistance for the self-charging structure.
resistance for the series and parallel electrode connection cases, respectively. A converse
behavior can be observed in the current output curves compared to the voltage output
curves. A horizontal asymptote exists for small load resistance values and the current
linearly decreases for larger load resistance values. The peak current output is predicted
by the model for the smallest load resistance at the short-circuit and open-circuit resonance
frequencies as 0.99 mA/g and 0.32 mA/g, respectively, for the series case and 1.93 mA/g
and 0.64 mA/g, respectively, for the parallel case.
64
Lastly, the variation of the power output with load resistance is given in Fig. 3.13(e)
and Fig. 3.13(f) for the series and parallel cases, respectively. For both cases, a peak power
output exists for each excitation frequency at different optimal load resistance values between short-circuit and open-circuit conditions. Additionally, the peak power output for
the short-circuit and open-circuit conditions are roughly equal. Furthermore, the peak
powers predicted for the series connection and parallel connection are identical. The peak
power output for excitation at the short-circuit resonance frequency is predicted by the
model as 2.68 mW/g2 for a load resistance of 10.55 kΩ for the series connection and 2.62
kΩ for the parallel connection. For excitation at the open-circuit resonance frequency, the
model predicts a peak power output of 2.70 mW/g2 for a load resistance of 100.1 kΩ for
the series connection and 27.77 kΩ for the parallel connection. Again, these peak values
are linear estimates based on low-amplitude excitation and are not necessarily accurate for
large amplitude excitation.
The results presented in Fig. 3.10 and Fig. 3.11 as well as Fig. 3.13 validate the electromechanical assumed modes model through comparison of simulation results to experimental measurements. Both the vibration and voltage response of the self-charging structure over a range of frequencies surrounding the fundamental frequency (which is of most
interest in piezoelectric energy harvesting) are successfully predicted by the model. The
coupled model can, therefore, be used to accurately predict the behavior of self-charging
structures excited via harmonic base excitation.
3.3.2
Self-Charging Structure Concept Validation
The ability of the self-charging structures to operate in a self-charging manner in which
energy is simultaneously harvested in the piezoelectric layers and stored in the battery
layers is explored. Charge/discharge experiments are conducted in which the device is
initially excited at resonance with the piezoelectric layers connected through the linear
regulator harvesting circuit to charge the battery while the voltage of and current flowing
into the thin-film battery layer is monitored. Subsequently, the thin-film battery is discharged while monitoring battery current and voltage. Charge and discharge profiles can
then be defined.
The resonance frequency of the self-charging structure when connected to the energy
harvesting circuit circuit has been obtained in the previous section as 210.0 Hz (Fig. 3.10(b)).
65
The experimental setup for charge/discharge measurements (similar to the setup used
for FRF measurements) is shown in Fig. 3.14. The new equipment used here includes a
National Instruments Corp. [110] CompactDAQ chassis with a NI 9215 4-channel analog
voltage input card with 16-bit precision (used to measure battery voltage and current measurement from transimpedance amplifier), a NI 9233 4-channel 24-bit analog input card
with IEPE coupling (used for accelerometer measurement), and a NI 9263 4-channel analog output card with 16-bit precision and ±10 V range (used to excite shaker). A custom
LabVIEW program is written to perform the data acquisition and signal generation using
the National Instruments hardware. The linear voltage regulator circuit is also pictured
in Fig. 3.14. For this experimentation, the two piezoelectric layers of the self-charging
(e)
(f)
(h)
(g)
(a)
(b)
(d)
(c)
(d)
(c)
(c)
(a)
(a) NI CompactDAQ data acquisition system and computer; (b) Dual power
supplies; (c) Voltage regulator circuit; (d) Electromagnetic shaker; (e) Fixed
gain amplifier; (f) Accelerometer; (g) Self-charging structure; (h) Clamp
Figure 3.14: Experimental setup used to obtain charge/discharge measurements of selfcharging structures.
66
structure are connected in series for increased voltage output and used to charge a single
battery layer. The input base excitation amplitude is set to ±1.0 g. The device is excited
at resonance for 1 hour and the battery voltage and current into the battery are measured
throughout the test. Once the test is complete, the battery is discharged using a 2749 Ω resistor in order to draw roughly 2C of current (1.4 mA) from the battery. Results from both
the charge and discharge tests are shown in Fig. 3.15. From Fig. 3.15(a), it can be seen that
the piezoelectric layers are able to supply an average of about 0.08 mA of current into the
battery. Using Eq. (3.2), the capacity during charging is found to be 0.0781 mAh. During
(a)
0.25
4.1
Current (mA)
0.2
3
0.15
Current
0.1
2
0.05
Voltage (V)
Voltage
1
Charge Capacity:
0.0781 mAh
0
0
1000
1500 2000
Time (sec)
2500
3000
0
3
4.1
3.8
Voltage
2
Current (mA)
3500
3
Load
Disconnected
Current
2
1
Voltage (V)
(b)
500
Discharge Capacity:
0.0663 mAh
1
0
0
50
100
150
Time (sec)
200
250
0
Figure 3.15: Experimental curves for self-charging structures in (a) charging and (b) discharging under ±1.0 g acceleration at 210.0 Hz.
67
discharging, the current output is held at 1.4 mA for about 120 seconds before beginning
to decay, as shown in Fig. 3.15(b), and a capacity of 0.0663 mAh is calculated. There is a
slight difference between the capacities calculated in charging and discharging, which is
likely a leakage effect where some of the energy during charging is dissipated, thus there
is a small decrease in capacity when discharging.
Although a simple linear regulator circuit is used here to evaluate the performance of
the self-charging structures (the goal of this analysis is to prove the ability to self-charge),
more advanced circuitry can be used to improve the amount of energy extracted from
the piezoelectric layers and transferred to the battery layers. Although not shown here, a
nonlinear switching circuit with impedance matching ability has been investigated and it
has been found that significant increases in efficiency can be obtained for high excitations
levels (the power draw of the self-powered switching circuit outweighs any advantages
under low level excitation) [51, 123].
The charge/discharge results presented in Fig. 3.15 prove the ability of the self-charging
structures to simultaneously generate and store electrical energy in a multifunctional manner, and validate the concept of self-charging. Furthermore, the current of 0.08 mA corresponds to an average power of around 0.306 mW during charging. This is a reasonable
value for piezoelectric energy harvesting, where typical harvested powers are in the microwatt to milliwatt range [4].
3.4
Chapter Summary
This chapter presents details of the fabrication and experimental testing of self-charging
structures. The evaluation and selection of thin-film batteries is first given. Fabrication of
a self-charging structure prototype is then reviewed. Lastly, results of experimental testing are presented including both frequency response function measurements and charge/
discharge measurements to validate the electromechanical model presented in Chapter 2
and the ability of the device to act in a self-charging manner.
R
R
Two types of thin-film batteries are investigated; NanoEnergy
and Thinergy
cells.
R
Based on preliminary testing results, Thinergy
batteries are selected for use in self-charg-
ing structures due to their superior robustness and packaging. Fabrication of the prototype
device is achieved through a vacuum bagging procedure using two part epoxy to bond
68
each layer, thus providing thin and uniform bonding layers.
Experimental testing is performed to validate both the electromechanical assumed modes model and the self-charging ability of the device. Frequency response measurements
are recorded for the self-charging structure mounted in a cantilever configuration and subjected to harmonic base excitation for a range of load resistances. Simulation results predicted by the model are compared to the experimental measurements and the model is
found to accurately predict the vibration and voltage response of the harvester for each
load resistance tested. Experimental charge/discharge testing is also conducted in which
the energy harvesting ability of the self-charging prototype is investigated. With the piezoelectric layers connected in series to charge a single battery, the device is excited at resonance for 1 hour and found to successfully charge the battery. Discharge measurements
confirm that the battery was charged during the test. Results of the charge/discharge measurements prove the ability of the device to act in a self-charging fashion.
69
C HAPTER
4
S TRENGTH A NALYSIS OF S ELF -C HARGING
S TRUCTURES
I
T
has been proposed that self-charging structures be directly integrated into host struc-
tures in a multifunctional manner. Inherent in this proposal is the fact that the self-
charging structures must act as load bearing members. Experimental testing is carried
out on self-charging structures in order to determine their strength properties. Results of
the strength testing can be used as a design tool in the development of embedded selfcharging structure systems. Both static three-point bend testing and dynamic harmonic
base excitation testing are considered. Additionally, static testing is performed on a variety of piezoelectric materials as the piezoelectric layer is found to be the critical layer in
static failure. The following sections outline the formulations and procedures used to define the failure strength of the self-charging structures and piezoelectric materials, as well
as the results of experimental failure testing.
4.1
Static Strength Analysis of Self-Charging Structures
Bending tests are typically employed to evaluate the tensile strength of brittle materials
(such as the piezoceramic layers in the case of self-charging structures) [124]. Classical
three-point bending tests are performed in order to experimentally evaluate the strength of
the individual components of the self-charging structures as well as the complete structure
under static loading.
70
4.1.1
Strength Calculations for 3-Point Bend Testing
Consider the schematic of a three-point bending test setup shown in Fig. 4.1. The rectangular test specimen is loaded in the transverse direction with a load of P . The load
is applied at the center of the support span (x = L/2), therefore, the maximum bending
moment occurs at this point and is equal to Mmax = P L/4. The load required for failure
of the specimen by transition from elastic material behavior to either plastic behavior (for
ductile materials) or abrupt failure (for brittle materials) can be defined as the mechanical
failure load, Pf , which can be used to define a mechanical failure strength. The maximum
bending moment that corresponds to the failure load, Pf , of the complete device is defined
as the failure bending moment, Mf .
The bending strength of a simple beam placed under three-point bending is defined
from Euler-Bernoulli beam theory as
σb =
3L
Pf
2bh2
(4.1)
where σb is the bending strength of the specimen, b is the specimen width and h is the
specimen thickness.
The maximum bending stress of a given layer of a multilayer composite device, such
as the complete self-charging structure, can be defined as
σbmax =
Yk hkn L
Yk hkn
Mf =
Pf
YI
4Y I
(4.2)
where σbmax is the maximum bending stress of a layer at a given failure load of the device,
Yk is the elastic modulus of the layer of interest (layer k), hkn is the distance from the
neutral axis to the outer surface of the kth layer, and Y I is the overall bending stiffness of
the multilayer beam. The calculation of Y I for a composite beam has been discussed in
Chapter 2, and details of the formulation for self-charging structures (both the root section
P
L
Figure 4.1: Schematic of three-point bending test.
71
without battery layers and the tip section with battery layers) are given in Appendix B.2. A
beam-like aspect ratio is assumed in the foregoing derivation, where thin-plate parameters
can be used for bending of plate-like configurations [30].
Equation (4.1) gives the failure strength of the self-charging structure layers when
tested individually, where Eq. (4.2) can be used to estimate the maximum stress of an
individual layer for the failure load of the total device. It is worth mentioning that the
maximum stress of a layer for the failure load of the entire structure might be lower than
its individual failure strength. For example, for the failure load that results in fracture of
a piezoceramic layer in a multi-layer assembly, the maximum stress in the substrate layer
could be lower than its individual failure strength. Nevertheless, the overall structure is
assumed to be failed when any layer starts exhibiting brittle or ductile failure behavior.
4.1.2
Experimental Testing and Results
R
Experimental testing is performed using an Instron
[125] 4204 universal test frame with
an MTS Systems Corp. [126] ReNew upgrade package equipped with a 1000 N load cell
and a small three-point bend fixture with adjustable supports, shown in Fig. 4.2. Each
specimen rests on the two lower support pins, which are spaced 20 mm apart, and the central pin is lowered at a rate of 0.3 mm/min until a prescribed displacement is reached. In
each case, the specimens fail before the maximum displacement is achieved. Both the load
and the crosshead displacement are recorded throughout each test using MTS TestWorks
4 software.
R
Three individual samples are tested for the aluminum substrate, QuickPack
QP10n
Upper
loading pin
Test
specimen
Bottom
adjustable
support pins
Figure 4.2: Experimental setup used for three-point bend testing including Instron/MTS
4204 test frame and fixture.
72
R
piezoceramic, and Thinergy
battery layers. Conventionally, three-point bend testing is
performed on beam-shaped samples in order to eliminate Poisson effects. It is desirable,
R
battery samples in an unmodified state as dicing the bathowever, to test the Thinergy
teries could result in damage to the packaging or delamination, therefore, plate-like samples are tested here. The aluminum specimens are cut to 25.4 mm x 25.4 mm, and the
R
QuickPack
samples are cut in half to fit in the test fixture (resulting in two identical sam-
ples of about 25.4 mm x 25.4 mm). A single self-charging structure is tested and cut in
half such that each section can be tested separately. Photographs of each component after failure testing are shown in Fig. 4.3. Typical load-deflection curves for the individual
layers as well as the complete structure are shown in Fig. 4.4. From the results presented
R
in Fig. 4.4(a), it is clear that the individual QuickPack
piezoceramic layers exhibit brittle
R
failure and the individual aluminum substrate and Thinergy
battery layers exhibit duc-
tile failure. In the case of the aluminum sample, the failure load is taken where a slight,
prolonged drop in the force is observed, as noted in the figure. From Fig. 4.4(b), it can be
seen that the root section of the self-charging structure experiences brittle failure, where
(a)
(b)
(c)
(d)
(e)
R
Figure 4.3: Various specimens after three-point failure testing including (a) Thinergy
R
battery, (b) QuickPack
QP10n piezoelectric, (c) aluminum substrate, (d) root section of
complete self-charging structure, and (e) tip section of complete self-charging structure.
73
(a)
14
QuickPack
12
Brittle Failure
Thinergy
Load (N)
10
8
Ductile Failure
6
Aluminum
4
Aluminum
QuickPack
Thinergy
Linear Curve Fit
2
0
-2
0
0.5
1
1.5
2
2.5
Displacement (mm)
3
3.5
(b)
Simultaneous Ductile/Brittle Failure
180
250
160
Tip Section
Root Section
Linear Curve Fit
140
Load (N)
200
120
0.1 0.15 0.2 0.25
150
Tip
100
50
0
0
Root
0.5
1
1.5
2
2.5
Displacement (mm)
3
3.5
4
Figure 4.4: Load-deflection curves for (a) individual layers and (b) complete self-charging
structure sections.
the tip section exhibits simultaneous ductile and brittle failure behavior. This simultaneous failure phenomenon is likely due to failure occurring in the piezoceramic (brittle) and
battery (ductile) layers for nearly the same applied load. The failure load results for all of
the specimens tested are presented in Table 4.1.
With the failure loads obtained, Eq. (4.1) and Eq. (4.2) can be used to obtain the maximum bending stress values for each sample tested. The minimum failure load is used in
the calculations for the individual layers to give a conservative estimate. For the complete
self-charging structure, the overall bending stiffness, Y I of the root section (containing
only the aluminum substrate and piezoceramic layers) is calculated as Y I = 0.0647 Nm2 ,
74
Table 4.1: Failure loads for three-point bending tests.
Parameter
Failure Load (N)
Minimum (N)
Failure Load (N)
Aluminum
Substrate
QP10n
Device
R
Thinergy
Batteries
3.21
7.25
6.58
3.36
8.80
5.47
3.66
8.50
5.89
3.21
7.25
5.47
Complete Self-Charging Structure
Root
Section
Tip Section
39.9
165.3
and of the tip section (containing the aluminum substrate, piezoceramic layers, and battery
layers) is calculated as Y I = 0.2380 Nm2 . From the geometry of the device given previously in Section 3.2, the distances from the neutral axis to the outer layers of the aluminum
substrate, piezoelectric, and battery layers are calculated as hkn = 0.0762 mm, hkn = 0.4142
mm, and hkn = 0.6760 mm, respectively. The calculated failure strength values for each
of the specimens are listed in Table 4.2. It should be noted that the calculation of failure
R
stress in the QuickPack
layers considers the dimensions of only the active element in
calculating hkn , ignoring the outer Kapton layer, as the ceramic layer experiences failure.
From the results, it can be seen that failure in the root section of the self-charging strucTable 4.2: Failure strengths for three-point bending tests.
Parameter
Aluminum
Substrate
QP10n
Device
R
Thinergy
Batteries
Individual Layers
Failure Stress
(MPa)
229.27
159.82
199.33
Self-Charging Structure - Root
Failure Stress
(MPa)
15.84
83.62
-
Self-Charging Structure - Tip
Failure Stress
(MPa)
17.84
94.18
75
126.18
ture is due to failure of the piezoceramic layers. At the point of failure, the maximum stress
in the aluminum layer is much less than the failure stress observed in a single aluminum
R
is about half of the failure stress obtained
layer. The maximum stress in the QuickPack
for a single layer, however, it is on the same order of magnitude. Although there is a significant difference between the maximum stress of the single layer and composite device,
it is typical in brittle failure to observe a wide range of failure loads (thus failure stresses)
for a single material. Results for the tip section of the self-charging structure show failure
in both the piezoceramic and battery layers with stresses similar to the failure stress of the
individual layers in both cases. This result is confirmed by the simultaneous brittle and
ductile failure observed in Fig. 4.4(b). Overall, it can be concluded that the piezoceramic
and battery layers are the critical layers in three-point bending failure.
4.2
Dynamic Strength Analysis of Self-Charging Structures
Piezoelectric energy harvesters must be subjected to dynamic vibration excitation in order to create useful electrical energy. A series of dynamic strength tests are conducted to
gain an understanding of the dynamic loading that can be withstood by the self-charging
structures without failure. Specifically, harmonic base excitation at increasing acceleration
amplitudes is imposed on a cantilever structure which is monitored for signs of mechanical
and/or electrical failure.
4.2.1
Strength Calculations for Harmonic Base Excitation Testing
Based on the assumed modes model presented in Chapter 2, the maximum dynamic stress
of the kth layer of a self-charging structure excited under base excitation can be expressed
in terms of a stress - to - base acceleration FRF. Recall from Eq. (2.5) that the stress in a
structural layer is given as
σ(x, z, t) = Ys Sxx
(4.3)
∂ 2 wrel (x, t)
∂x2
Recognizing that the maximum stress in a particular layer of interest will occur at the
= −Ys z
outermost surface of the layer, the maximum stress in a structural layer can be defined as
∂ 2 wrel (x, t) max
σk (xcr , hkn , t) = −Yk hkn
(4.4)
∂x2
x=xcr
76
where Yk is the elastic modulus of the kth layer, hkn is the distance from the neutral axis to
the outermost surface of the kth layer, and xcr is the position along the length of the beam
where the curvature is maximum (e.g. it is the root for the fundamental mode of a uniform
cantilever beam).
Similarly, for a piezoelectric layer, Eq. (2.9) leads to the following expression for the
maximum stress in a piezoelectric layer
σkmax (xcr , hkn , t) = −cE
11 hkn
∂ 2 wrel (x, t) v(t)
− e31
∂x2
hkn
x=xcr
(4.5)
Note that the piezoelectric layers include a voltage induced stress contribution that always
adds to the stress. The sign of both terms in Eq. (4.5) varies depending on the poling
direction of the piezoceramic layer, however, the absolute value of both terms can be used
for simplification.
The relative beam displacement expression given in Eq. (2.87) leads to
∂ 2 wrel (x, t)
00
= wrel (x, t) = Φ” T (x) −ω 2 [M] + jω [C] + [K]
2
∂x
!−1
−1
1
Fejωt
Θeq Θeq T
+jω jωCpeq +
Rl
(4.6)
Using Eq. (4.6) and the expression previously given for the voltage output of the selfcharging structures (Eq. (2.86)) along with Eq. (4.4) and Eq. (4.5), the following expression
can be defined to describe the maximum stress FRF in any layer of the self-charging structure
00
σkmax (xcr , hkn , t) wrel (xcr , t) vk (t) = Yk hkn
+ λk e31
ab ejωt
ab ejωt hp ab ejωt (4.7)
where Yk is the elastic modulus of the kth layer (the elastic modulus measured at constant electric field in the case of a piezoelectric layer) and λk is equal to 1 if layer k is a
piezoelectric layer, otherwise it is zero.
Using Eq. (4.7), the maximum dynamic stress values of the individual layers can be predicted for a given value of base acceleration. The same self-charging structure prototype
described and tested in Chapter 3 (shown in Fig. 3.8) is tested here for dynamic failure.
Again an overhang length of 44.2 mm is achieved with the device clamped to the shaker.
Both the aluminum and the piezoelectric layers are clamped at the root of the device, therefore, the maximum stress in those layers is expected to occur at the root (i.e. xcr = 0 mm).
77
The battery layers, however, are near the free end of the beam, therefore, the maximum
stress occurs at the edge of the battery closest to the root of the cantilever (xcr = 18.8 mm).
Estimates of the maximum stress - to - base acceleration FRFs of the aluminum, piezoceramic, and battery layers are given in Fig. 4.5. From the results, the maximum bending
stress of the aluminum, piezoelectric, and battery layers are found to be 3.1 MPa/g, 17.7
MPa/g, and 4.6 MPa/g, respectively. These linear estimates of the stress per base acceleration provide insight into the amount of excitation that can be safely imposed on the device.
For large amplitude excitations, however, both geometric and material nonlinearities may
exist in the cantilever piezoelectric structure, thus the linear estimates must be used with
caution.
4.2.2
Experimental Testing and Results
Dynamic failure testing is conducted using the same experimental setup used previously
