Electromechanical Modeling of Piezoelectric Energy Harvesters

319 Pages·2009·8.87 MB·English

Electromechanical Modeling of Piezoelectric Energy Harvesters

Alper Erturk

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
Engineering Mechanics

Daniel J. Inman, Chair
Scott L. Hendricks
Michael W. Hyer
Ishwar K. Puri
Liviu Librescu (deceased)

November 20, 2009
Blacksburg, VA

Keywords: Piezoelectricity, vibration energy harvesting, structural dynamics,
electromechanical modeling

Copyright © 2009 Alper Erturk

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Electromechanical Modeling of Piezoelectric Energy Harvesters

Alper Erturk

Abstract

Vibration-based energy harvesting has been investigated by several researchers over the last
decade. The ultimate goal in this research field is to power small electronic components
(such as wireless sensors) by using the vibration energy available in their environment.
Among the basic transduction mechanisms that can be used for vibration-to-electricity
conversion, piezoelectric transduction has received the most attention in the literature.
Piezoelectric materials are preferred in energy harvesting due to their large power densities
and ease of application. Typically, piezoelectric energy harvesters are cantilevered structures
with piezoceramic layers that generate alternating voltage output due to base excitation. This
work presents distributed-parameter electromechanical models that can accurately predict the
coupled dynamics of piezoelectric energy harvesters. First the issues in the existing models
are addressed and the lumped-parameter electromechanical formulation is corrected by
introducing a dimensionless correction factor derived from the electromechanically
uncoupled distributed-parameter solution. Then the electromechanically coupled closed-form
analytical solution is obtained based on the thin-beam theory since piezoelectric energy
harvesters are typically thin structures. The multi-mode electromechanical frequency
response expressions obtained from the analytical solution are reduced to single-mode
expressions for modal vibrations. The analytical solutions for the electromechanically
coupled voltage response and vibration response are validated experimentally for various
cases. The single-mode analytical equations are then used for deriving closed-form relations
for parameter identification and optimization. Asymptotic analyses of the electromechanical
frequency response functions are given along with expressions for the short-circuit and the
open-circuit resonance frequencies. A simple experimental technique is presented to identify
the optimum load resistance using only a single resistor and an open-circuit voltage
measurement. A case study is given to compare the power generation performances of
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commonly used monolithic piezoceramics and novel single crystals with a focus on the
effects of plane-stress material constants and mechanical damping. The effects of strain
nodes and electrode configuration on piezoelectric energy harvesting are discussed
theoretically and demonstrated experimentally. An approximate electromechanical solution
using the assumed-modes method is presented and it can be used for modeling of asymmetric
and moderately thick energy harvester configurations. Finally, a piezo-magneto-elastic
energy harvester is introduced as a non-conventional broadband energy harvester.

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Acknowledgements

First and foremost, I would like to extend my deepest gratitude to Prof. Daniel J. Inman for he
has been more than an academic advisor over the last three years. Prof. Inman has been a great
advisor who was always available to discuss and support the technical problems came to my
mind. He did make me feel like his colleague, more than a graduate student, throughout my
entire PhD study. He provides a very pleasant research environment in the lab and he really
knows how to communicate with his students. He has been a great mentor who was available to
discuss and advise on non-technical problems of life as well. Prof. Inman has given me the
opportunity to work on several research problems other than the subject of this dissertation and
he has offered the freedom to start developing my own academic style. He has given me the
opportunity to co-teach classes with him, to attend conferences several times and to take
responsibilities in the research community. I cannot think of a more fruitful and joyful post-
masters research