in Chapter 3 for charge/discharge measurements (shown in Fig. 3.14) by subjecting the
cantilever harvester to resonant base excitations of increasing amplitude until electrical
or mechanical failure is observed. Electrical failure is defined as a 10% decrease in either
the charge or discharge behavior of the self-charging structure as compared to a baseline
charge/discharge profile, signifying failure in the thin-film battery layers. Mechanical failure is defined as fracture or cracking of the aluminum or piezoceramic layers evident from
|Maximum Stress FRF| (MPa/g)
20
Aluminum
Piezoceramic
Battery
15
10
5
0
180
190
200
210
220
Frequency (Hz)
230
240
250
Figure 4.5: Estimates of the maximum dynamic bending stress in the aluminum, piezoceramic, and battery layers.
78
changes in the dynamic behavior of the system and from the ability of the piezoelectric
layers to deliver energy to the battery layers. The self-charging structure is clamped to the
shaker and remains undisturbed throughout the duration of the dynamic testing.
An initial baseline charge/discharge measurement is obtained for the device by following the procedure outlined previously in Subsection 3.1.3 and all future measurements for
battery failure are compared to this baseline. Once the baseline is obtained, the device is
first excited at resonance at an initial acceleration input level of ±0.2 g for 1 hour. During
the test, the piezoceramic layers are connected in series to the linear regulator circuit and
used to charge a single thin-film battery layer (which is initially fully discharged to 3.0
V). The battery voltage and current are monitored and recorded in order to evaluate the
health of the piezoelectric layers (mechanical failure will cause the output of the piezoelectric layers to vary). After 1 hour, the excitation is ceased and a discharge test is performed
on the battery by drawing 2C of current through a 2749 Ω load. The self-charging structure is then allowed to sit for 24 hours before testing is resumed, as chemical failure in the
battery (potentially due to delaminations) may require time to take effect. After 24 hours,
the thin-film battery is charged using the power supply and subsequently discharged. This
charge/discharge data is compared to the baseline charge/discharge profile recorded prior
to failure testing and significant changes indicate electrical failure in the battery (caused by
the excitation the previous day). Finally, the acceleration amplitude is increased and the
process is repeated. It is expected that for larger excitation amplitudes, the piezoelectric
layers will provide more power. Deviations in this trend indicate mechanical failure in the
piezoelectric layers. Complete results from the dynamic failure testing for the power supply charge/discharge (used to indicate electrical failure) are given in Fig. 4.6(a) for base
acceleration values from 0.2 g to 7.0 g. Additionally, the charge/discharge results with the
piezoceramic layers charging the battery (used to indicate mechanical failure) are given in
Fig. 4.6(b). All of the charge/discharge curves for both the power supply and piezoelectric
charging of the battery are given in Appendix D for reference. Based on the maximum
stress predictions given in Fig. 4.5, an upper limit of 7.0 g (corresponding to 125 MPa of
stress in the piezoceramic layers) is chosen. Recall from Subsection 4.1.2 that the piezoelectric layer exhibited a maximum stress of around 100-125 MPa in static failure. Although
linear estimates are used here, they provide a reasonable basis for limiting the dynamic
excitation level.
79
From the dynamic failure testing results presented in Fig. 4.6(a), it can be seen that
as the excitation amplitude is increased from 0.2 g to 7.0 g, there is no significant change
in the power supply charge or discharge behavior. In each case, the charge amplitude
is slightly higher than the discharge amplitude, likely due to leakage in the battery. The
power supply charge after 5.5 g excitation is abnormally high, thus the battery initially
appears damaged, but continuation of testing at higher excitation levels shows that the
battery functions properly. This phenomenon may be attributed to experimental variation.
Although it was expected that electrical failure would occur in the batteries at the acceler(a)
1
0.9
Charge
Discharge
Capacity (mAh)
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Baseline 0.2 0.4 0.6 0.8 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
Acceleration (g)
(b)
0.35
Charge
Capacity (mAh)
0.3
Discharge
0.25
0.2
0.15
0.1
0.05
0
0.2 0.4 0.6 0.8 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
Acceleration (g)
Figure 4.6: Charge/discharge capacities measured for (a) power supply and (b) piezoelectric charging for dynamic failure analysis of self-charging structures.
80
ation levels tested, no electrical failure was observed, hence an electrical failure strength
cannot be defined for the device.
The piezoelectric charge/discharge results presented in Fig. 4.6(b) show that the piezoelectric layers are able to partially charge the thin-film battery. As the excitation amplitude
is increased, the total charge capacity (as well as discharge capacity) monotonically increases. This is expected as higher excitation levels induce more vibration energy into
the harvester. The steady increase in charge capacity shows that no mechanical failure is
observed in the piezoelectric layers at the acceleration levels tested. The results given in
Fig. 4.6(b) also show a difference in the charge and discharge capacities for each test. This
variability is likely due to leakage in the battery as the current input from the piezoceramic
layers is quite low. Overall, the experimental results show that no electrical battery failure
or mechanical piezoelectric failure is observed for any of the excitation levels tested.
4.3
Static Strength Testing of Various Piezoceramic Materials
The results presented in Subsection 4.1.2 suggest that the piezoceramic and battery layers
are critical in static failure. While the selection of thin-film batteries is limited, a wide variety of piezoelectric ceramics exist for consideration in vibration energy harvesting systems
including self-charging structures. In this section, a more rigorous investigation of the
static bending strength of several commercially available piezoceramic materials through
three-point bend testing is performed to provide a design tool in the development of vibration energy harvesting systems in which the active device is subjected to bending loads.
Previous research exists in both the ceramics and smart structures fields in which the
bending strength of various types of piezoceramic materials has been investigated [127–
133]. Testing of soft piezoceramics [127–130], hard ceramics [131, 132], and single crystal
ceramics [133] has been reported. Researchers have investigated the effects of poling state,
surface condition, applied electric field, and temperature on the bending strength of the
materials. Although some data exists on the bending strength of piezoceramic materials,
the goal of the research presented in this section is to present a more comprehensive study
of the bending strength of several commonly used piezoceramic materials as well as more
recently developed PMN-PT and PMN-PZT single crystal piezoelectric materials.
81
4.3.1
Materials
Several commonly used piezoelectric materials are investigated. Perhaps some of the most
commonly used piezoelectric materials in vibration energy harvesting systems, two types
of soft PZT ceramic materials are tested. PZT-5A (DOD Type II) and PZT-5H (DOD Type
VI) ceramics (PSI-5A4E and PSI-5H4E, respectively) manufactured by Piezo Systems, Inc.
[134] are investigated. Both materials utilize vacuum sputtered nickel electrodes and are
poled through the thickness. Several researchers have used these PZT ceramics in energy
harvesting applications including PZT-5A for self-powered sensor nodes [135] and gunfire
shock munitions harvesting [136], and PZT-5H for microgenerators in wireless electronics
[22].
Two types of hard PZT ceramic are also tested. PZT-4 (DOD Type I) and PZT-8 (DOD
Type III) ceramics (PZT-844 and PZT-881, respectively) manufactured by APC International, Inc. [137] are studied. The materials utilize silver electrodes and are poled through
the thickness. While hard PZT is not typically used in energy harvesting applications,
PZT-4 has recently been proposed for use in harvesting magnetic energy through the combination of piezoelectric and piezomagnetic material layers in a composite device [138].
In addition to conventional PZT ceramics, two types of single crystal piezoelectric ceramics are also investigated. PMN-PT single crystal ceramics and PMN-PZT single crystals
(CPSC 160-95) produced by Ceracomp Co., Ltd. [139] are tested. The materials contain gold
electrodes and are poled through the thickness. Several researchers have investigated the
use of PMN-PT in energy harvesting applications to utilize its large piezoelectric coupling
[140, 141]. Although a relatively new material, some research has also been conducted to
investigate the use of PMN-PZT single crystal ceramics for energy harvesting applications
[142, 143].
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The final types of piezoelectric material investigated are QuickPack
devices. Both
QP10n (used in self-charging structures) and QP16n devices are investigated. QP16n are
similar to QP10n, with the difference being in the thickness of the devices (0.254 mm thickness for QP16n as opposed to 0.381 mm thickness for QP10n).
82
4.3.2
Sample Preparation
All of the materials investigated are supplied as bulk material from the manufacturers,
with the exception of the PMN-PT samples which are supplied as beam like specimens
that do not require further processing. The physical dimensions as well as the manufacturer part numbers for all of the materials tested in this work are given in Table 4.3. Typical
bend test specimens are beam like with large aspect ratios, thus processing of the bulk material is required to obtain appropriate samples for testing. The ASTM C 1161-02c standard,
titled “Standard Test Method for Flexural Strength of Advanced Ceramics at Room Temperature” was consulted in preparing the test samples [144]. Samples with dimensions
of 4 mm x 3 mm x 45 mm are specified in the standard, however, the thicknesses of the
materials acquired dictate a deviation from the standard. In order to maintain beam like
samples, the bulk material is cut using a dicing saw to provide samples of approximately
2 mm width. A MicroAutomation 1006 dicing saw is used with 2-6 µm grit, 75 µm wide
diamond dicing blades, a spindle speed of 28000 rpm, and a feed rate of 2 mm/sec. The
length of the diced samples varies from 20.0 mm to 46.0 mm across the different material types. After completion of the dicing operation, the samples are investigated using a
Nikon Instruments, Inc. [145] Eclipse LV100 optical microscope in order to obtain a precise measurement of the width of each sample, and additionally to determine the average
flaw size induced in the edges from the dicing process. Figure 4.7 shows an example of
an image acquired using the optical microscope in which the width of a PMN-PZT sample
Table 4.3: Physical dimensions of various piezoceramic materials tested.
Material
Manufacturer
Part Number
Bulk Dimensions
(mm)
Final Sample
Dimensions (mm)
PZT-5A
T110-A4E-602
72.4 x 72.4 x 0.267
36.18 x 1.959 x 0.267
PZT-5H
T110-H4E-602
72.4 x 72.4 x 0.267
36.18 x 1.959 x 0.267
PZT-4
PZT-844
40.0 x 10.0 x 0.5
20.0 x 1.950 x 0.5
PZT-8
PZT-881
25.4 x 25.4 x 1.0
25.4 x 1.914 x 1.0
PMN-29PT
30.0 x 2.012 x 0.28
30.0 x 2.012 x 0.28
PMN-PT
PMN-PZT
CPSC 160-95
40.0 x 10.0 x 0.28
40.0 x 1.967 x 0.28
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QuickPack
QP10n
46.0 x 20.6 x 0.254
46.0 x 1.962 x 0.254
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QuickPack
QP16n
46.0 x 20.6 x 0.152
46.0 x 1.960 x 0.152
83
Dicing cut
Piezoelectric
material
Adhesive
substrate
Figure 4.7: Optical microscope image of PMN-PZT sample after dicing showing width
measurement (5X objective lens).
has been measured. Characteristic images of the flaws induced during diced are shown in
Fig. 4.8, and the average flaw size is found to be about 6 µm, 8 µm, 40 µm, 35 µm, 19 µm,
and 19 µm for PZT-5A, PZT-5H, PZT-4, PZT-8, PMN-PT, and PMN-PZT, respectively. The
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outer Kapton layers of the QuickPack
devices prohibited optical measurement of flaw
size in the piezoelectric layers. Figure 4.9 shows the final prepared samples ready for bend
testing. The final sample dimensions for each material are also listed in Table 4.3.
4.3.3
Experimental Setup
Three-point bend testing is carried out using an Instron [125] 5848 MicroTester frame
equipped with a 50 N load cell, along with the same 3-point bend fixture used previously.
The experimental setup is shown in Fig. 4.10. Per the ASTM C 1161-02c test standard, the
crosshead rate for each test is specified in order to provide a strain rate of 1×10−4 s−1 using
(a)
(b)
Flaw
Flaw
Figure 4.8: Optical microscope images of (a) PMN-PZT and (b) PZT-5H samples showing
flaw sizes (20X objective lens).
84
(a)
(b)
(d)
(e)
(c)
(f)
(g)
(h)
Figure 4.9: Diced piezoelectric samples prepared for bending tests including (a) PZT-5H,
(b) PZT-5A, (c) PZT-8, (d) PZT-4, (e) PMN-PZT, (f) PMN-PT, (g) QP10n, and (h) QP16n.
the following relation [144]
ε=
6hs
L2
(4.8)
where ε is the strain rate, h is the thickness of the sample, s is the crosshead rate, and L
is the support span. The number of samples, crosshead rate and support span for each
type of material tested are listed in Table 4.4. During each test, the load and crosshead
displacement are recorded at a sampling rate of 50 Hz using Instron Bluehill 2 software.
85
(a)
(b)
Instron test
frame
Test sample
3-point
bend fixture
Figure 4.10: Three-point bend testing experimental setup including (a) Instron 5848 MicroTester frame and (b) fixture.
4.3.4
Experimental Results
Typical load-displacement curves recorded during the bending tests for each material investigated are shown in Fig. 4.11. Separate graphs are presented for the ‘thin’ samples
(PZT-5A, PZT-5H, PMN-PT, PMN-PZT, and QP10n) and for the ‘thick’ samples (PZT-4
and PZT-8) for clarity. It should be noted that the QP16n samples did not exhibit any internal cracking or failure throughout the entire displacement range of the test. Although both
R
QuickPack
devices contain identical PZT-5A active material, the piezoceramic layer con-
tained in the QP16n samples is considerably thinner than that of the QP10n devices, thus
Table 4.4: Various test parameters for the piezoelectric materials investigated.
Material
Number of
Samples
Crosshead Rate
(mm/min)
Support Span
(mm)
PZT-5A
50
2.341
25
PZT-5H
50
2.341
25
PZT-4
30
0.450
15
PZT-8
30
0.400
20
PMN-PT
30
2.232
25
PMN-PZT
10
3.214
30
QP10n
30
4.823
35
QP16n
30
8.059
35
86
(a)
0.6
0.5
Load (N)
0.4
0.3
PZT-5A
PZT-5H
PMN-PT
PMN-PZT
QP10n
0.2
0.1
0
0
1
2
3
4
5
Crosshead Displacement (mm)
6
7
(b)
8
Load (N)
6
4
2
PZT-4
PZT-8
0
0
0.05
0.1
0.15
Crosshead Displacement (mm)
0.2
Figure 4.11: Load-deflection curves for (a) thin samples (PZT-5A, PZT-5H, PMN-PT, PMNPZT, and QP10n) and (b) thick samples (PZT-4 and PZT-8).
the maximum curvature obtained during testing is not large enough to induce the strains
required for failure of the piezoceramic layer in the QP16n samples. No failure data is
obtained for the QP16n samples and the material will not be discussed further. The failure
loads observed during testing are used in Eq. (4.1) to determine the bending strength of
each sample. The failure load, Pf , is taken as the maximum load observed during the test
for the materials exhibiting classic brittle failure (PZT-5A, PZT-5H, PZT-4, PZT-8, PMN-PT,
and PMN-PZT), and as the first peak in the load, which occurs immediately before the initial cracking, in the QP10n material. Bending strengths calculated using the failure loads
obtained from the experiments are presented in Fig. 4.12 for all materials tested. Load87
displacement plots for all samples of each material tested are given in Appendix E for
reference. A summary of the experimental results for each material investigated is given
below.
PZT-5A and PZT-5H
Both PZT-5A and PZT-5H materials exhibit similar brittle behavior during 3-point bending tests, which can be observed from the results shown in Fig. 4.11(a). Both materials
(whose samples are of identical dimensions) follow a similar load-displacement trend with
an abrupt failure typical of brittle materials, however, PZT-5H fails at a slightly lower load
than PZT-5A. From the bending strength results shown in Fig. 4.12, it can be seen that the
PZT-5H samples, on average, have lower strength values than PZT-5A over the entire sample set. Strength testing of ceramic materials often exhibits significant variability between
samples [146]. The variability observed in the PZT-5A and PZT-5H strength data is quite
reasonable, and the results are considered to be favorable.
PZT-4 and PZT-8
From Fig. 4.11(b) it can be seen that both PZT-4 and PZT-8 samples exhibit linear loaddisplacement trends and classic brittle failure. The sample sizes for both materials differ,
therefore, direct comparison of the load-deflection curves cannot be made. Based on the
300
PZT-5A
PZT-5H
PZT-4
PZT-8
PMN-PT
PMN-PZT
QP10n
Failure Strength b (MPa)
250
200
150
100
50
0
Sample
Figure 4.12: Bending strength values calculated for various piezoelectric materials tested.
88
calculated failure strength values presented in Fig. 4.12, however, it can be shown that
the two material exhibit very similar average bending strengths, which lie between the
bending strengths of PZT-5A and PZT-5H.
PMN-PT and PMN-PZT Single Crystals
As with the other monolithic ceramics tested, both PMN-PT and PMN-PZT materials fail
in a brittle fashion, as shown in Fig. 4.11(a). Again, the sample sizes differ between materials, so comparison of the load-deflection diagrams will not be made. The bending strength
values given in Fig. 4.12 show that the bending strengths of the single crystal materials
are significantly lower than those of the other materials tested. This difference in strength
can be explained due to the unique crystal structure of the PMN-PT and PMN-PZT samples which allows for rapid propagation of cracks originating at flaw sites. The single
crystal samples are expected to give lower bending strengths as compared to the other ceramic materials investigated (the appeal of single crystal piezoelectric material for energy
harvesting purposes is the significant increase in piezoelectric coefficients over conventional piezoceramic materials). Variability in the bending strength results for single crystal
samples is also rather low and compares well to the results observed for the monolithic
samples. Additionally, the bending strength of PMN-PT samples is slightly higher than
the bending strength of PMN-PZT samples.
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QuickPack
QP10n
Load-displacement data for the QP10n samples exhibits unique behavior. Figure 4.11(a)
shows a typical load-displacement curve in which the load is observed to drop sharply
after an initial fracture, however, the samples recover and continue to exhibit several brittle
failures while maintaining a fairly constant average load. This unique behavior occurs
because of the outer Kapton layers surrounding the inner PZT-5A layer. The brittle failures
observed during the test are a result of the inner piezoceramic layer cracking, however,
unlike the traditional monolithic and single crystal ceramic samples, the QP10n sample
does not completely fail upon cracking of the PZT-5A layer. The outer Kapton is able to
maintain the integrity of the sample after cracking, and in fact, the samples never exhibit
complete fracture as the Kapton is able to resist complete failure throughout the entire
displacement range of the test. The maximum load observed immediately before the initial
89
R
cracking is taken as the failure load of the QuickPack
samples. Based on the failure
strength data presented in Fig. 4.12, it can be observed that the average bending strength
for the QP10n samples is notably higher than that of the conventional ceramic materials.
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This can possibly be attributed to the composite construction of the QuickPack
devices
which includes high shear strength epoxy on the piezoceramic surfaces. Additionally, the
variability between samples is significantly higher compared to that of the other materials
tested. The increased variability is likely due to the non-uniformity of the test samples
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as the electrode layers of the QuickPack
devices do not cover the entire surface of the
device, hence some diced samples contained outer electrode layers while others lacked
such layers. It was observed that samples with outer electrode layers exhibited higher
strength values than samples absent of electrode layers.
4.3.5
Statistical Analysis of Bending Strength Results
As a means of quantifying the average bending strength and sample variability of each
material investigated, the bending strength data presented in Fig. 4.12 is fit to a Weibull
distribution, which is common practice for tensile and bending failure strengths of ceramic
materials [146]. The Weibull distribution is described by the following cumulative distribution function
−
F =1−e
σb
σθ
m
(4.9)
where F is the probability of failure, σb is the failure strength, σθ is the Weibull characteristic strength, and m is the Weibull modulus. The Weibull distribution is left-skewed which
better represents the flaw dependent failure mode of ceramic materials as opposed to the
standard normal distribution. The Weibull characteristic strength, σθ , provides an estimate
of the strength observed over the entire sample set, and the Weibull modulus, m, gives a
measure of the variability in the strength data, with a larger value of m corresponding to a
small amount of variation in the data. As suggested in the ASTM C 1161-02c test standard,
materials with sample sizes of 30 or more (PZT-5A, PZT-5H, PZT-4, PZT-8, PMN-PT, and
QP10n) are fit to the Weibull distribution, where materials with sample sizes less than 30
(PMN-PZT) are fit to the Normal distribution with simple mean and standard deviation
calculations made [144]. Results of the statistical analysis for all materials tested are given
in Fig. 4.13 as well as Table 4.5 where both the Weibull characteristic strength and Weibull
90
250
Weibull Distribution
Bending Strength b (MPa)
Normal Distribution
200
150
100
50
0
PZT-5A
PZT-5H
PZT-4
PZT-8
PMN-PT PMN-PZT
QP10n
Material
Figure 4.13: Bending strength statistical comparison for all materials tested with error bars
representing 95% confidence interval.
modulus (mean strength and standard deviation for PMN-PZT samples) are given along
with a 95% confidence interval on all terms. The results confirm the strength and variability trends described previously and provide numerical measures of those properties.
The Weibull characteristic strength of PZT-5A is slightly higher than the strength of PZT5H with values of 140.4 MPa and 114.8 MPa, respectively. PZT-4 and PZT-8 have nearly
identical strengths at 123.2 MPa and 127.5 MPa, respectively, which lie between PZT-5A
and PZT-5H. The strengths of the PMN-PT and PMN-PZT single crystal samples are considerably lower than the other materials tested, with values of 60.6 MPa and 44.9 MPa,
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respectively. Lastly, the QuickPack
QP10n samples exhibit the highest strength with a
value of 206.4 MPa.
4.4
Chapter Summary
Analysis of the static and dynamic strength of self-charging structures is presented. Classic
three-point bend testing is first performed on the individual self-charging structure material layers as well as the complete structure. Expressions are derived to predict the maximum stress in each layer for a given failure load in three-point bending. Dynamic testing
is then performed in which a cantilever self-charging structure is excited at resonance with
increasing base excitation amplitude while monitoring for signs of dynamic mechanical or
91
electrical failure. An expression to predict the maximum dynamic stress FRF per base acceleration input is also given. Lastly, a comprehensive study of the static bending strength
of several commercially available piezoelectric materials is given as a design tool to aid in
the creation of self-charging structures.
Results of the static three-point bend testing show that the complete self-charging structure has a bending strength of around 100 - 125 MPa. Additionally, it is shown that the
piezoceramic and battery layers are the critical layers in bending failure, with both layers
failing at roughly the same load in the tip section of the self-charging structure. These
results provide insight into which material layers must be chosen carefully in order to
avoid failure in a specific application. Dynamic testing results show that neither mechanical failure in the aluminum or piezoelectric layers or electrical failure of the battery layer is
experienced for up to 7.0 g of harmonic excitation. The strength testing results presented in
this chapter provide a basis of the static and dynamic loads that can be safely imposed on
self-charging structures. Overall, self-charging structures exhibit reasonable static strength
and prove to be robust under dynamic resonant excitation.
A comprehensive experimental study of the static bending strength of various commercially available piezoceramic materials including hard, soft, single crystal, and composite
piezoelectric material/devices is presented. All samples are prepared and tested using the
Table 4.5: Bending strength parameters for all materials tested with 95% confidence interval given in brackets.
Material
Weibull
Characteristic
Strength (MPa) σb
Weibull Modulus
m
PZT-5A
140.4 [137.6;143.2]
14.6 [11.7;18.2]
PZT-5H
114.8 [112.8;116.9]
16.6 [13.5;20.3]
PZT-4
123.2 [121.0;125.5]
20.6 [15.6;27.1]
PZT-8
127.5 [124.6;130.5]
16.3 [12.6;21.1]
60.6 [58.7;62.5]
12.1 [9.3;15.6]
206.4 [195.2;218.2]
Mean Strength
(MPa)
6.8 [5.2;8.9]
PMN-PT
QP10n
PMN-PZT
44.9 [42.1;47.7]
92
Standard Deviation
3.9 [2.7;7.1]
same equipment. Bending strength results show a relatively small amount of variability for
the monolithic piezoceramic samples tested. The strength of the soft (PZT-5A and PZT-5H)
and hard (PZT-4 and PZT-8) ceramics are relatively similar, with the strengths of the single
crystal materials (PMN-PT and PMN-PZT) considerably lower than the other monolithic
materials tested. This result is expected considering the susceptibility to crack propagation
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found in single crystal materials. The strength values of the QuickPack
QP10n samples
exhibit a considerable amount of variation, which is due to the non-uniformity in electrode
layers between samples. On average, the strength values of the QP10n samples are much
greater than the monolithic materials, likely due to the composite structure of the device.
Overall, the results provide a foundation for the selection of active material in piezoelectric energy harvesters in which load bearing capability is required, such as self-charging
structures.
93
C HAPTER
5
P ERFORMANCE M ODELING OF U NMANNED
A ERIAL V EHICLES WITH O N B OARD E NERGY
H ARVESTING
U
NMANNED
aerial vehicles present a potential application in which the benefits of a
multifunctional energy harvesting solution could enable the use of vibration energy
harvesting. As suggested in Chapter 2, self-charging structures can be incorporated into
UAVs with the goal of creating local autonomous power sources for low-power sensors. In
the following chapters, the use of self-charging structures for vibration energy harvesting
in UAVs is explored as a case study. This chapter presents theoretical performance modeling of unmanned aircraft with on board energy harvesting systems. A system level model
is first described in which changes to the flight endurance of a UAV with the addition of an
energy harvesting system can be predicted. The model is then applied to a small Multiplex
USA [147] EasyGlider remote control (RC) aircraft which is chosen as the test platform for
the case study. Theoretical predictions are first made to describe flight endurance changes
with the addition of a piezoelectric harvesting system. Experimental flight testing is then
performed in which the EasyGlider aircraft is flown with on board vibration harvesting
and the power output of the piezoelectric harvesters is measured during flight. The results
are used to update the initial predictions. Lastly, the flight endurance formulation is used
to predict the effects of adding self-charging structures to the EasyGlider aircraft.
94
5.1
Piezoelectric Energy Harvesting in Unmanned Aerial Vehicles
Previous energy harvesting approaches for small aircraft mainly focus on harvesting solar
energy through the use of photovoltaic panels in order to provide propulsive power for
aircraft flight. More recently, however, researchers have begun to investigate alternate
sources of energy that can be harvested in UAVs. Thomas et al. [91] explore the energy
available from solar, wind, thermal, and electromagnetic radiation sources during flight.
Vibration energy harvesting has also been investigated by several researchers [29, 94, 95,
97, 98]. Piezoelectric vibration harvesting is attractive because, where other sources may
be dependent on environmental conditions, piezoelectric harvesting is effective any time
the aircraft is in flight. Additionally, piezoelectric harvesters are typically small and can be
easily integrated into an aircraft structure. The power available from piezoelectric energy
harvesters, however, is inadequate for providing propulsive power to an aircraft. Instead,
the focus of piezoelectric harvesting in UAVs is to provide a local power source for lowpower sensors, enabling the concept of self-powered sensors that do not draw from the
main power source of the aircraft.
A critical aspect of the integration of any energy harvesting system into a UAV is the
potential effect that the added system can have on the performance of the aircraft. While
the additional energy available from harvesting provides a clear benefit, the added mass
and volume of a harvesting system can have a detrimental effect on the flight performance
of the host aircraft. Both mass and drag are significant parameters in determining the
power required for flight, and increases in these quantities present obvious challenges. It
proves beneficial to have a means of predicting the impact of an energy harvesting system
on a host aircraft in order to help develop harvesting solutions that provide optimal power
output without negative effects on the flight performance. A system level approach is investigated here for analyzing the effects of adding piezoelectric energy harvesting systems
to UAVs on the flight performance of the aircraft.
95
5.2
System Level Flight Endurance Model
A mathematical model used to predict the change in flight endurance of an electric unmanned aerial vehicle with on board harvesting as a result of changes to subsystem masses,
amount of battery power, and amount of harvested power has recently been presented in
the literature by Thomas et al. [91]. The formulation utilizes a basic aerodynamic model of
aircraft flight in order to relate the power required for flight to the power available from
battery and harvester sources, allowing for calculation of the flight endurance of an aircraft
that contains energy harvesting systems. Details of the derivation of the flight endurance
formulation derived by Thomas et al. [91] are given in this section. Formulations for both
the flight endurance and the normalized change in flight endurance based on changes to
subsystem masses and harvester parameters are presented.
5.2.1
Flight Endurance of an Electric Powered UAV with On Board Energy Harvesting
The flight endurance of an electric powered UAV with on board energy harvesting is given
by Thomas et al. [91] as
tE =
3/2
WT
EB ηB
− PHarv
3
ρ∞ SCL
2
2CD
1/2
ηP
ρ∞ SCL3
2
2CD
1/2
ηP
(5.1)
where tE is the flight endurance, EB is the energy available in the battery, ηB is the energy
extraction efficiency factor of the battery, WT is the total weight of the aircraft, PHarv is
the power supplied by the harvester, ρ∞ is the air density, S is the wing area, CL is the
coefficient of lift, CD is the coefficient of drag, and ηP is the motor and propeller efficiency.
This formulation can be derived by considering a basic aerodynamic model to describe the
flight of an aircraft as a balance between the energy required for steady, level flight and
the energy available from all power sources. The basic relationships can be found in most
introductory aerodynamics texts, such as Anderson [148]. In steady level flight, thrust is
equal to drag and lift is equal to weight as follows
T =D
(5.2)
L=W
(5.3)
96
where T is thrust, W is the aircraft weight, L is lift, and D is drag. The thrust required for
flight can be expressed as
D
D
W
W = W =
W
L
L/D
TR = D =
(5.4)
The thrust required for flight is, therefore, dependent on the weight of the aircraft as well
as the lift-to-drag ratio. Lift-to-drag is a direct measure of the aerodynamic efficiency of
an aircraft, hence it is a very important parameter affecting aircraft performance. The liftto-drag ratio can be shown to be equal to the ratio of coefficient of lift, CL , to coefficient
of drag, CD , by considering lift and drag in terms of air density, aircraft speed, and the
respective coefficients in the following
1
2 SC
ρ∞ V∞
L
CL
L
= 12
=
2
D
CD
2 ρ∞ V∞ SCD
(5.5)
where V∞ is the aircraft velocity. Next, the power required for steady level flight is given
by
(5.6)
PR = TR V∞
Substituting Eq. (5.4) and Eq. (5.5) into Eq. (5.6), the power required for flight can be rewritten as
PR =
W
V∞
CL /CD
(5.7)
2 SC , the aircraft velocity can be written as
Recalling that L = W = 21 ρ∞ V∞
L
s
2W
V∞ =
ρ∞ SCL
(5.8)
Substituting Eq. (5.8) into Eq. (5.7) gives an expression for the power required for steady
level flight
W
PR =
CL /CD
s
2W
=
ρ∞ SCL
s
2
2W 3 CD
ρ∞ SCL3
(5.9)
If the efficiency of the aircraft propeller and motor in converting electrical energy to thrust,
which can be expressed as ηP , is considered, Eq. (5.9) can be rewritten as
s
2
2W 3 CD
PR =
/ηP
ρ∞ SCL3
(5.10)
In order to determine the flight endurance of an aircraft, one can first consider the
definition of power to yield the following relationship between power, energy, and time
Power =
Energy
Energy
⇒ Time =
Time
Power
97
(5.11)
The energy available in an electric aircraft with a battery as well as a harvesting system
that supplements the propulsive power supply can be written as
(5.12)
Eavail = EB ηB + PHarv tE
A balance between the energy available in the aircraft for flight with the power required
for flight leads to
EB ηB + PHarv tE
Eavail
EB ηB + PHarv tE
tE =
= r
=
2
3
PR
W 3/2
2W CD
/η
P
3
ρ SC
∞
ρ∞ SCL3
2
2CD
1/2
ηP
(5.13)
L
Solving for tE yields Eq. (5.1), as given by Thomas et al. [91]
tE =
3/2
WT
EB ηB
− PHarv
3
ρ∞ SCL
2
2CD
1/2
ηP
ρ∞ SCL3
2
2CD
1/2
ηP
(5.14)
where W has been replaced with WT .
5.2.2
Normalized Change in Flight Endurance
The flight endurance of an electric UAV with energy harvesting is given in Eq. (5.14). It
is of interest to determine the effects of an added energy harvesting system on the flight
endurance of an aircraft, therefore, a formulation for the change in flight time (which can
be normalized for comparison purposes) is desired. Following the work of Thomas et al.
[91], a linear Taylor series expansion of the flight time, tE , about the point PHarv = 0 can
be used to formulate the normalized change in flight endurance.
By taking the aerodynamic terms in Eq. (5.14) as constant
ψ=
ρ∞ SCL3
2
2CD
1/2
ηP
(5.15)
the flight time can be rewritten as
tE =
EB ηB
3/2
WT
− PHarv ψ
−1
3/2
ψ
ψ = EB ηB WT − PHarv ψ
(5.16)
Recalling the linear multivariable Taylor expansion, given by
∂f ∂f ∂f f (x, y, z) ≈ f (x0 , y0 , z0 )+ (x−x0 )+ (y −y0 )+ (z −z0 ) (5.17)
∂x x0 ,y0 ,z0
∂y x0 ,y0 ,z0
∂z x0 ,y0 ,z0
98
and defining x − x0 = ∆x, the flight endurance can be estimated with the linear Taylor
series as
∂tE ∂tE ∆tE = tE −
≈
∆EB ηB +
∆WT
∂EB ηB PHarv =0
∂WT PHarv =0
∂tE +
∆PHarv
∂PHarv toE
(5.18)
PHarv =0
Evaluating the partial derivative terms, Eq. (5.18) can be rewritten as
∆tE ≈
ψ
∆EB ηB −
3/2
WT
3 EB ηB ψ
EB ηB ψ 2
∆PSC
∆W
+
T
2 W 5/2
WT3
(5.19)
T
The change in flight time can be normalized by the flight time for an electric aircraft without added energy harvesting (non-harvesting design), which can be expressed by setting
PHarv = 0 in Eq. (5.14) giving
tE =
EB ηB ψ
3/2
WT
(5.20)
Normalizing Eq. (5.19) by Eq. (5.20) gives
∆EB ηB
3 ∆WT
ψ
∆tE
≈
+
+ 3/2 ∆PSC
tE
EB ηB
2 WT
WT
(5.21)
From the non-harvesting design given by Eq. (5.20), the following relationship can be defined
ψ
3/2
WT
=
tE
EB ηB
(5.22)
Using Eq. (5.22) in Eq. (5.21) the normalized change in flight time can be written as
∆tE
∆EB ηB
3 ∆WT
∆PHarv
≈
+
+
tE
EB ηB
2 WT
EB ηB /tE
(5.23)
The total weight of the aircraft, WT , can be expanded to include several subsystem
weights as
WT = WB + WST + WHarv
(5.24)
where WB is the weight of the battery, WST is the structural weight including the motor, electronics, and control surface actuators, and WHarv is the weight of the harvesting
system. Using Eq. (5.24) in Eq. (5.23), the normalized change in flight endurance becomes
∆tE
∆EB ηB
3 ∆WB + ∆WST + ∆WHarv
∆PHarv
≈
+
+
tE
EB ηB
2 WB + WST + WHarv
EB ηB /tE
99
(5.25)
Equation (5.25) can be written in terms of changes in mass by using the following relations
EB = eB mB
W = mg
PHarv = pHarv mHarv
EB ηB /tE = pave mB
(5.26)
(5.27)
(5.28)
(5.29)
where eB is the specific energy of the battery (i.e. J/kg), mB is the mass of the battery, g is
the acceleration of gravity, pHarv is the specific power of the harvester (i.e. W/kg), mHarv
is the mass of the harvesting system, and pave is the average specific power supplied by
the battery in the non-harvesting design, i.e. pave = EB ηB /tE mB . Using these relations,
the normalized change in flight endurance can be written in terms of changes in mass, as
given in Thomas et al. [91] as follows
∆tE
∆mB
3 ∆mB + ∆mST + ∆mHarv
pHarv ∆mHarv
≈
−
+
tE
mB
2
mT
pave mB
(5.30)
where mT is the total mass of the aircraft. Equation (5.30) is valid for an electric aircraft
in steady, level flight containing any energy harvesting system (vibration, solar, thermal,
etc.) with the assumption of a constant power output from the harvester. The simple
aerodynamic model used in the derivation naturally adds several assumptions about the
flight of the aircraft and the ambient environmental conditions, namely imposing constant
conditions. The formulation, however, can be used to provide insight into the effects of
adding energy harvesting systems to electric powered UAVs.
5.3
Theoretical Case Study
In this section, a theoretical case study is performed in which the normalized change in
flight time of the EasyGlider aircraft test platform, shown in Fig. 5.1, with the addition of
piezoelectric vibration energy harvesting is investigated using the formulation presented
in the previous sections. It is proposed that flexible fiber-based piezoelectric devices be
installed onto the wing spar of the EasyGlider aircraft to harvest vibration energy in the
wings caused by aerodynamic loading during flight. Multifunctional energy harvesting
approaches will be considered in which structural mass is removed from the aircraft structure to compensate for the addition of a harvesting system. Changes to flight endurance
100
are calculated for varying degrees of multifunctionality and with increasing size of the
harvesting system. Although the focus of piezoelectric harvesting in UAVs is to provide
power to low-power sensors as significant increases in flight time are not expected due
to the low power output of piezoelectric harvesting, the power output of the piezoelectric
system is modeled as contributing to the propulsive power supply of the aircraft for analysis purposes. The goal of this case study is to identify harvester designs in which the flight
endurance does not significantly suffer with the addition of piezoelectric harvesting.
The EasyGlider aircraft has a wing span of 1.8 m, length of 1.1 m, and is powered
by an 11.1 V, 2100 mAh lithium polymer rechargeable battery. Several parameters can be
defined for the standard, non-harvesting EasyGlider aircraft, as given in Table 5.1, where
the total mass of the aircraft is simply the sum of the battery mass and the structure mass,
the battery efficiency factor is estimated as 0.8, the original flight endurance for steady,
level flight is given as 10 minutes based on previous flight testing estimates [149], and the
wing spar area is used to limit the number of piezoelectric devices that can be installed in
the aircraft. Using these parameters, the average specific power supplied by the battery in
the non-harvesting design, pave , can be calculated as
pave =
EB ηB
= 388.5 W/kg
mB tE
(5.31)
Two flexible fiber-based piezoelectric devices are considered for use in the EasyGlider
wing spar, namely the M8507-P1 Macro-Fiber Composite (MFC) manufactured by Smart
Figure 5.1: Multiplex USA EasyGlider remote control aircraft test platform.
101
Table 5.1: Physical properties of the EasyGlider aircraft.
Property
Symbol
Value
Battery mass
mB
144 g
Structure mass
mST
945.2 g
Total aircraft mass
mT
1089.2 g
Battery energy
EB
83916 J
Battery efficiency factor
ηB
0.80
Original flight endurance
tE
600 s
Wing area
SW
0.416 m2
Wing spar area
SW S
0.0252 m2
Material, Corp. [112] and the Piezoelectric Fiber Composite (PFC) manufactured by Advanced Cerametrics, Inc., [113] both shown in Fig. 5.2. Relevant physical properties of the
devices are given in Table 5.2
The specific power output of a piezoelectric harvester will vary considerably based
on the physical configuration of the device and the vibration environment in which it is
placed. Theoretical specific power values are, therefore, generally not given by device
manufacturers. In order to perform the piezoelectric simulation, a single specific power
value can be assigned for a representative piezoelectric fiber-based device. Sodano et al.
R
[34] performed a study in which a QuickPack
QP10ni fiber-based device of dimensions
50.8 mm x 25.4 mm x 0.381 mm and mass of about 3.0 g was surface mounted near the root
of a 83.8 cm long aluminum cantilever beam and excited harmonically. The specific power
of this device can be used as a generic reference value in this simulation. At the beam’s
third resonance frequency of 64 Hz, the device was found to perform optimally and harvest
(a)
(b)
Figure 5.2: Piezoelectric devices including (a) M8507-P1 Macro-Fiber Composite, and (b)
Piezoelectric Fiber Composite.
102
Table 5.2: Physical properties of the piezoelectric devices considered.
Symbol
Macro-Fiber
Composite
Piezoelectric Fiber
Composite
Device length
lP
102 mm
145 mm
Device width
wP
16 mm
15 mm
Device thickness
tP
0.3 mm
0.3 mm
Device mass
mP
3.0 g
2.0 g
Property
a maximum power of 29.5 µW, corresponding to a specific power of pHarv = 0.0098 W/kg.
The specific power of the piezoelectric device is several orders of magnitude lower than
that average specific power supplied by the battery, hence the goal of vibration energy
harvesting is to create local power sources for low-power electronics and not to provide
propulsive power for the aircraft.
In the simulation, the mass of the battery is assumed to stay fixed, ∆mB = 0, and only
changes to the piezoelectric harvester mass and structure mass are allowed. The number of
piezoelectric devices (correspondingly, harvester mass) added to the aircraft is varied, with
an upper bound based on half of the total available surface area of the wing spar, SW S /2.
Voltage generation of a piezoelectric material is based on induced strain and the strain in
the wing spar of an aircraft is largest at the fuselage and decreases towards the wing tip.
It is, therefore, proposed to install piezoelectric devices only near the root of each wing
as harvesters located near the wing tip would not produce significant amounts of energy.
For each discrete number of piezoelectric devices added, the change in structure mass (to
compensate for the addition of the harvesters) is allowed to vary from ∆mST = 0, which
corresponds to the completely non-multifunctional design in which no structure mass is
removed, to ∆mST = −∆mHarv , which is the fully multifunctional design in which all of
the added piezoelectric mass is removed from structure mass, thus the total mass of the
aircraft remains unchanged with the addition of the harvesting system. Using Eq. (5.30),
simulation results for the normalized change in flight time based on the parameters described above are given in Fig. 5.3 for various discrete numbers of piezoelectric devices
used in the harvesting system. Upon investigating the results, it appears as if there is
only a single line, however, the magnified view shows that indeed there are multiple lines,
each corresponding to a different number of piezoelectric devices. Values along the y-axis
103
0.005
-0.0138
0
-0.0138
 tE/tE
-0.01
-0.0138
9.996
9.9985
10.001
-0.02
Increasing Number
of Piezo Devices
-0.03
-0.04
0
5
10
15
20
Total Added Mass (g)
25
30
Figure 5.3: Simulation results for normalized change in flight time of the EasyGlider aircraft based on the addition of piezoelectric harvesting with varying degrees of multifunctionality.
(total added mass = 0) indicate fully multifunctional solutions, where progressing along
the x-axis indicates decreasing degrees of multifunctionality. It should be noted that in
this simulation, only the mass of the piezoelectric material contributes to mHarv and the
mass of any requisite harvesting components, such as circuitry or mounting hardware, are
ignored as these quantities would depend on specific applications and may also perform
additional functions on the aircraft.
The results of the simulation show that a net loss in flight time occurs for all but the
fully multifunctional case where the overall aircraft mass remains unchanged. While the
focus of a piezoelectric system is not on achieving an increase in flight time, the goal is to
choose a design in which the flight endurance does not suffer. These results clearly show
that a fully multifunctional design is necessary for the integration of piezoelectric vibration
harvesting into the EasyGlider aircraft. The loss in flight time observed in the simulation
results from the mass loading and low specific power of piezoelectric energy harvesting.
5.4
Experimental Case Study
In this section, data collected from experimental flight testing is used to update the simulation results with an accurate measure of the power output of the harvesting devices
during flight to provide more realistic estimates of the change in flight endurance with the
104
addition of energy harvesting systems to the EasyGlider aircraft.
5.4.1
EasyGlider Aircraft Configuration
The EasyGlider aircraft is modified to include both MFC and PFC piezoelectric devices on
a custom made fiberglass wing spar. The original round carbon fiber reinforced plastic
wing spar rod is replaced by a rectangular fiberglass spar with a foam core. Replacing the
original round spar with a rectangular spar helps facilitate mounting of the MFC and PFC
devices. The stiffness of the wing spar is an important factor in the design and performance
of the aircraft, therefore, the stiffness of the fiberglass spar is designed to closely match the
stiffness of the original carbon fiber spar. Fabrication of the custom spar is accomplished
by first modifying the round channels in the aircraft wings to accept a rectangular spar,
then shaping a foam core with a hot wire cutter to fit in the modified channel, and finally
using a vacuum bagging procedure to apply fiberglass layers over the foam core with West
System [150] 105/205 two part epoxy and allowing the spar to cure under vacuum. Once
cured, the MFC M8507-P1 piezoelectric device and PFC device are bonded to the surface of
the spar using the same epoxy and vacuum bagging procedure. The original foam covers
used to secure the carbon fiber spar are replaced with two fiberglass covers molded to the
underside surface of the wings and attached with cyanoacrylate adhesive. Photographs of
the MFC and PFC patches attached to the fiberglass spar as well as the spar inserted into
the wing are shown in Fig. 5.4.
The EasyGlider aircraft is also modified to house two data loggers used to record measurements during flight. An XR5-SE-M-50mv eight channel, 0-5 V data logger from Pace
Scientific, Inc. [151] is used to measure the voltage output of the piezoelectric harvesters
and a UA-004-64 Pendant G 3-axis accelerometer data logger with a range of ±3 g from
Onset Computer Corp. [152] is used to measure the aircraft accelerations. Each data logger
is battery powered and stores data on internal memory. The XR5-SE-M-50mv logger has
dimensions of 12.4 cm x 5.6 cm x 3.3 cm and weighs 120 g, where the UA-004-64 logger
has dimensions of 3.7 cm x 2.5 cm x 0.7 cm and weighs 8 g. The XR5-SE-M-50mv unit is
installed in the cockpit of the aircraft and a custom fiberglass canopy is created to allow its
installation. The data logger and canopy are shown installed in the aircraft in Fig. 5.5. In
order to compensate for the added mass of the logger, an additional steel mass is added
at the tail of the aircraft inside the fuselage to maintain the proper center of gravity. The
105
UA-004-64 accelerometer logger is installed on the underside of the aircraft at the center of
gravity and an aerodynamic fiberglass canopy is created to cover and protect the logger.
The accelerometer data logger and protective canopy are shown in Fig. 5.6.
5.4.2
Flight Testing Results
Flight testing is performed where the voltage output of both piezoelectric devices as well
as the accelerations experienced by the aircraft are recorded during flight. The piezoelectric
(a)
(b)
Figure 5.4: Fiberglass wing spar with (a) surface mounted Macro-Fiber Composite and
Piezoelectric Fiber Composite devices (b) inserted in wing.
(a)
(b)
Figure 5.5: XR5-SE-M-50mv voltage data logger shown (a) installed into cockpit of
EasyGlider and (b) with fiberglass canopy installed.
106
(a)
(b)
Figure 5.6: UA-004-64 accelerometer data logger shown (a) installed on the underside of
the EasyGlider and (b) with fiberglass canopy installed.
devices are each connected through a full bridge diode rectifier to an optimal resistive load
(found as 330 kΩ for both devices). The voltage across the load is measured at a sampling
rate of 200 Hz. The accelerometer data logger is configured to record accelerations in all
three axes at a sampling frequency of 50 Hz. During the flight test, the aircraft is flown
for about 6 minutes on a sunny day with winds approximately 10-15 km/h. The aircraft is
hand launched and a simple wide circle pattern is flown. An airspeed of approximately 45
km/h at an altitude of around 30 m is achieved. Results of the flight testing are shown in
Fig. 5.7 and Fig. 5.8 for the acceleration and voltage, respectively. It should be noted that
both data loggers begin recording data approximately 60 seconds prior to the launch of the
aircraft.
From the acceleration measurements presented in Fig. 5.7, it can be estimated that the
average acceleration in the x- and y- axis, which correspond to the horizontal directions
from propeller to tail and from wing tip to wing tip, respectively, is about ±1.5 g, and the
average acceleration in the z-axis is about ±1 g. The z-axis acceleration is centered around
-1 g due to the acceleration of gravity and the orientation of the logger when attached to
the underside of the fuselage. The measurements give insight into the typical acceleration
levels experienced in a small unmanned aircraft.
The voltage measurements given in Fig. 5.8 show that both piezoelectric devices exceed
the 5 V limit of the data logger several times during flight. The data can still be used,
however, to give a rough estimate of the power harvested during flight. The average power
107
X-Axis
Acceleration (g)
(a)
2
0
-2
0
50
100
150
200
Time (sec)
250
300
350
50
100
150
200
Time (sec)
250
300
350
50
100
150
200
Time (sec)
250
300
350
Y-Axis
Acceleration (g)
(b)
2
0
-2
0
Z-Axis
Acceleration (g)
(c)
2
0
-2
0
Figure 5.7: Acceleration measurements recorded during flight in the (a) x-axis, (b) y-axis,
and (c) z-axis directions.
output of the piezoelectric devices is calculated from the flight measurements using the
following relation
Pave =
2
vRM
S
R
(5.32)
where Pave is the average power, vRM S is the root mean square (RMS) voltage, and R is
the load resistance. The average power output of the MFC and PFC devices is found to
be 11.3 µW, and 10.1 µW, respectively. The exact power output values are expected to be
larger than these averages because of the voltage clipping effects of the data logger.
5.4.3
Simulation Results
The power density value used in Section 5.3 is based on high frequency sinusoidal excitation, which is not likely attainable during flight, and is expected to give an overestimate
108
(a)
MFC Piezoelectric Voltage (V)
5
4
3
2
1
0
0
(b)
50
100
150
200
Time (sec)
250
300
350
50
100
150
200
Time (sec)
250
300
350
AFC Piezoelectric Voltage (V)
5
4
3
2
1
0
0
Figure 5.8: Voltage measurements recorded during flight for (a) MFC and (b) PFC piezoelectric devices.
of the normalized change in flight time. Results of the experimental flight testing are used
here to update the simulation results with a more accurate measure of the power harvested
during flight. Based on the average power values calculated during flight, the following
specific power values can be defined for each piezoelectric device
pM F C = 0.00377 W/kg
(5.33)
pP F C = 0.00505 W/kg
(5.34)
where pM F C is the specific power of the MFC and pP F C is the specific power of the PFC.
Using these updated specific power values, the normalized change in flight time is recalcu109
lated, giving the results presented in Fig. 5.9, where the effect of adding each piezoelectric
device is considered individually. Although the results are similar to the those presented
in Fig. 5.3 for the theoretical case study, a slight decrease in the normalized change in
flight time is observed. The results clearly confirm that a fully multifunctional solution is
required to avoid a loss in flight time when adding piezoelectric harvesting to UAVs.
5.5
Flight Endurance Modeling of Self-Charging Structures in
UAVs
The results presented in the previous sections have shown that a multifunctional design is
needed in order to incorporate piezoelectric vibration harvesting into a UAV without adversely affecting the flight time. The integration of multifunctional self-charging structures
into UAVs is investigated in this section. From the experimental testing performed on the
self-charging structure prototype in Chapter 3, a power output of 0.306 mW is obtained for
a cantilever self-charging structure excited at resonance (210 Hz) under ±1.0 g of harmonic
base excitation. The excitation frequency experienced by the device when installed in the
wing spar of the EasyGlider will likely not match this optimal condition, and the excitation
amplitude will be considerably lower, therefore, a power output of 10 times less (30.6 µW)
-3
1
x 10
0
-3
MFC Patch
PFC Patch
-1.377
x 10
-1.3771
 tE/tE
-1
-1.3771
1
-2
1
1
-3
-4
-5
0
0.5
1
1.5
2
2.5
Total Added Mass (g)
3
3.5
Figure 5.9: Updated simulation results for normalized change in flight time of the
EasyGlider aircraft based on the addition of piezoelectric harvesting with varying degrees
of multifunctionality.
110
is assumed for the self-charging structure when integrated into the wing spar of a UAV.
With a power output of 30.6 µW, the specific power of the self-charging structure (which
has a mass of 6.2 g) is calculated as pSCS = 0.00494 W/kg.
As discussed in Chapter 2, self-charging structures can be integrated into the wing spar
of a UAV in a multifunctional manner by supporting aerodynamic loading on the wing
such that structure mass can be removed to compensate for the added mass of the harvester. The flight endurance formulation is used here to predict the effects of integrating
self-charging structures into the EasyGlider aircraft. Consistent with the previous simulations, the battery mass is assumed to remain constant, ∆mB = 0, and only changes to
the harvester and structure masses are allowed. Discrete numbers of self-charging structures are considered with a limit based on half of the wing spar area. Varying degrees
of multifunctionality are considered by changing the amount of structure mass removed
from the aircraft to compensate for the addition of the piezoelectric harvesting system.
Results of the simulation are presented in Fig. 5.10. As in the theoretical case study presented in Section 5.3, it again appears as if there is only a single line, however, multiple
lines corresponding to different numbers of devices are present although each gives almost identical results. As expected, the simulation results agree with those previously
presented for piezoelectric harvesting and suggest that a fully multifunctional design is
required in order to avoid potential losses in flight endurance. Incorporating a fully multi0.01
-0.0207
0
-0.0207
-0.01
 tE/tE
-0.02
-0.0207
14.9987
15.0001
15.0016
-0.03
-0.04
Increasing Number of
Self Charging Structures
-0.05
-0.06
-0.07
0
10
20
30
Total Added Mass (g)
40
50
Figure 5.10: Simulation results for normalized change in flight time of the EasyGlider
aircraft based on the addition of self-charging structures.
111
functional self-charging structure harvesting system into the aircraft, however, can enable
self-powered sensors in the aircraft wings while the flight endurance remains unchanged.
5.6
Chapter Summary
A system level approach for examining the effects of adding energy harvesting systems to
unmanned aerial vehicles is investigated. A formulation recently presented by Thomas et
al. [91] is applied to an EasyGlider remote control aircraft in order to predict the effects of
adding energy harvesting systems on the aircraft flight endurance. Details of the derivation of the flight endurance formulation are first given. A theoretical case study is then
performed where simulations are carried out to predict the change in flight endurance
based on the addition of flexible piezoelectric energy harvesting devices to the aircraft.
An experimental case study is then given in which flight test data is used to update the
specific power values used in the theoretical simulation. Lastly, the addition of multifunctional self-charging structures to the EasyGlider aircraft is investigated through the use of
the flight endurance formulation.
Results of both the theoretical and experimental simulations show that a fully multifunctional design, where all harvester mass added to the aircraft is removed in structure
mass, is required for piezoelectric harvesting systems in order to avoid a loss in flight endurance. With a fully multifunctional design, however, self-charging structures can be
used to create self-powered sensor systems on board UAVs. The results presented in this
chapter enforce the need for multifunctional piezoelectric harvesting designs for use in
unmanned aerial vehicle applications.
112
C HAPTER
6
E LECTROMECHANICAL M ODELING OF
M ULTIFUNCTIONAL E NERGY H ARVESTING
W ING S PAR FOR U NMANNED A ERIAL
V EHICLES
T
HIS
chapter presents the electromechanical modeling of a multifunctional energy har-
vesting wing spar for unmanned aerial vehicles utilizing embedded self-charging
structures. The model described here is an adaptation of the experimentally validated
assumed modes model given by Erturk and Inman [30–32] and described previously in
Chapter 2. Details of the derivation are first given for both the series and parallel electrode connection of the piezoelectric layers. Following the procedure outlined in Chapter 2, energy formulations are derived for the structure, an assumed solution composed
of trial functions multiplied by generalized coordinates is substituted into the energy expressions, and Hamilton’s principle is applied yielding the coupled electromechanical Lagrange equations. The Lagrange equations are solved, giving the mechanical and electrical
governing equations, and finally the displacement - to - base acceleration and voltage - to
- base acceleration frequency response functions are defined. Experimental testing of a
representative wing spar with an embedded self-charging structure is also performed in
order to validate the model. A cantilever wing spar is subjected to harmonic base excitation in order to obtain experimentally measured frequency response functions which are
then compared to those predicted by the model for validation purposes.
113
6.1
Electromechanical Modeling
The electromechanical assumed modes model presented in this section for a multifunctional energy harvesting wing spar is based on the previously described assumed modes
model presented by Erturk and Inman [30–32]. Differences between the formulation given
here and the model presented in Chapter 2 are highlighted.
6.1.1
Modeling Assumptions and Device Configuration
Consider the piezoelectric energy harvesting wing spar configurations shown in Fig. 6.1 for
a 1/2 spar model with the moving base representing the aircraft fuselage. The spar contains an embedded self-charging structure, hence it represents a bimorph configuration.
Both series (Fig. 6.1(a)) and parallel (Fig. 6.1(b)) connections of the piezoelectric devices are
considered. Translational base acceleration, ab (t), is imposed in the transverse direction at
the clamped end. The structures are assumed sufficiently thin such that Euler-Bernoulli
assumptions hold. The perfectly conductive electrodes covering the surfaces of the piezoelectric layers allow a single electric potential to be defined across them. Once again, the
R
piezoelectric layers used in the spar are QuickPack
QP10n devices and the thin-film batR
teries used are Thinergy
MEC 101-7SES devices. 3M [100] ScotchWeldTM DP460 two part
epoxy is used to bond all layers, and each bonding layer is assumed to be perfect with
identical thickness. In order to prevent electrical shorting of the batteries (recall, whose
outer substrate serve as the electrodes), a Kapton film layer is placed between the substrate and the batteries. Lastly, a sensor node is modeled as a lumped mass at an arbitrary
location near the free end of the spar.
6.1.2
Energy Formulations for the Electromechanical Spar
As with the assumed modes model presented for the self-charging structures in Chapter 2,
the motion of the spar is a combination of the base motion and the relative motion of the
beam, given by Eq. (2.2) as
wabs (x, t) = wb (t) + wrel (x, t)
(6.1)
where wabs is the absolute displacement of the beam, wb is the base displacement and wrel
is the displacement of the beam relative to the moving base.
114
(a)
z
ab (t )
x0
(b)
L1
L2
L3
L4
x
Rl
vs (t )
x
Rl
v p (t )
L
z
ab (t )
x0
L1
L2
L3
L4
L
L1  L  L2
(c)
hp
hk
hs1
he
hp
Piezoceramic Layer
h
Thin-Film Battery Layer
Substrate Layer
b
Kapton Layer
L2  L  L3
hb
hki
hs 2
Epoxy Layer
Electrodes
h
Poling Direction
hb
b
Figure 6.1: Multifunctional piezoelectric energy harvesting wing spar configurations
showing (a) series and (b) parallel connection of the piezoelectric layers along with the
(c) cross-sectional views of both composite sections.
The strain energy in the structure layers is given by Eq. (2.7) as
Us =
1
2
Z
Ys (x)z 2
∂ 2 wrel (x, t)
∂x2
2
dVs
(6.2)
dx
(6.3)
Vs
1
=
2
ZL
Y Is (x)
∂ 2 wrel (x, t)
∂x2
2
0
where Ys is the elastic modulus of the structure layers, I is the area moment of inertia,
and Y Is (x) is the bending stiffness of the structure materials, which varies along the xdirection. The bending stiffness of the structure layers for the wing spar differs from the
115
self-charging structure and is given by
Y Is (x) = Y Is [H(L1 − x) + H(x − L3 )] + Y Ic1 H(x − L1 )H(L2 − x)
(6.4)
+ Y Ic2 H(x − L2 )H(L3 − x)
where H(x) is the Heaviside step function and Y Is , Y Ic1 , and Y Ic2 are the bending stiffnesses of the substrate material in the uniform sections from 0 ≤ x ≤ L1 and L3 ≤ x ≤ L,
in the first composite section from L1 < x < L2 , and in the second composite section
from L2 < x < L3 , respectively. Again, the symmetry of the spar about the x-axis allows
the use of the parallel axis theorem in calculating the bending stiffness of the composite
sections. Calculation of the bending stiffnesses of the multifunctional spar are given in
Appendix B.3.
The total strain energy in the piezoelectric layers is given by Eq. (2.10) and Eq. (2.11),
and when combined with the strain energy in the structure layers given by Eq. (6.3), yields
the following total strain energy in the structure
ZL
1
U =
2
s
∂ 2 wrel (x, t)
∂x2
2
∂ 2 wrel (x, t)
∂x2
2
∂ 2 wrel (x, t)
∂x2
2
∂ 2 wrel (x, t)
∂x2
2
Y Is (x)
0
ZL2
dx
(6.5)
cE
11 Ip
+
∂ 2 wrel (x, t)
− Jps vs (t)
∂x2
!
dx
L1
ZL
1
U =
2
p
Y Is (x)
dx
0
ZL2
+
(6.6)
cE
11 Ip
∂ 2 wrel (x, t)
− Jpp vp (t)
∂x2
!
dx
L1
where cE
11 is the elastic modulus of the piezoelectric measured at constant electric field,
p
s
cE
11 Ip is the bending stiffness of the piezoelectric layer (see Appendix B.3), Jp and Jp are
the piezoelectric coupling terms given by Eq. (2.14) and Eq. (2.15), v(t) is the voltage output
of the piezoelectric layers, and superscript s and p stand for series and parallel connection
of the piezoelectric layers.
The total kinetic energy of the beam is given by Eq. (2.16) as
Tbeam
1
=
2
ZL
(ρAs (x) + ρAp (x))
(6.7)
0
"
×
∂wb (t)
∂t
2
+2
∂wb (t) ∂wrel (x, t)
+
∂t
∂t
116
∂wrel (x, t)
∂t
2 #
dx
where ρAs (x) and ρAp (x) are the mass density functions of the structure and piezoelectric
layers, which again differ from the self-charging structure derivation, and are given by
ρAs (x) = ρAs [H(L1 − x) + H(x − L3 )] + ρAc1 H(x − L1 )H(L2 − x)
(6.8)
+ ρAc2 H(x − L2 )H(L3 − x)
ρAp (x) = ρAp H(x − L1 )H(L2 − x)
(6.9)
where ρAs , ρAc1 , and ρAc2 are the mass densities of the substrate material in the uniform
sections from 0 ≤ x ≤ L1 and L3 ≤ x ≤ L, in the first composite section from L1 < x <
L2 , and in the second composite section from L2 < x < L3 , respectively, and ρAp is the
mass density of the piezoelectric layers in the first composite section from L1 < x < L2 .
Calculation of the mass densities is given in Appendix B.3.
The kinetic energy of the lumped mass must be taken into consideration (again differing from the self-charging structure derivation), and is given by
"
#
1
∂wb (t) ∂wrel (L4 , t)
∂wrel (L4 , t) 2
∂wb (t) 2
Tmass = M
+2
+
2
∂t
∂t
∂t
∂t
(6.10)
where M is the mass of the lumped mass. The total kinetic energy of the system can be
expressed as a sum of the kinetic energy of the beam and the kinetic energy of the lumped
mass, given by
1
T =
2
ZL
(ρAs (x) + ρAp (x))
0
#
∂wb (t) 2
∂wb (t) ∂wrel (x, t)
∂wrel (x, t) 2
×
+2
+
dx
∂t
∂t
∂t
∂t
"
#
1
∂wb (t) 2
∂wb (t) ∂wrel (L4 , t)
∂wrel (L4 , t) 2
+ M
+2
+
2
∂t
∂t
∂t
∂t
"
(6.11)
The internal electrical energy is given by Eq. (2.22) and Eq. (2.23), and can be simplified
for the wing spar as
Wies
1
=
2
ZL2
2Jps vs (t)
∂ 2 wrel (x, t)
1
dx + Cp vs2 (t)
2
∂x
4
(6.12)
∂ 2 wrel (x, t)
dx + Cp vp2 (t)
∂x2
(6.13)
L1
Wiep
1
=
2
ZL2
2Jpp vp (t)
L1
117
where the internal capacitance of a piezoelectric layer, Cp , is given by Eq. (2.26).
Lastly, the work of non-conservative forces is given by Eq. (2.27) as
(6.14)
Wnce = Q(t)v(t)
where Q(t) is the electric charge output of the piezoelectric layers.
6.1.3
Substitution of the Assumed Solution
Substitution of the assumed solution given by Eq. (2.28) as
wrel (x, t) =
N
X
(6.15)
φr (x)ηr (t)
r=1
which consists of a series of trail functions (φr (x)) multiplied by generalized coordinates
(ηr (t)), leads to the following expressions for the potential, kinetic, and internal electrical
energy terms
N
Us =
Up =
N
T =
N
1 XX
(ηr (t)ηl (t)krl − 2ηr (t)vs (t)θrs )
2
(6.16)
1
2
(6.17)
r=1 l=1
N X
N
X
(ηr (t)ηl (t)krl − 2ηr (t)vp (t)θrp )
r=1 l=1
N
1 XX
(η˙r (t)η̇l (t)mrl + 2η˙r (t)pr )
2
r=1 l=1
ZL
1
+
2
(ρAs (x) + ρAp (x))
∂wb (t)
∂t
2
1
dx + M
2
∂wb (t)
∂t
2
(6.18)
0
Wies
N 1X
1
s
2
=
2ηr (t)vs (t)θr + Cp vs (t)
2
2
Wiep =
r=1
N
X
1
2
2ηr (t)vp (t)θrp + 2Cp vp2 (t)
(6.19)
(6.20)
r=1
where
ZL
krl =
00
ZL2
00
Y Is (x)φr (x)φl (x) dx + 2
0
00
00
cE
11 Ip φr (x)φl (x) dx
(6.21)
L1
θrs =
ZL2
00
Jps φr (x) dx
(6.22)
L1
θrp
ZL2
=
00
Jpp φr (x) dx
L1
118
(6.23)
ZL
mrl =
(ρAs (x) + ρAp (x)) φr (x)φl (x) dx + M φr (L4 )φl (L4 )
(6.24)
∂wb (t)
∂wb (t)
dx + M φr (L4 )
∂t
∂t
(6.25)
0
L
Z
pr =
(ρAs (x) + ρAp (x)) φr (x)
0
where prime represents ordinary differentiation with respect to the spatial variable, x, and
an overdot represents ordinary differentiation with respect to the temporal variable, t.
6.1.4
Lagrange Equations with Electromechanical Coupling
The electromechanical Lagrange equations for the energy harvesting wing spar follow
from the extended Hamilton’s principle. The first set of Lagrange equations are given
by Eq. (2.55) and Eq. (2.56) as
N
X
l=1
N
X
(mil η̈l + kil ηl − 2θis vs − fi ) = 0
(6.26)
(mil η̈l + kil ηl − 2θip vp − fi ) = 0
(6.27)
l=1
where the forcing due to base excitation for the spar configuration (which also differs from
the self-charging structure derivation), fi , is
∂pi
=−
fi = −
∂t
ZL
(ρAs (x) + ρAp (x)) φi (x)
∂ 2 wb (t)
∂ 2 wb (t)
dx
+
M
φ
(L
)
i
4
∂t2
∂t2
(6.28)
0
The second set of Lagrange equations are given by Eq. (2.67) and Eq. (2.68) as
N
1
vs X
Cp v̇s +
+
(2η̇r θrs ) = 0
2
Rl
2Cp v̇p +
vp
+
Rl
r=1
N
X
(2η̇r θrp ) = 0
(6.29)
(6.30)
r=1
As described in Subsection 2.3.5, the equivalent series/parallel representation of the
Lagrange equations (as suggested by Erturk and Inman [32]) is given by Eq. (2.69) and
Eq. (2.70) as
N
X
(mil η̈l + kil ηl − θieq v − fi ) = 0
(6.31)
l=1
N
Cpeq v̇ +
X
v
(η̇r θreq ) = 0
+
Rl
r=1
119
(6.32)
where the equivalent electromechanical coupling, θeq , and capacitance, Cpeq , terms are selected from Table 2.1 based on the series or parallel electrode connection.
6.1.5
Solution of the Equivalent Representation of the Lagrange Equations
Following the solution procedure outlined in Subsection 2.3.6, the equivalent form of the
Lagrange equations can be rewritten in matrix form with proportional damping introduced in the first equation, given in Eq. (2.72) and Eq. (2.73) as
[M] η̈ + [C] η̇ + [K] η − Θeq v = f
v
Cpeq v̇ +
+ Θeq η̇ = 0
Rl
(6.33)
(6.34)
where [M], [K], and [C] are the N × N mass, stiffness, and damping matrices, η, f , and Θeq
are the N × 1 vectors of generalized coordinates, forcing functions, and electromechanical
coupling terms. The damping matrix is given by
[C] = α [M] + β [K]
(6.35)
where α and β are constants of proportionality.
Solution of the matrix equations leads to the frequency response functions given by
Eq. (2.88) and Eq. (2.89), as follows
wrel (x, t)
= ΦT (x) −ω 2 [M] + jω [C] + [K]
ab ejωt
!−1
1 −1 eq eq T
eq
+ jω jωCp +
Θ Θ
F̃
Rl
1 −1 eq T
v(t)
eq
Θ
−ω 2 [M] + jω [C] + [K]
= −jω jωCp +
ab ejωt
Rl
!−1
1 −1 eq eq T
eq
+jω jωCp +
Θ Θ
F̃
Rl
(6.36)
(6.37)
where the components of the forcing vector for the harvester spar include the effects of the
lumped mass, and are given by
 L

Z
F̃i = −  (ρAs (x) + ρAp (x)) φi (x) dx + M φi (L4 )
0
120
(6.38)
6.2
Experimental Validation of the Assumed Modes Formulation
In order to validate the electromechanical assumed modes formulation developed in the
previous section for a piezoelectric energy harvesting wing spar, a representative wing
spar with embedded piezoelectric and thin-film battery layers is experimentally tested
and the results are compared to the model predictions.
6.2.1
Representative Energy Harvesting Wing Spar Configuration
A 3003-H14 aluminum alloy beam is selected as the substrate layer of the representative
wing spar. The overall dimensions of the aluminum substrate, which has a modulus of
69 GPa and a mass density of 2730 kg/m3 , are 304.80 mm x 26.62 mm x 3.237 mm. Two
opposite faces of the beam are precisely machined to allow bonding of the piezoelectric
and thin-film battery layers on the symmetric structure, such that the thickness along the
R
length of the beam is constant. QuickPack
QP10n piezoelectric devices are bonded near
R
the root of the beam and Thinergy
MEC 101-7SES thin-film batteries are bonded just
after the piezoelectric layers. Each layer is bonded using a vacuum bagging procedure, as
described previously in Subsection 3.2.2, with ScotchWeldTM DP460 two part epoxy used
as the bonding layer. The fabricated representative wing spar is shown in Fig. 6.2 where
two magnets placed near the end of the beam create a lumped mass representative of a
sensor node. The geometric and material properties of the piezoelectric and battery layers
have been given previously in Table 3.2, and the material properties of the Kapton and
epoxy layers are discussed in Subsection 3.2.1.
Figure 6.2: Representative aluminum wing spar with embedded self-charging structure.
121
6.2.2
Experimental Setup
The device is clamped to an APS Dynamics, Inc. [153] 113 long-stroke shaker, which is
powered by an APS Dynamics, Inc. 125 power amplifier, with an overhang length of 25.40
cm, as shown in Fig. 6.3. In the clamped configuration, the device becomes a 5-segment
beam with an initial 25.40 mm segment containing only the aluminum substrate, followed
by a 9.53 transition section containing only the reduced thickness substrate to allow for
the electrode connection of the QP10n device, a 50.80 mm composite section containing
the substrate layer and symmetric piezoelectric layers bonded with epoxy to the substrate,
a 25.40 mm composite section containing the substrate and symmetric thin-film battery
layers insulated with Kapton and bonded with epoxy, and finally a 142.88 mm segment
which again contains only the aluminum substrate. A 15.6 g lumped mass in the form of
two rectangular magnets of dimension 25.4 mm x 6.35 mm x 6.35 mm placed on opposite
faces of the beam is fixed at a distance of 203.2 mm from the root of the beam. A DSP
Technology, Inc. [119] SigLab 20-42 data acquisition system is used for all FRF measurements. Low amplitude chirp signals are used to excite the shaker, and 5 averages are taken
for each measurement. The input acceleration is measured using a PCB Piezotronics, Inc.
(a)
(b)
(h)
(f)
(g)
(c)
(c)
(d)
(d)
(a) SigLab data acquisition system
(b) Laser vibrometer
(c) Clamp
(d) Electromagnetic shaker
(e) Amplifier
(f) Load resistances
(g) Energy harvester wing spar
(h) Accelerometer
(e)
Figure 6.3: Experimental setup used to obtain frequency response measurements of the
aluminum wing spar.
122
[120] U352C67 accelerometer attached to the base of the clamp, the tip velocity is measured
at the tip of the beam using a Polytec, Inc. [121] PDV-100 laser Doppler vibrometer, and the
voltage output of the self-charging structure is measured directly with the SigLab data acquisition system using an Agilent Technologies, Inc. [109] N2862A 10:1 probe. The overall
test setup is shown in Fig. 6.3. As with the FRF measurements made on the self-charging
structure in Subsection 3.3.1, the laser vibrometer measures the absolute tip velocity of the
beam, thus Eq. (3.3) can be used to predict the absolute tip velocity - to - base acceleration
FRFs.
6.2.3
Experimental Results
Both the voltage - to - base acceleration and tip velocity - to - base acceleration FRFs are
measured for the series and parallel electrode connections for a set of resistive loads ranging from 100 Ω to 1 MΩ. The Agilent 10:1 probe used to measure the piezoelectric voltage
output introduces a 10 MΩ resistance in parallel with the load resistances used, therefore,
it must be considered in calculating the total effective load seen by the piezoelectric layTable 6.1: Load resistance values used in FRF measurements of aluminum spar along with
effective load resistances.
Load Resistor
Values (Ω)
Effective
Resistance Seen
by Piezoelectric
Layers (Ω)
100
470
1200
6800
10000
22000
47000
100000
330000
470000
680000
1000000
99.99
469.98
1199.86
6795.38
9990.01
21951.71
46780.13
99009.90
319457.89
448901.62
636704.12
909090.91
123
ers. Table 6.1 lists the load resistor values used as well as the effective loads seen by the
piezoelectric layers. The voltage and tip velocity FRFs for the series and parallel electrode
cases are shown in Fig. 6.4 and Fig. 6.5, respectively, where the model predicted FRFs are
plotted over the experimental results in order to validate the model. Forty modes are used
in the assumed modes formulation (N = 40) to ensure convergence of the fundamental
natural frequency. The admissible functions given by Eq. (2.29) are used in the model. The
FRF results show good correlation between the model predictions and the experimental
measurements and confirm the ability of the model to accurately predict the vibration and
(a)
4
Experiment
Model
|Voltage FRF| (V/g)
10
Increasing R
l
2
10
0
10
-2
10
24
26
28
30
Frequency (Hz)
32
34
|Tip Velocity FRF| ((m/s)/g)
(b)
Experiment
Model
1
10
Increasing R
l
0
10
27
27.5
28
28.5
29
Frequency (Hz)
29.5
30
Figure 6.4: Experimental and numerical (a) voltage - to - base acceleration FRFs and (b)
tip velocity - to - base acceleration FRFs of the aluminum wing spar for series electrode
connection case for various load resistances.
124
(a)
4
Experiment
Model
|Voltage FRF| (V/g)
10
Increasing Rl
2
10
0
10
-2
10
24
26
28
30
Frequency (Hz)
32
34
|Tip Velocity FRF| ((m/s)/g)
(b)
Experiment
Model
1
10
Increasing R
l
0
10
27
27.5
28
28.5
29
Frequency (Hz)
29.5
30
Figure 6.5: Experimental and numerical (a) voltage - to - base acceleration FRFs and (b)
tip velocity - to - base acceleration FRFs of the aluminum wing spar for parallel electrode
connection case for various load resistances.
voltage response of the multifunctional energy harvester spar. As the load resistance increases from near the short-circuit condition at 100 Ω to near the open-circuit condition
at 1 MΩ, the experimentally measured fundamental natural frequency shifts from 28.13
Hz to 28.38 Hz for both series and parallel connection cases. These frequencies are predicted by the assumed modes formulation as 28.10 Hz and 28.40 Hz, respectively. The
amplitude predictions of the model also fit the experimental measurements well. The experimental voltage FRFs show a maximum peak voltage output of 968.1 V/g for the series
connection and 674.1 V/g for the parallel connection at the largest load resistance. As
125
discussed in Subsection 3.3.1, these FRF measurements are linear estimates obtained from
low-amplitude excitation and are not necessarily accurate for large amplitude excitations.
The model predicts these peak voltage outputs as 875.0 V/g and 593.9 V/g for the series
and parallel case, respectively. The maximum tip velocity is measured as 10.57 (m/s)/g
and 10.30 (m/s)/g for the series and parallel cases, respectively, when excited at shortcircuit conditions. The model predicts these peak velocities as 10.64 (m/s)/g and 10.59
(m/s)/g. For excitation at the open-circuit condition, the experimentally measured peak
tip velocities are 6.83 (m/s)/g and 9.15 (m/s)/g for the series and parallel cases, respectively, and are predicted by the model as 6.67 (m/s)/g and 9.21 (m/s)/g.
Using the voltage FRF data given in Fig. 6.4(a) and Fig. 6.5(a), electrical performance
curves for the peak voltage, current, and power output with varying load resistance can
be created. Excitation at both the short-circuit and open-circuit resonance frequencies is
considered. It should be noted again that the performance curves are based on linear FRF
measurements and may be inaccurate for predicting the response under large amplitude
excitations. The variation of the peak voltage with load resistance is given in Fig. 6.6(a)
and Fig. 6.6(b) for the series and parallel cases, respectively. The model agrees well with
the experimental data. The peak voltage output values predicted by the model at the
short-circuit and open-circuit frequencies for the largest load resistance are 439.5 V/g and
875.0 V/g, respectively for the series case and 230.4 V/g and 593.9 V/g, respectively for
the parallel case.
The variation of the peak current output with load resistance is presented in Fig. 6.4(c)
and Fig. 6.4(d) for the series and parallel electrode connections, respectively. The peak
current output occurs for both the short-circut and open-circuit resonance frequencies at
the smallest load resistance. The peak currents are predicted by the model for excitation at
the short-circuit and open-circuit resonance as 7.32 mA/g and 2.71 mA/g, respectively, for
the series case and 14.58 mA/g and 5.42 mA/g, respectively, for the parallel case.
Figure 6.4(e) and Fig. 6.4(f) present the variation of the peak power output with load resistance for the series and parallel cases, respectively. For both electrode connection cases,
peak power outputs (which are roughly equal) exist for separate load resistance values for
excitation at the short-circuit and open-circuit resonance frequencies. The peak power outputs predicted for the series and parallel cases are identical. The model predictions for the
peak power output for excitation at the short-circuit resonance are 884.9 mW/g2 for a load
126
4
10
(b)
Experiment (28.38 Hz)
Experiment (28.13 Hz)
Model (28.40 Hz)
Model (28.10 Hz)
2
Voltage (V/g)
Voltage (V/g)
(a)
10
0
4
10
Experiment (28.38 Hz)
Experiment (28.13 Hz)
Model (28.40 Hz)
Model (28.10 Hz)
2
10
0
10
10
2
10
4
10
Load Resistance ()
10
6
(c)
2
10
(d)
4
10
Load Resistance ()
10
6
2
10
1
Current (mA/ g)
0
10
Experiment (28.38 Hz)
Experiment (28.13 Hz)
Model (28.40 Hz)
Model (28.10 Hz)
-1
10
2
10
Power (mW/g2)
(e)
1
10
0
10
10
10
6
4
2
10
Experiment (28.38 Hz)
Experiment (28.13 Hz)
Model (28.40 Hz)
Model (28.10 Hz)
0
10
2
4
10
Load Resistance ()
10
2
10
4
10
Load Resistance ()
6
10
2
10
Experiment (28.38 Hz)
Experiment (28.13 Hz)
Model (28.40 Hz)
Model (28.10 Hz)
0
10
6
10
4
(f)
10
10
Experiment (28.38 Hz)
Experiment (28.13 Hz)
Model (28.40 Hz)
Model (28.10 Hz)
-1
4
10
Load Resistance ()
Power (mW/g2)
Current (mA/ g)
10
10
2
4
10
Load Resistance ()
10
6
Figure 6.6: Experimental and numerical electrical performance curves of aluminum wing
spar for the peak voltage output in the (a) series and (b) parallel case, current output in the
(c) series and (d) parallel case, and power output in the (e) series and (f) parallel case with
varying load resistance.
resistance of 64.04 kΩ for the series connection and 16.09 kΩ for the parallel connection.
For excitation at the open-circuit condition, the model predicts the peak power outputs as
913.9 mW/g2 for a load resistance of 503.3 kΩ for the series connection and 126.1 kΩ for
the parallel connection.
127
6.3
Chapter Summary
Electromechanical modeling is presented for a multifunctional energy harvesting wing
spar with embedded self-charging structures. The assumed modes model presented in
Chapter 2 is modified to include the effects of the mass and stiffness distribution of the
wing spar, including the effects of a lumped mass, in order to predict the coupled vibration
and voltage response of the cantilever wing spar model. The energy harvesting spar configuration is first described in which both series and parallel electrode connections of the
bimorph structure are considered. Details of the assumed modes formulation are then reviewed. Experimental testing is performed on a representative aluminum wing spar with
embedded piezoelectric and thin-film battery layers in order to validate the model. Experimentally measured voltage and tip velocity frequency response functions are compared
to those predicted by the assumed modes model. Lastly, several electrical performance
curves are created, again comparing experimental data with model predicted results.
The assumed modes model developed in this chapter is based on the model previously
described in Chapter 2 and developed by Erturk and Inman. Differences in the models result from the presence of a lumped mass placed at an arbitrary position in the x-direction
as well as the unique bending stiffness and mass density functions of the wing spar configuration. Comparison of the experimentally measured voltage and tip velocity FRFs to
those predicted by the model show good correlation and confirm the ability of the model
to predict the vibration and voltage response of the representative wing spar.
128
C HAPTER
7
E NERGY H ARVESTING W ING S PAR FOR
E LECTRIC U NMANNED A ERIAL V EHICLE
P
IEZOELECTRIC
vibration energy harvesting in unmanned aerial vehicles has been in-
troduced as a case study in this dissertation. Chapter 5 presents the performance
modeling of electric UAVs with on board energy harvesting systems while Chapter 6 describes the electromechanical modeling of a UAV wing spar with embedded self-charging
structures along with experimental validation of the model in which a representative aluminum spar is tested. In this chapter, the development and testing of a realistic composite
multifunctional energy harvesting wing spar with embedded self-charging structures for
use in the EasyGlider RC aircraft test platform is given. A stiff, lightweight fiberglass wing
spar is developed here with piezoelectric and thin-film battery layers embedded in the
layup of the spar. Details of the design and fabrication of the wing spar are first given. Experimental bench testing is then performed on the spar in which frequency response measurements are made to gain information about the fundamental natural frequency of the
spar, and charge/discharge testing is performed in which the spar is excited harmonically
at resonance while the piezoelectric layers are connected to directly charge the thin-film
battery layers. Lastly, experimental flight testing is performed in which the fiberglass spar
is used in the EasyGlider aircraft and the energy harvesting performance during flight is
investigated.
129
7.1
Design and Fabrication of Multifunctional Composite Wing
Spar
A fiberglass wing spar with embedded self-charging structures is designed for use in the
EasyGlider aircraft. The physical configuration of the wing spar is shown in Fig. 7.1. The
main construction of the spar consists of a polystyrene foam core (which provides shape
and is extremely lightweight) wrapped in several layers of fiberglass. The wing spar is
shaped to fit in the tapered wing cavity of the EasyGlider aircraft (recall, a fiberglass spar
with surface mounted piezoelectric devices has been previously tested in the EasyGlider
aircraft, as described in Section 5.4, hence the wings have been previously modified to
accept a rectangular spar). The foam core has a length of 99.0 cm, a width of 2.87 cm,
and a maximum thickness of 1.4 cm at the center of the spar and 0.8 cm at the ends of
R
the spar. Two piezoelectric layers (QuickPack
QP10n devices) and two thin-film battery
R
layers (Thinergy
MEC-1017SES batteries) are used on both the top and bottom surfaces of
the spar, effectively creating two self-charging structures with the foam core becoming the
substrate layer. The self-charging structures are embedded between layers of fiberglass,
thus they are integral to the wing spar. Each self-charging structure is placed 3.8 cm from
the center of the wing spar, which corresponds to the root of the wing against the fuselage
when the spar is inserted into the wings and the aircraft is fully assembled. This location
will experience the largest strain during flight, hence the placement of the piezoelectric
devices at the root of the wings. Additionally, extra fiberglass layers are included in the
region between the piezoelectric devices to help increase the stiffness of the spar section
that passes through the fuselage and for maximum strain to be induced in the piezoelectric
devices.
Based on preliminary testing of various foam cores and fiberglass layers, extruded
Fiberglass Layers
Thin-Film
Battery Layers
Piezoelectric Layers
Foam Core
7.6 cm
1.4 cm
99 cm
Figure 7.1: Fiberglass wing spar schematic showing foam core with fiberglass layers and
embedded self-charging structures.
130
polystyrene insulating foam from The Dow Chemical Company [154] is selected as the
foam core material and 1080-50 E-Glass fiberglass cloth (which has a density of 1.45 oz/sq.
yd.) from Aircraft Spruce and Specialty Co. [155] is selected as the fiberglass layer. Additionally West System [150] 105/205 two part epoxy is chosen as the epoxy layer in the
fiberglass layup. Three layers of 1080-50 cloth are chosen in order to best match the stiffness of the original round carbon fiber reinforced plastic wing spar rod. The self-charging
structures are placed between the second and third layer of fiberglass. An additional layer
of fiberglass is added to the center section of the spar between the piezoelectric devices.
Electrodes are attached to each piezoelectric device and battery prior to fabrication of the
wing spar.
Fabrication of the spar involves first cutting a foam core using a hot wire cutter to the
dimensions shown in Fig. 7.1. A single piece of fiberglass cloth large enough to wrap the
entire foam core three times is then cut. The fiberglass is saturated in epoxy and wrapped
R
around the foam core. Between the second and third fiberglass layer, the QuickPack
and
R
Thinergy
devices are placed on the upper and lower surfaces of the spar. Small slits
are cut in the fiberglass cloth such that the device electrodes can be passed through the
fiberglass. With three layers of fiberglass covering the spar, an additional layer is added to
the center of the spar. The wing spar is cured using a vacuum bagging procedure similar
to that described in Subsection 3.2.2. A schematic of the vacuum bagging setup used for
the wing spar fabrication is shown in Fig. 7.2. The spar is surrounded on all sides by a peel
ply layer, a perforated ply layer (which controls the amount of epoxy absorbed away from
the spar), and a bleeder/breather layer because of the large amount of epoxy which must
be removed from the saturated fiberglass cloth to yield a lightweight spar. Once placed in
the vacuum bag, the spar is allowed to cure under approximately 20 in-Hg of vacuum for
6 hours. Photographs of the wing spar placed in the vacuum bag for curing are shown in
Fig. 7.3.
The fabricated fiberglass wing spar is shown in Fig. 7.4. The original carbon fiber reinforced plastic spar weighs 53 g, and the fiberglass spar weighs 50 g, comparing favorably.
Additionally, the stiffness of the fiberglass spar is found to be similar to the original spar.
131
Vacuum Gauge
15
20
10
25
5
0
30
in Hg
Vacuum Pump
Wing Spar
Work Surface
Breather/Bleeder
Cloth
Peel Ply
Perforated Ply
Vacuum Bag
Tape
Figure 7.2: Schematic of vacuum bagging procedure for fiberglass wing spar fabrication.
7.2
Experimental Evaluation of Energy Harvesting/Energy Storage Performance
Experiments are performed on the fiberglass wing spar in order to evaluate its ability to
simultaneously harvest and store energy in a self-charging manner. Harmonic base excitation at the fundamental natural frequency of the spar will be used, therefore, the resonance frequency must first be obtained. Frequency response function measurements are
made on the spar mounted in a double cantilever configuration (clamped at the center of
the spar) for this purpose. Once the fundamental natural frequency of the spar is found,
charge/discharge measurements are made in which the spar is excited at resonance while
the piezoelectric layers are used to charge the battery layers.
Figure 7.3: Photographs of fiberglass wing spar curing under vacuum.
132
Figure 7.4: Complete self-charging fiberglass wing spar.
7.2.1
Frequency Response Measurements
The frequency response of the fiberglass wing spar is first obtained experimentally. The
experimental setup used for the tests, shown in Fig. 7.5, is similar to the setup used for testing the aluminum wing spar (see Fig. 6.3). The spar is mounted to an an APS Dynamics,
Inc. [153] 113 long-stroke shaker, powered by an APS Dynamics, Inc. 125 power amplifier.
A double cantilever configuration is used in which the spar is clamped at the center, effectively creating a cantilever beam on either side of the clamp, as shown in Fig. 7.5. This
configuration is used to simulate the operating condition of the spar when inserted into the
EasyGlider aircraft (where the fuselage effectively becomes a rigid base used to excite each
side of the spar). In the clamped configuration, each cantilever has an overhang length
of 47.0 cm. A DSP Technology, Inc. [119] SigLab 20-42 data acquisition system is used for
the FRF measurements, which are obtained using low amplitude chirp excitation. Five averages are taken for each measurement. The input acceleration is measured using a PCB
Piezotronics, Inc. [120] U352C67 accelerometer attached to the base of the clamp, the tip
velocity is measured at the tip of the spar using a Polytec, Inc. [121] PDV-100 laser Doppler
vibrometer, and the voltage output of the self-charging structure is measured directly with
the SigLab data acquisition system using an Agilent Technologies, Inc. [109] N2862A 10:1
probe.
The frequency response functions of the spar with two piezoelectric layers of a self-
133
(b)
(a)
(a) SigLab data acquisition system
(b) Laser vibrometer
(c) Fiberglass wing spar
(d) Electromagnetic shaker
(e) Amplifier
(f) Accelerometer
(g) Clamp
(c)
(d)
(c)
(e)
(f)
(g)
(d)
Figure 7.5: Experimental setup used to obtain frequency response measurements of fiberglass wing spar.
charging structure connected in series to a 1 MΩ load resistance (creating an effective
load resistance of 909090.91 Ω when taking the 10 MΩ Agilent probe into consideration)
are recorded and are given in Fig. 7.6, where Fig. 7.6(a) shows the entire spectrum and
Fig. 7.6(b) shows the results near the fundamental resonance frequency. Recalling the results presented in Fig. 3.10(b), where the response of the self-charging structure connected
to the linear regulator energy harvesting circuit was found to closely match the case with
the largest resistive load, a load of 1 MΩ is chosen here as the same linear regulator circuit
will be used for charge/discharge measurements. The frequency response measurements
given in Fig. 7.6 show that the fundamental resonance frequency of the spar is around
43 Hz. It can be seen that the measurements contain several peaks, and while atypical
for a uniform cantilever beam, this behavior is expected for the non-uniform composite
fiberglass spar. In addition to various bending modes, such a structure may experience
134
(a)
2
Voltage FRF (V/g)
Tip Velocity FRF ((m/s)/g)
FRF Magnitude
10
0
10
-2
10
-4
10
0
100
200
300
Frequency (Hz)
400
500
(b)
Voltage FRF (V/g)
Tip Velocity FRF ((m/s)/g)
2
FRF Magnitude
10
0
10
10
20
30
40
50
Frequency (Hz)
60
70
80
Figure 7.6: Frequency response functions of fiberglass wing spar (series electrode connection, 1 MΩ load) showing (a) broad spectrum and (b) detail near the fundamental resonance frequency.
torsional vibration modes due to non-uniformity. Additionally, the clamp condition is not
perfect as a large clamping force would cause mechanical failure of the wing spar. Thin
pieces of foam are also inserted between the spar and clamp interface to help protect the
spar. The non-ideal clamping condition likely introduces several of the peaks observed
in the measurement. It can be observed from Fig. 7.6(b) that two peaks exist at the fundamental frequency, making the exact resonance frequency difficult to assess. These FRF
measurements, however, give a basis for the fundamental natural frequency of the fiberglass spar. Precise tuning can be performed during the charge/discharge measurements.
135
7.2.2
Self-Charging Charge/Discharge Measurements
With a general measure of the fundamental natural frequency obtained, the energy harvesting performance of the self-charging fiberglass wing spar is investigated. The spar is
excited at resonance with two piezoelectric layers connected through the linear regulator
energy harvesting circuit, shown previously in Fig. 3.12, to charge a single thin-film battery
layer. A base excitation level of ±0.5 g is chosen for the testing based on the flight acceleration measurements found previously in Chapter 5 (shown in Fig. 5.7(c)), which show an
average acceleration of around ±1.0 g in the z-direction. Considering that the excitation
frequency experienced on the aircraft does not match the resonance of the harvester, an acceleration level of half of the measured acceleration is chosen, however, the resonance tests
performed here will likely still overestimate the harvesting ability of the spar during flight.
After some preliminary tests, the two piezoelectric layers of a single self-charging structure
will be connected in parallel to the energy harvesting circuit for increased current output
as the voltage level under ±0.5 g excitation is sufficient for charging the battery layer. The
experimental setup used for the charge/discharge measurements is shown in Fig. 7.7. The
setup is similar to that used in Subsection 3.3.2, however, a Keithley Instruments Inc. [156]
2611A SourceMeter is used for the current measurement in charging and for both current
and voltage measurements in discharging. The SourceMeter is capable of simultaneously
sourcing or sinking either current or voltage while measuring both quantities. In discharging, the SourceMeter is used to draw a constant current out of the battery throughout the
entire discharge test (which is not possible by using a simple resistive load to discharge the
battery, as done previously, because the voltage of the battery drops slightly throughout
the test). In charging, a National Instruments Corp. [110] CompactDAQ chassis utilizing
a NI 9215 4-channel analog voltage input card is used measure the battery voltage. Additionally, a NI 9233 4-channel analog input card with IEPE coupling is used to measure the
accelerometer signal to ensure ±0.5 g excitation is achieved, and a NI 9263 4-channel analog output card is used to excite the shaker. Both the Keithley SourceMeter and National
Instruments CompactDAQ chassis are controlled by custom written LabVIEW programs
running on separate computers.
Prior to running the charge/discharge test, the exact resonance frequency of the fiberglass spar when connected to the energy harvesting circuit is found by fine tuning the
136
(a) Keithley SourceMeter
(b) NI CompactDAQ data
acquisition system
(c) Fiberglass spar
(d) Clamp
(e) Accelerometer
(f) Electromagnetic shaker
(a)
(b)
(e)
(c)
(d)
(a)
(f)
Figure 7.7: Experimental setup used to obtain charge/discharge measurements of selfcharging fiberglass wing spar.
excitation frequency around 43 Hz while monitoring for the largest power output from
the piezoelectric layers, resulting in a resonance frequency of 43.5 Hz. The fiberglass spar
is excited for 3 hours while the piezoelectric layers are used to charge the battery layer.
Both the Keithley SourceMeter and the NI CompactDAQ data acquisition system are configured to record the current and voltage data at 10 Hz sampling rates. Results of the test
are presented in Fig. 7.8. The results of the charge test are given in Fig. 7.8(a), where it can
be seen that an average of about 110 µA of current flows into the battery at a voltage of 3.9
V, corresponding to an average power of 0.430 mW. Both the current and the voltage remain fairly constant throughout the duration of the test. The charge capacity is calculated
using Eq. (3.2) as 0.323 mAh. From the discharge test results presented in Fig. 7.8(b), it can
be seen that a constant current of 2C (1.4 mA) is drawn from the battery throughout the
test. The voltage remains constant at the beginning of the test and tapers off quickly near
the end. In discharging, a capacity of 0.324 mAh is calculated. The discharge capacity is
slightly higher than the charge capacity, likely because some small residual charge existed
in the battery prior to the test. The results of the charge/discharge testing show the ability
of the fiberglass spar to operating in a self-charging manner, and give an estimate of the
power that can be harvested by the spar.
137
(a)
0.25
4.1
Current (mA)
0.2
3
0.15
Current
0.1
2
Charge Capacity:
0.323 mAh
0.05
Voltage (V)
Voltage
1
0
0
8000
10000
0
5
4
Current (mA)
4000
6000
Time (sec)
4.1
3.8
Voltage
3
3
Load
Disconnected
2
Current
2
Voltage (V)
(b)
2000
1
Discharge Capacity:
0.324 mAh
0
0
200
400
Time (sec)
600
1
800
0
Figure 7.8: Experimental curves for fiberglass wing spar in (a) charging and (b) discharging
under ±0.5 g acceleration at 43.5 Hz.
7.3
Experimental Flight Testing of Self-Charging Fiberglass Wing
Spar
Flight testing is performed in which the self-charging fiberglass wing spar is used in the
EasyGlider aircraft test platform. Vibration energy harvesting is performed during flight
and several measurements are made using data loggers. Details of the EasyGlider aircraft
configuration as well as the flight testing measurement results are reviewed.
138
7.3.1
EasyGlider Aircraft Configuration
The EasyGlider aircraft configuration used for flight testing with the self-charging fiberglass spar is similar to the aircraft configuration described previously in Subsection 5.4.1.
No further modification is required to the aircraft wings to allow insertion of the selfcharging fiberglass spar, which is shown in Fig. 7.9. The two fiberglass canopies developed previously are used again to facilitate the installation of the Pace Scientific, Inc. [151]
XR5-SE-M-50mv eight channel, 0-5 V data logger (Fig. 5.5) and the UA-004-64 Pendant G
3-axis accelerometer data logger from Onset Computer Corp. [152] (Fig. 5.6).
7.3.2
Flight Testing Results
Flight testing is performed in which the energy harvesting performance of the self-charging
fiberglass spar is evaluated in flight. Although it is desirable to measure the operation of
the wing spar in a self-charging manner during flight (in which the piezoelectric layers are
used to charge a battery layer), several challenges exist making this test condition impractical with the current configuration. The energy harvesting circuitry required to allow the
transfer of energy from the piezoelectric layers to the battery layers would require miniaturization, perhaps incorporating the entire circuit on a single chip, in order to physically
fit in the aircraft. The linear regulator circuit has been constructed on a breadboard for de-
Figure 7.9: Self-charging fiberglass wing spar inserted into EasyGlider aircraft wings.
139
velopment purposes, shown previously in Fig. 3.14(c), and is too large to fit in the cockpit
of the aircraft. Additionally, the available data loggers lack current measuring capability for monitoring the current transfered from the piezoelectric layers to the battery layer.
Furthermore, in the short flight time of the aircraft (on the order of 5-10 minutes), it is
expected that the amount of energy harvested by the piezoelectrics may not show appreciable increase in the battery capacity beyond any recovered capacity the battery naturally
achieves after being discharged. Therefore, during the flight test, the output of a single
R
QuickPack
piezoelectric device across a matched load of 130 kΩ is measured with the
XR5-SE-M-50mv data logger at a sampling frequency of 100 Hz. The aircraft acceleration
in the z-axis (vertical axis) is also measured during flight with the UA-004-64 accelerometer data logger at a sampling rate of 100 Hz. During the flight test, the aircraft was hand
launched and flew on a sunny day with light and variable winds experiencing gusts up to
10 km/h for about 10 minutes, of which about 7 minutes of data was recorded with the
data loggers. Figure 7.10 shows a photograph of the EasyGlider aircraft in flight during
testing. The aircraft followed a wide circle flight pattern with several climb, cruise, descend cycles. In cruise conditions, a maximum altitude of around 75 m and a cruise speed
of around 30 km/h was achieved, with maximum speeds reaching around 40 km/h.
Results of the flight testing are presented in Fig. 7.11. It should be noted that the air-
Figure 7.10: Photograph of EasyGlider aircraft during flight testing of multifunctional
fiberglass wing spar.
140
craft is launched after about 60 seconds of data logging (evident from the acceleration
measurement). From the voltage results presented in Fig. 7.11(a), it can be seen that the
R
device exceeds the 5 V limit of the data logger at various points
output of the QuickPack
throughout the flight. The results, however, can still be used to provide a rough estimate
of the power harvested during flight. Using Eq. (5.32), the average power output of the
piezoelectric device is calculated as 3.22 µW. Due to the voltage clipping experienced by
the data logger, it is expected that the results underestimate the actual power harvested
during flight.
(a)
Piezoelectric Voltage (V)
5
4
3
2
1
0
0
(b)
100
200
300
Time (sec)
400
500
100
200
300
Time (sec)
400
500
4
Z-Axis
Acceleration (g)
3
2
1
0
-1
0
Figure 7.11: Flight testing results of fiberglass self-charing spar showing (a) voltage output
of piezoelectric device and (b) z-axis acceleration.
141
The acceleration measured in the z-axis during flight is given in Fig. 7.11(b). The results
show an an average acceleration of around ±0.75 g (note the acceleration during flight is
centered around 1.0 g due to the acceleration of gravity acting on the accelerometer). Peak
acceleration values reach up to ±2.0 g at times during the flight test. These results provide
a baseline for the excitation levels experienced during flight.
7.4
Chapter Summary
Details of the design, fabrication, and experimental testing of a multifunctional energy
harvesting/energy storage fiberglass UAV wing spar are presented. The fiberglass spar,
which contains two self-charging structures, is designed for use in the EasyGlider test
platform. Piezoelectric and battery elements are embedded near the center of the spar
at the root of the wings for maximum induced strain in the piezoelectrics during flight.
Experimental testing is performed on the fabricated spar to evaluate its energy harvesting
ability. Frequency response measurements are first obtained to determine the fundamental
resonance frequency of the spar. Charge/discharge measurements are then performed by
subjecting the spar to resonant base excitation while the piezoelectric layers are used to
charge the battery layers. Lastly, flight testing is performed in which the spar is used in
the EasyGlider aircraft and the output of a piezoelectric layer is monitored during flight.
Experimental evaluation of the charge/discharge performance of the multifunctional
spar under harmonic base excitation at the fundamental natural frequency confirms the
ability of the spar to simultaneously harvest energy in the piezoelectric layers and store
energy in the battery layers. At an excitation level of ±0.5 g, an average power of 0.430
mW transfered from the piezoelectric layers to the battery layers through a simple linear
regulator harvesting circuit is found. Through the use of more efficient energy harvesting
circuity, the amount of harvested/transfered power is expected to increase. Flight testing
of the wing spar installed in the EasyGlider aircraft provides a realistic example of the
use of self-charging structures in an application where multifunctional solutions are necessary. Results of the flight testing show that an average of about 3.22 µW of energy can be
harvested from each piezoelectric device. Although the harvested energy is small, recall
the focus of piezoelectric harvesting in UAVs is to provide autonomous power sources for
low-power sensors, which is likely possible with this level of harvested energy.
142
C HAPTER
8
C ONCLUSIONS AND R ECOMMENDATIONS
E
NERGY
harvesting is a critical technology for the development of autonomous, self-
powered electronic devices. As the energy requirements of low-power electronics
decrease, the use of energy harvesting systems in providing them power is becoming
more promising. Energy harvesting technology has the potential to revolutionize the way
in which power is provided to small electronics; transitioning from a reliance on battery
power to self-powered, autonomous systems. In this dissertation, multifunctional piezoelectric vibration energy harvesting concepts have been investigated. Perhaps one of the
most promising applications of piezoelectric vibration energy harvesting is in providing
power to autonomous sensor nodes. Piezoelectric vibration energy harvesting utilizes the
electromechanical coupling exhibited in piezoelectric materials to convert ambient vibration energy surrounding a system into useful electrical energy. The harvested energy is
typically stored in a battery or capacitor for use in powering a small electronic device. The
multifunctional concepts explored in this dissertation aim to add functionality to classical
piezoelectric energy harvesting systems such that a single harvester can not only generate useful electrical energy, but store the harvested energy and support structural load as
an integrated component in a host structure. Multifunctional energy harvesting solutions
work to enable the use of energy harvesting in systems where added mass and volume are
critical and a fully integrated solution is desirable. This final chapter provides a summary
of the research presented in this dissertation. Significant contributions made to the field
of piezoelectric energy harvesting are reviewed. Finally, recommendations are made to
extend the reach of this research in the future.
143
8.1
Research Summary
Several aspects of the development of multifunctional piezoelectric self-charging structures
are investigated in this dissertation. Introduction of the self-charging structure concept as
well as the application of an electromechanical model used to predict the electrical and
mechanical response of self-charging structures is given in Chapter 2. Self-charging structures consist of piezoelectric layers directly bonded to thin-film battery layers in a bimorph
configuration with a passive substrate layer such that excitation of the device allows for
simultaneous energy generation in the piezoelectric layers and energy storage in the battery layers. Self-charging structures can also be embedded into a host structure and used
to support structural load, thus providing energy harvesting, energy storage, and load
bearing ability. An approximate distributed parameter electromechanical assumed modes
model developed by Erturk and Inman is applied to cantilever self-charging structures
subjected to harmonic base excitation. Electromechanical displacement - to - base acceleration and voltage - to - base acceleration frequency response function expressions are
derived to describe the mechanical and electrical response of the structures.
Experimental evaluation of a self-charging structure prototype is presented in Chapter 3 in order to validate the electromechanical model presented in Chapter 2 and to confirm the ability of the device to simultaneously harvest and store electrical energy. Simulation results using the assumed modes model are found to closely match the experimentally
measured electromechanical frequency response functions over a range of load resistances
from 100 Ω to 1 MΩ for both the series and parallel connection of the piezoelectric layers.
Both the voltage FRF and the tip velocity FRF show good correlation between model and
experiment, thus validating the electromechanical model. Results of charge/discharge
testing in which the self-charging structure prototype is excited harmonically while the
piezoelectric layers are used to charge the battery layers through a linear regulator energy
harvesting circuit show simultaneous energy generation and energy storage ability. Under
±1.0 g of base excitation at the fundamental resonance frequency of the device, approximately 0.3 mW of power is harvested and stored in a thin-film battery layer.
It is proposed that self-charging structures be integrated into host structures in a multifunctional manner to support structural load. Chapter 4 presents analysis of the strength of
self-charging structures including both static strength testing and dynamic strength test-
144
ing. Static three-point bend testing is performed on a prototype device. Results of the
testing show that the bending strength of a complete self-charging structure is around 100
- 125 MPa. Dynamic testing is performed in which a cantilever self-charging structure is
subjected to harmonic base excitation of increasing amplitude while monitoring for mechanical or electrical failure of the device. Self-charging structures are found to be quite
robust under dynamic excitation, experiencing no failure at excitation levels up to ±7.0
g of base acceleration. Static three-point bend testing is also performed on a variety of
piezoceramic materials including monolithic, single crystal, and composite piezoelectrics.
Results of the comprehensive testing can be used as a design tool to aid in the development of self-charging structures, allowing the most appropriate piezoelectric material to
be selected for various applications.
Vibration energy harvesting in unmanned aerial vehicles is explored as a case study
in this dissertation in which multifunctional self-charging structures are used in a mass
and volume critical application. The goal of piezoelectric vibration harvesting in UAVs
is to provide local power sources for autonomous sensors. Modeling of the flight performance of electric UAVs with on board energy harvesting systems is presented in Chapter 5. A flight endurance formulation recently published by Thomas et al. is applied to the
EasyGlider remote control aircraft, which is chosen as the test platform explored in this
dissertation. Simulation results of the normalized change in flight time with the addition
of piezoelectric vibration energy harvesting indicate that a fully multifunctional design
in which all mass added to the aircraft by the harvesting system be removed from the
structural mass of the aircraft in a multifunctional manner is required to prevent loss in
flight time. This result is mainly attributed to the large density and low specific power
output of piezoelectric materials. Simulations performed to predict the effects of adding
self-charging structures to the EasyGlider aircraft confirm the need for a multifunctional
solution to prevent significant loss in flight endurance.
Electromechanical modeling of a multifunctional UAV wing spar with embedded selfcharging structures is described in Chapter 6. An electromechanical assumed modes formulation for a cantilever 1/2 spar model (where the moving base represents the aircraft
fuselage) based on the model presented by Erturk and Inman is developed to predict the
response of a UAV wing spar with integrated piezoelectric and thin-film battery layers. Experimental testing of a representative aluminum wing spar with embedded self-charging
145
structures is performed in order to validate the model. Comparison of experimental voltage and tip velocity FRFs to those predicted by the model show good correlation for both
the series and parallel electrode connection of the piezoelectric layers over a range of load
resistances, thus validating the model.
Fabrication and experimental testing of a realistic fiberglass wing spar with embedded
self-charging structures for use in the EasyGlider test platform is presented in Chapter 7.
A lightweight wing spar composed of several layers of fiberglass surrounding a foam core
is fabricated with piezoelectric and thin-film battery layers integrated between layers of
fiberglass. Experimental charge/discharge testing is performed where the spar is clamped
at the center and excited harmonically at resonance. Results show that the spar is able to
operate in a self-charging manner, with two piezoelectric layers delivering an average of
0.43 mW of power to a thin-film battery layer at ±0.5 g of excitation. Experimental flight
testing is also performed in which the fiberglass spar is used in the EasyGlider aircraft and
the energy harvesting performance of the spar is evaluated during flight. Although it was
not feasible to connect the piezoelectric layers to charge a thin-film battery layer during
the flight testing, the output of a single piezoelectric layer was measured across a matched
load resistance and an average power of about 3 µW was recorded during flight. These
results present the use of self-charging structures in a realistic application that can benefit
from a multifunctional approach.
8.2
Contributions of the Research
The self-charging structure concept presented in this dissertation is the first multifunctional vibration energy harvesting system to combine energy harvesting, energy storage,
and load bearing ability in a single device. A summary of the main contributions of this
work and reference to the corresponding publications is given below.
• A multifunctional energy harvesting concept that directly combines piezoelectric elements for energy generation and thin-film battery elements for energy storage has
been investigated. The proposed self-charging structures work to mitigate the drawbacks of conventional vibration energy harvesting approaches, in which the harvesting system is retrofit to an existing host structure, potentially hindering the performance of the original system, to enable more synergistic solutions to piezoelectric
146
energy harvesting [157].
• Comprehensive design, modeling, and experimental evaluation of the self-charging
structure concept has been performed. An electromechanical assumed modes model
has been presented to predict the coupled mechanical and electrical response of cantilever self-charging structures subject to harmonic base excitation. Experimental
testing of a prototype device validates the electromechanical model and also confirms the ability of the device to simultaneously harvest and store electrical energy
in a self-charging manner [115, 123, 157–159].
• Strength properties of prototype self-charging structures have been determined through both static three-point bend testing and dynamic resonance excitation testing.
Test results can be used in the design of embedded self-charging structure applications [123, 157].
• Experimental analysis of the bending strength of a variety of commercially available piezoceramic materials including hard, soft, single crystal, and composite piezoelectrics is performed. Previously lacking in the literature, the results can not only
be used in the design of optimal self-charging structures, but in any piezoelectric
system in which the active material is subject to bending loads [160].
• Multifunctional vibration harvesting in unmanned aerial vehicles using embedded
self-charging structures is explored as a case study. The electromechanical modeling,
design, fabrication, and experimental testing of a multifunctional energy harvesting/energy storage UAV wing spar provides one of the first works of piezoelectric
energy harvesting in UAVs in which extensive modeling and experimental studies
are carried out [161, 162].
8.3
Recommendations and Future Work
The multifunctional energy harvesting concepts explored in this dissertation provide an
initial investigation into the potential use and benefits of a multifunctional approach to
vibration energy harvesting. The results presented here show successful application of
simultaneous energy harvesting and energy storage, however, further investigation is required before such a multifunctional solution is seen as a practical replacement for bat147
teries. A more comprehensive understanding of the energy harvesting/energy storage
ability of self-charging structures when subjected to realistic excitation profiles is required.
Future investigation should consider a variety of ambient excitation sources consistent
with potential applications of self-charging structures. The design of self-charging structures, including selection of piezoelectric material, substrate material, and geometry, can
also be tailored to best suit specific applications. Another key issue that must be addressed
involves the development of more efficient energy harvesting and management circuitry
for the transfer of harvested energy from the piezoelectric layers to the battery layers for
storage as well as management of the energy consumed by the sensor node. In addition to
improving the efficiency of harvesting circuitry, future work should investigate the development of electronics that can be directly embedded in the energy harvesting device. In
this work, the harvesting circuit is external to the multifunctional self-charging structures,
however, the system can be improved with the integration of all required components
into a single device. Novel flexible electronic circuitry can be investigated for this purpose. Lastly, a critical element to be addressed in the future involves the investigation of
the longevity of a multifunctional piezoelectric energy harvesting system. Investigation
of the ability of the harvesting system to operate autonomously for the lifetime of a sensor node, including robustness against varying ambient vibration conditions, temperature,
and other environmental factors, is paramount.
While the electromechanical modeling presented in this dissertation can be used to accurately predict the response of cantilever self-charging structures subjected to harmonic
base excitation, new models can be developed to describe the behavior of the devices in a
variety of configurations and applications. Accurate models that can predict the response
of surface mounted devices and devices subject to stochastic ambient vibration excitation,
for example, should be explored. Although the strength of self-charging structures themselves has been thoroughly investigated, the effects of embedding self-charging structures
on the strength of host structures should also be considered. Mechanical models that can
predict the change in strength of a host system with the addition of self-charging structures would prove useful in the development of integrated multifunctional energy harvesting solutions. Finally, design optimization should be considered in the future. The
self-charging fiberglass wing spar developed in this work has not been optimized for maximum energy harvesting performance, and optimization of the spar is expected to provide
148
significant improvements in the performance of the device.
The development of multifunctional piezoelectric self-charging structures presented in
this dissertation holds promise for enabling the creation of autonomous, self-powered electronics. The groundwork for multifunctional energy harvesting systems is presented here
and future work addressing the key development issues of the technology can ultimately
lead to its widespread use.
149
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165
A PPENDIX
A
P IEZOELECTRIC C ONSTITUTIVE E QUATIONS
A.1
Constitutive Equations for Bulk Monolithic Piezoelectric Material
The constitutive equations for linear piezoelectric material are well documented in the
IEEE Standard on Piezoelectricity [163]. Both the direct and converse piezoelectric effects
can be expressed mathematically as a relationship between four field variables including
stress, S, strain, T , electric field, E, and electric displacement, D. Four forms of the constitutive equations can be written by considering any two field variables as independent
variables. The most common representation of the constitutive relations can be expressed
in tensor (or indicial) notation as [163]
Sij = sE
ijkl Tkl + dkij Ek
(A.1)
Di = dikl Tkl + εTik Ek
(A.2)
where Sij is the mechanical strain, Tkl is the mechanical stress, Ek is the electric field, Di
is the electric displacement, sE
ijkl is the mechanical compliance (reciprocal of the elastic
modulus) measured at constant electric field, dikl is the piezoelectric strain coefficient, and
εTik is the dielectric permittivity measured at zero mechanical stress.
Based on the symmetry of tensors, the constitutive equations can be written in compact
matrix form as
where a superscript
 
S

 
0
sE d T

=
D
d εT E
0
(A.3)
denotes transpose. Here the following notation (known as Voigt
166
notation) is adopted
S1 = S11
T1 = T11
(A.4)
S2 = S22
T2 = T22
(A.5)
S3 = S33
T3 = T33
(A.6)
S4 = S23 + S32
T4 = T23 + T32
(A.7)
S5 = S13 + S31
T5 = T13 + T31
(A.8)
S6 = S12 + S21
T6 = T12 + T21
(A.9)
The fully populated constitutive matrices can be further reduced for orthotropic materials (most piezoelectric materials are orthotropic) as follows
 

S1 









S2 










S3 


1
Y1E

 ν12
− Y E
 1
 ν
− 31E

=  Y3


 0

S4 















S


 0
5



 

 

S6
0
 



D 

 1


− Yν12E
0
0
0
1
− Yν13E
1
Y1E
− Yν23E
1
0
0
0
− Yν32E
1
Y3E
0
0
0
0
0
1
GE
23
0
0
0
0
0
1
GE
13
0
0
0
0
0
1
GE
12
3
1
0
0
0
0 d15


D2 =  0
0
0 d24 0







D 
d13 d23 d33 0
0
3
  



T
0
0
d
 1
13











 

T2 
  0
0
d
23 







 

E1 




T3 
  0



0
d
33



+

E
2




 0 d24 0  


T4 













E
3










T5 
d15 0
0 



 


 

T6
0
0
0
 

T1 













 
T


2


T




E1 

0 
ε
0
0

T3 
  11

 




+  0 εT22 0  E2
0



T4 
 






E 

T


0 
0
0
ε
3


33



T5 







 

T6
(A.10)
where Y is the elastic modulus, ν is Poisson’s ratio, G is the shear modulus, and the components of the elastic compliance matrix, sE
ijkl , have been expressed in terms of these quantities.
A.2
Alternative Representations of the Constitutive Equations
As noted above, four unique forms of the constitutive equations can be written based on
the choice of independent variables. In addition to Eq. (A.3), the three alternative forms
167
are given as
 
T

 
cE −e  S 

= 0
D
e εS E
  
 
D
S
s
g T

= 0
T  
E
D
−g β
  
 
T
cD −h  S 

= 0
S  
E
−h β
D
e = cE d
0
εS = εT − d cE d
(A.11)
g = dβ T
0
sD = sE − dβ T d
(A.12)
h = cD g
0
β S = β T + g cD g
(A.13)
The transformations listed in Eq. (A.11), Eq. (A.12), and Eq. (A.13) can be used to help
move between each form of the constitutive equations.
A.3
Reduced Form of the Constitutive Equations for a Thin Beam
Operating in ‘31’ Mode
A common operating mode of a piezoelectric material is the ‘31’ mode, in which the electric
field is applied/produced in the 3-direction (it is common practice to align the 3-direction
axis of the material with the poling direction, which is assumed here) and the voltage is
applied/generated in the 1-direction. For a thin beam allowing Euler-Bernoulli beam assumptions operating in ‘31’ mode, the only non-zero stress component is in the 1-direction,
giving
T2 = T3 = T4 = T5 = T6 = 0
(A.14)
Focusing on the electric field in the 3-direction, this leads to the following reduced
form of the constitutive equations (applying the first alternative representation given in
Eq. (A.11))
T1 = cE
11 S1 − e31 E3
D3 = e31 S1 + εS33 E3
168
(A.15)
A PPENDIX
B
B ENDING S TIFFNESS AND M ASS D ENSITY
C ALCULATIONS FOR C OMPOSITE
C ROSS -S ECTIONS
B.1
Bending Stiffness Calculation using the Parallel Axis Theorem
For a symmetric composite structure, the neutral axis of the cross-section lies in plane with
the x-axis at the center of the structure. The bending stiffness of a symmetric composite
section can, therefore, be found using the parallel axis theorem to calculate the area moment of inertia of each material section about the neutral axis. The parallel axis theorem
allows the moment of inertial of a rigid body to be found about any axis. Consider the
rigid body shown in Fig. B.1. The parallel axis theorem states
Ix = Ix0 + Az 2
(B.1)
where Ix is the area moment of inertia of the rigid body about the arbitrary axis, x, Ix0 is
the area moment of inertia of the body about its neutral axis, x0 , A is the area of the body,
and z is the distance of the neutral axis to the arbitrary axis.
x'
z
x
Figure B.1: Rigid body with neutral axis x0 .
169
B.2
Self-Charging Structure Sections
Consider the self-charging structure configuration shown in Fig. 2.4(c) and repeated again
in Fig. B.2. The neutral axis lies in the geometric center of the substrate layer in both
sections.
0  x  L1
hp
hs
hp
hk
Piezoceramic Layer
he h1
Thin-Film Battery Layer
Substrate Layer
L1  x  L
Kapton Layer
hb
Epoxy Layer
Electrodes
h2
Poling Direction
hb
Figure B.2: Cross-sectional views of the two composite sections of the self-charging structure.
For the first composite section from 0 < x < L1 , the bending stiffness of each material
section is given using Eq. (B.1) as follows
Outer Kapton Layer
Y Iko = Yk
Piezoelectric Layer
Y Ip = Yp
Inner Kapton Layer
Y Iki = Yk
Epoxy Layer
Y Ie = Ye
Substrate Layer
Y Is = Ys
!
hs
3hk 2
+ he + hp +
2
2
!
bh3p
hp 2
hs
+ bhp
+ he + hk +
12
2
2
!
bh3k
hs
hk 2
+ bhk
+ he +
12
2
2
!
bh3e
hs he 2
+ bhe
+
12
2
2
bh3k
+ bhk
12
bh3s
12
(B.2)
(B.3)
(B.4)
(B.5)
(B.6)
where Yk , Yp , Ye , and Ys are the elastic moduli of the Kapton, piezoelectric, epoxy, and
structure materals, respectively.
The total bending stiffness is the sum of the bending stiffness of each material section.
For the structural materials, the bending stiffness is given by
Y Ic1 = Y Is + 2Y Ie + 2Y Iki + 2Y Iko
170
(B.7)
For the piezoelectric materials, the bending stiffness is simply
cE
11 Ip = 2Y Ip
(B.8)
The mass density of the first composite section is simply
(B.9)
ρAc1 = ρs bhs + 2ρe bhe + 4ρk bhk + 2ρp bhp
For the second composite section from L1 < x < L, the individual bending stiffness of
each material section is
Battery Layer
Y Ib = Yb
Outer Epoxy Layer
Y Ieo = Ye
Outer Kapton Layer
Y Iko = Yk
Piezoelectric Layer
Y I p = Yp
Inner Kapton Layer
Y Iki = Yk
Inner Epoxy Layer
Y Iei = Ye
Substrate Layer
Y Is = Ys
bh3b
+ bhb
12
hs
hb
+ 2he + 2hk + hp +
2
2
!
3he 2
hs
bh3e
+ bhe
+ 2hk + hp +
12
2
2
!
bh3k
hs
3hk 2
+ bhk
+ he + hp +
12
2
2
!
bh3p
hp 2
hs
+ bhp
+ he + hk +
12
2
2
!
bh3k
hs
hk 2
+ bhk
+ he +
12
2
2
!
bh3e
hs he 2
+ bhe
+
12
2
2
bh3s
12
2 !
(B.10)
(B.11)
(B.12)
(B.13)
(B.14)
(B.15)
(B.16)
where Yb is the elastic modulus of the battery layer.
The total bending stiffness of the structure materials is then
Y Ic2 = Y Is + 2Y Iei + 2Y Iki + 2Y Iko + 2Y Ieo + 2Y Ib
(B.17)
The bending stiffness of the piezoelectric layers is once again
cE
11 Ip = 2Y Ip
(B.18)
The mass density of the second composite section is
ρAc2 = ρs bhs + 4ρe bhe + 4ρk bhk + 2ρp bhp + 2ρb bhb
171
(B.19)
B.3
Self-Charging Wing Spar Sections
Consider the self-charging structure wing spar configuration shown in Fig. 6.1(c) and repeated again in Fig. B.3. The neutral axis lies in the geometric center of the substrate layer
in both sections.
L1  L  L2
hp
hk
hs1
he
hp
Piezoceramic Layer
h
Thin-Film Battery Layer
b
Substrate Layer
Kapton Layer
L2  L  L3
hb
hki
hs 2
Epoxy Layer
Electrodes
h
Poling Direction
hb
b
Figure B.3: Cross-sectional views of the two composite sections of the self-charging wing
spar.
The bending stiffness and mass density of the uniform structure sections from 0 < x <
L1 and L3 < x < L are simply
Y Is = Ys
bh3
12
(B.20)
(B.21)
ρAs = ρs bh
The bending stiffness of the materials in the first composite section from 0 < x < L1 ,
are given using Eq. (B.1) as
Outer Kapton Layer
Y Iko = Yk
Piezoelectric Layer
Y Ip = Yp
Inner Kapton Layer
Y Iki = Yk
Epoxy Layer
Substrate Layer
Y Ie = Ye
Y Is1 = Ys
!
hs1
3hk 2
+ he + hp +
2
2
!
bh3p
hp 2
hs1
+ bhp
+ he + hk +
12
2
2
!
bh3k
hs1
hk 2
+ bhk
+ he +
12
2
2
!
bh3e
hs1 he 2
+ bhe
+
12
2
2
bh3k
+ bhk
12
bh3s1
12
172
(B.22)
(B.23)
(B.24)
(B.25)
(B.26)
The total bending stiffness is the sum of the bending stiffness of each material section. For
the structure materials in the first composite section, this gives
(B.27)
Y Ic1 = Y Is1 + 2Y Ie + 2Y Iki + 2Y Iko
The bending stiffness of the piezoelectric layers is simply
cE
11 Ip = 2Y Ip
(B.28)
The mass density of the first composite section is
(B.29)
ρAc1 = ρs bhs1 + 2ρe bhe + 4ρk bhk + 2ρp bhp
For the second composite section from L1 < x < L, the individual bending stiffness of
each material section is
bh3b
+ bhb
12
Battery Layer
Y Ib = Yb
Epoxy Layer
Y Ie = Ye
Substrate Layer
Y Is = Ys
hs2
hb
+ he +
2
2
!
bh3e
hs2 he 2
+ bhe
+
12
2
2
bh3s2
12
2 !
(B.30)
(B.31)
(B.32)
The total bending stiffness is then
Y Ic2 = Y Is2 + 2Y Ie + 2Y Ib
(B.33)
The mass density of the second composite section is
ρAc2 = ρs bhs2 + 2ρe bhe + 2ρb bhb
173
(B.34)
A PPENDIX
C
T RANSFORMATION FROM R ELATIVE TO
A BSOLUTE F RAME OF R EFERENCE FOR
F REQUENCY R ESPONSE F UNCTIONS
The relative displacement frequency response function given in Eq. (2.88) is valid for the
displacement of the beam relative to the moving base. It is common practice in laboratory experiments to measure the absolute displacement or velocity of the beam (where the
measurement system is stationary while the beam vibrates).
Recalling Eq. (2.2), the absolute displacement of the beam is the sum of the base displacement and the relative displacement given by
wabs (x, t) = wb (t) + wrel (x, t)
(C.1)
which leads to
wabs (x, t) = wb ejωt + ΦT (x) −ω 2 [M] + jω [C] + [K]
!−1
1 −1 eq eq T
eq
+jω jωCp +
Θ Θ
Fejωt
Rl
(C.2)
Considering this relationship, the relative displacement FRF given in Eq. (2.88) can be
modified to give the following expression for the absolute displacement FRF
wabs (x, t)
1
= − 2 + ΦT (x) −ω 2 [M] + jω [C] + [K]
ab ejωt
ω
!−1
1 −1 eq eq T
eq
+ jω jωCp +
Θ Θ
F̃
Rl
(C.3)
The absolute velocity can be found by simply taking the temporal derivative of the
174
absolute displacement as follows
Vabs (x, t) =
d
(wb (t) + wrel (t))
dt
(C.4)
which gives
Vabs (x, t) = jωwb ejωt + jωΦT (x) −ω 2 [M] + jω [C] + [K]
!−1
−1
1
Θeq Θeq T
+jω jωCpeq +
Fejωt
Rl
(C.5)
The absolute velocity FRF can then be written as
Vabs (L, t)
1
=
+ jωΦT (L) −ω 2 [M] + jω [C] + [K]
jωt
ab e
jω
!−1
−1
1
+ jω jωCpeq +
Θeq Θeq T
F̃
Rl
175
(C.6)
A PPENDIX
D
C HARGE /D ISCHARGE C URVES FOR
D YNAMIC FAILURE T ESTING
The following figures provide the charge/discharge curves measured for both power supply and piezoelectric charging of the battery layers in dynamic failure testing.
Voltage
Current (mA)
2
2
Voltage (V)
Current (mA)
3
Current
4.1
3.9
Voltage
6
4
3
4.1
2
3
Current
2
1
1
1
0
0
0
(c)
500
1000
Time (sec)
1500
2000
0
0
(d)
8
2
2
Current (mA)
4
Voltage (V)
3
1000
Time (sec)
1500
0
2000
4.1
3.9
Voltage
6
Current
500
3
4.1
Voltage
Current (mA)
Voltage (V)
(b)
8
2
3
Current
2
1
1
Voltage (V)
(a)
1
0
0
0
500
1000
1500
Time (sec)
2000
0
0
500
1000
Time (sec)
1500
0
2000
Figure D.1: Power supply charge and discharge curves for excitation at (a-b) ±0.2 g and
(c-d) ±0.4 g.
176
Voltage
2
2
Current (mA)
4
Voltage (V)
3
Current
4.1
3.9
Voltage
6
2
3
Current
2
1
1
1
0
0
0
(c)
500
1000
Time (sec)
1500
0
0
(d)
8
2
2
Current (mA)
4
Voltage (V)
Current (mA)
3
1500
2
3
Current
2
1
1
1
0
0
0
(e)
500
1000
Time (sec)
1500
0
0
(f)
8
Voltage
2
2
Current (mA)
4
1000
Time (sec)
1500
0
2000
4.1
3.9
Voltage
Voltage (V)
3
Current
500
3
4.1
6
Current (mA)
0
2000
4.1
3.9
Voltage
6
Current
1000
Time (sec)
3
4.1
Voltage
500
Voltage (V)
Current (mA)
3
4.1
Voltage (V)
(b)
8
2
3
Current
2
1
1
Voltage (V)
(a)
1
0
0
0
500
1000
Time (sec)
1500
0
0
500
1000
Time (sec)
1500
0
2000
Figure D.2: Power supply charge and discharge curves for excitation at (a-b) ±0.6 g, (c-d)
±0.8 g, and (e-f) ±1.0 g.
177
Voltage
Current (mA)
2
2
Voltage (V)
2
3
Current
2
1
1
1
0
0
0
(c)
500
1000
Time (sec)
1500
0
0
(d)
8
2
2
Current (mA)
4
Voltage (V)
Current (mA)
3
1500
2
3
Current
2
1
1
1
0
0
0
(e)
500
1000
Time (sec)
1500
0
0
(f)
8
Voltage
2
2
Current (mA)
4
1000
Time (sec)
1500
0
2000
4.1
3.9
Voltage
Voltage (V)
3
Current
500
3
4.1
6
Current (mA)
0
2000
4.1
3.9
Voltage
6
Current
1000
Time (sec)
3
4.1
Voltage
500
Voltage (V)
Current (mA)
3
Current
4.1
3.9
Voltage
6
4
3
4.1
Voltage (V)
(b)
8
2
3
Current
2
1
1
Voltage (V)
(a)
1
0
0
0
500
1000
Time (sec)
1500
0
2000
0
500
1000
Time (sec)
1500
0
2000
Figure D.3: Power supply charge and discharge curves for excitation at (a-b) ±1.5 g, (c-d)
±2.0 g, and (e-f) ±2.5 g.
178
Voltage
Current (mA)
2
2
Voltage (V)
2
3
Current
2
1
1
1
0
0
0
(c)
500
1000
Time (sec)
1500
0
0
(d)
8
2
2
Current (mA)
4
Voltage (V)
Current (mA)
3
1500
2
3
Current
2
1
1
1
0
0
0
(e)
500
1000
Time (sec)
1500
0
0
(f)
8
Voltage
2
2
Current (mA)
4
1000
Time (sec)
1500
0
2000
4.1
3.9
Voltage
Voltage (V)
3
Current
500
3
4.1
6
Current (mA)
0
2000
4.1
3.9
Voltage
6
Current
1000
Time (sec)
3
4.1
Voltage
500
Voltage (V)
Current (mA)
3
Current
4.1
3.9
Voltage
6
4
3
4.1
Voltage (V)
(b)
8
2
3
Current
2
1
1
Voltage (V)
(a)
1
0
0
0
500
1000
Time (sec)
1500
0
0
500
1000
Time (sec)
1500
0
2000
Figure D.4: Power supply charge and discharge curves for excitation at (a-b) ±3.0 g, (c-d)
±3.5 g, and (e-f) ±4.0 g.
179
Voltage
Current (mA)
2
2
Voltage (V)
2
3
Current
2
1
1
1
0
0
0
(c)
500
1000
Time (sec)
0
1500
0
(d)
8
2
2
Current (mA)
4
Voltage (V)
Current (mA)
3
1500
2
3
Current
2
1
1
1
0
0
0
(e)
500
1000
Time (sec)
0
1500
0
(f)
8
Voltage
2
2
Current (mA)
4
1000
Time (sec)
1500
0
2000
4.1
3.9
Voltage
Voltage (V)
3
Current
500
3
4.1
6
Current (mA)
0
2000
4.1
3.9
Voltage
6
Current
1000
Time (sec)
3
4.1
Voltage
500
Voltage (V)
Current (mA)
3
Current
4.1
3.9
Voltage
6
4
3
4.1
Voltage (V)
(b)
8
2
3
Current
2
1
1
Voltage (V)
(a)
1
0
0
0
500
1000
Time (sec)
1500
2000
0
0
500
1000
Time (sec)
1500
0
2000
Figure D.5: Power supply charge and discharge curves for excitation at (a-b) ±4.5 g, (c-d)
±5.0 g, and (e-f) ±5.5 g.
180
Voltage
Current (mA)
2
2
Voltage (V)
2
3
Current
2
1
1
1
0
0
0
(c)
500
1000
Time (sec)
1500
0
2000
0
(d)
8
2
2
Current (mA)
4
Voltage (V)
Current (mA)
3
1500
2
3
Current
2
1
1
1
0
0
0
1000
Time (sec)
1500
0
0
(f)
8
4
2
2
1000
Time (sec)
1500
0
2000
4.1
3.9
Voltage
Current (mA)
3
Current
500
3
4.1
Voltage
6
Current (mA)
500
Voltage (V)
(e)
0
2000
4.1
3.9
Voltage
6
Current
1000
Time (sec)
3
4.1
Voltage
500
Voltage (V)
Current (mA)
3
Current
4.1
3.9
Voltage
6
4
3
4.1
Voltage (V)
(b)
8
2
3
Current
2
1
1
Voltage (V)
(a)
1
0
0
0
500
1000
Time (sec)
1500
0
0
500
1000
Time (sec)
1500
0
2000
Figure D.6: Power supply charge and discharge curves for excitation at (a-b) ±6.0 g, (c-d)
±6.5 g, and (e-f) ±7.0 g.
181
3
4.1
3.9
4.1
Voltage
3
0.1
2
0.05
(c)
1000
2000
Time (sec)
3000
2
1
1
0
0
0
3
Current
1
Current
2
0
0
(d)
0.25
50
100
Time (sec)
150
3
4.1
3.9
4.1
Voltage
0.15
3
0.1
2
Current
0.05
Current (mA)
Voltage
Voltage (V)
Current (mA)
0.2
(e)
1000
2000
Time (sec)
3000
3
2
1
1
0
0
0
2
Current
1
0
0
(f)
0.25
50
100
Time (sec)
150
200
4.1
3.9
3
0.1
2
Current
0.05
2000
Time (sec)
3000
3
2
1
1
0
0
1000
2
Current
1
0
Current (mA)
Voltage
Voltage (V)
Current (mA)
Voltage
0.15
0
3
4.1
0.2
0
Voltage (V)
0.15
Current (mA)
Voltage
Voltage (V)
Current (mA)
0.2
Voltage (V)
(b)
0.25
Voltage (V)
(a)
0
0
50
100
Time (sec)
150
0
Figure D.7: Piezoelectric charge and discharge curves for excitation at (a-b) ±0.2 g, (c-d)
±0.4 g, and (e-f) ±0.6 g.
182
3
4.1
3.9
4.1
Voltage
Current
0.1
2
0.05
(c)
2000
Time (sec)
3000
2
1
1
0
0
1000
3
Current
1
0
2
0
0
(d)
0.25
50
100
150
Time (sec)
200
3
4.1
3.9
4.1
Voltage
3
0.15
Current
0.1
2
0.05
Current (mA)
Voltage
Voltage (V)
Current (mA)
0.2
(e)
2000
Time (sec)
3000
3
Current
2
1
0
0
1000
2
1
1
0
0
0
(f)
0.25
50
100
150
Time (sec)
200
250
4.1
3.9
3
Current
0.1
2
0.05
2000
Time (sec)
3000
3
Current
2
1
0
0
1000
2
1
1
0
Current (mA)
Voltage
Voltage (V)
Current (mA)
Voltage
0.15
0
3
4.1
0.2
0
Voltage (V)
3
0.15
Current (mA)
Voltage
Voltage (V)
Current (mA)
0.2
Voltage (V)
(b)
0.25
Voltage (V)
(a)
0
0
50
100
150
Time (sec)
200
250
0
Figure D.8: Piezoelectric charge and discharge curves for excitation at (a-b) ±0.8 g, (c-d)
±1.0 g, and (e-f) ±1.5 g.
183
3
4.1
3.9
4.1
Voltage
Current
0.1
2
0.05
(c)
2000
Time (sec)
3000
2
1
0
0
1000
3
Current
1
1
0
2
0
0
(d)
0.25
50
100
150 200
Time (sec)
250
300
3
4.1
3.9
4.1
Voltage
0.15
3
Current
0.1
2
0.05
Current (mA)
Voltage
Voltage (V)
Current (mA)
0.2
(e)
2000
Time (sec)
3000
3
Current
2
1
0
0
1000
2
1
1
0
0
0
(f)
0.25
50
100
150
200
Time (sec)
0
250
3
4.1
3.9
0.15
Current
Voltage
3
0.1
2
0.05
2000
Time (sec)
3000
3
Current
2
1
0
0
1000
2
1
1
0
Current (mA)
Voltage
Voltage (V)
Current (mA)
4.1
0.2
0
Voltage (V)
3
0.15
Current (mA)
Voltage
Voltage (V)
Current (mA)
0.2
Voltage (V)
(b)
0.25
Voltage (V)
(a)
0
0
50
100 150 200
Time (sec)
250
300
0
Figure D.9: Piezoelectric charge and discharge curves for excitation at (a-b) ±2.0 g, (c-d)
±2.5 g, and (e-f) ±3.0 g.
184
3
4.1
3.9
3
0.2
Current
2
0.1
0
(c)
1000
2000
Time (sec)
3000
3
Current
2
1
1
1
0
2
0
0
0
(d)
0.4
100
200
Time (sec)
300
3
4.1
3.9
Voltage
3
0.2
Current
2
0.1
0
1000
2000
Time (sec)
3000
2
3
Current
2
1
1
1
0
(e)
Current (mA)
Voltage
Voltage (V)
Current (mA)
4.1
0.3
0
0
0
(f)
0.4
100
200
Time (sec)
300
4.1
3.9
0.2
2
0.1
0
1000
2000
Time (sec)
3000
2
3
Current
2
1
1
1
0
Current (mA)
3
Voltage (V)
Current (mA)
Voltage
Voltage
Current
0
3
4.1
0.3
0
400
Voltage (V)
0.3
Voltage
Current (mA)
Voltage
Voltage (V)
Current (mA)
4.1
Voltage (V)
(b)
0.4
Voltage (V)
(a)
0
0
0
100
200
300
Time (sec)
400
0
500
Figure D.10: Piezoelectric charge and discharge curves for excitation at (a-b) ±3.5 g, (c-d)
±4.0 g, and (e-f) ±4.5 g.
185
3
4.1
3.9
3
0.2
Current
2
0.1
0
(c)
1000
2000
Time (sec)
3000
3
Current
2
1
1
1
0
2
0
0
0
(d)
0.4
100
200
300
Time (sec)
400
500
3
4.1
3.9
Voltage
3
0.2
Current
2
0.1
0
1000
2000
Time (sec)
3000
2
3
Current
2
1
1
1
0
(e)
Current (mA)
Voltage
Voltage (V)
Current (mA)
4.1
0.3
0
0
0
(f)
0.4
100
200
300
400
Time (sec)
500
4.1
3.9
Current
0.2
2
0.1
0
1000
2000
Time (sec)
3000
2
3
Current
2
1
1
1
0
Current (mA)
Voltage
Voltage (V)
Current (mA)
Voltage
3
0
600
3
4.1
0.3
0
Voltage (V)
0.3
Voltage
Current (mA)
Voltage
Voltage (V)
Current (mA)
4.1
Voltage (V)
(b)
0.4
Voltage (V)
(a)
0
0
0
100
200
300 400
Time (sec)
500
600
0
Figure D.11: Piezoelectric charge and discharge curves for excitation at (a-b) ±5.0 g, (c-d)
±5.5 g, and (e-f) ±6.0 g.
186
Current
0.2
2
0.1
Voltage (V)
3
Current (mA)
0.3
1000
2000
Time (sec)
3000
Voltage
3
Current
2
1
0
0
0
(d)
0.4
2
1
1
0
100
200
300 400
Time (sec)
500
3
Current
0.2
2
0.1
0
1000
2000
Time (sec)
3000
2
3
Current
2
1
1
1
0
0
4.1
3.9
Voltage
0.3
600
3
4.1
Current (mA)
Current (mA)
4.1
3.9
Voltage
0
(c)
3
4.1
Voltage (V)
Voltage
Voltage (V)
(b)
0.4
Voltage (V)
Current (mA)
(a)
0
0
0
200
400
Time (sec)
600
0
Figure D.12: Piezoelectric charge and discharge curves for excitation at (a-b) ±6.5 g and
(c-d) ±7.0 g.
187
A PPENDIX
E
L OAD -D EFLECTION C URVES FOR ALL
P IEZOELECTRIC S AMPLES T ESTED
The following figures provide the load-deflection curves measured for all piezoelectric
samples tested. The point at which the sample is considered to be failed is marked by a
red circle for each trace.
0.6
PZT-5A
0.5
Load (N)
0.4
0.3
0.2
0.1
0
-0.1
0
0.2
0.4
0.6
0.8
1
1.2
Crosshead Displacement (mm)
1.4
1.6
Figure E.1: Load-deflection curves for PZT-5A samples.
188
(a)
0.6
PZT-5H
0.5
Load (N)
0.4
0.3
0.2
0.1
0
-0.1
0
(b)
0.2
0.4
0.6
0.8
1
Crosshead Displacement (mm)
1.2
1.4
3
PZT-4
2.5
Load (N)
2
1.5
1
0.5
0
0
(c)
0.05
0.1
0.15
Crosshead Displacement (mm)
0.2
10
PZT-8
Load (N)
8
6
4
2
0
0
0.02
0.04 0.06 0.08 0.1 0.12 0.14
Crosshead Displacement (mm)
0.16
0.18
Figure E.2: Load-deflection curves for (a) PZT-5H, (b) PZT-4, and (c) PZT-8 samples.
189
(a)
0.35
PMN-PT
0.3
Load (N)
0.25
0.2
0.15
0.1
0.05
0
0
(b)
0.2
0.4
0.6
0.8
1
1.2
1.4
Crosshead Displacement (mm)
1.6
1.8
0.2
PMN-PZT
Load (N)
0.15
0.1
0.05
0
0
0.5
1
1.5
Crosshead Displacement (mm)
2
(c)
0.8
QP10n
Load (N)
0.6
0.4
0.2
0
0
1
2
3
4
5
6
Crosshead Displacement (mm)
7
8
Figure E.3: Load-deflection curves for (a) PMN-PT, (b) PMN-PZT, and (c) QP10n samples.
190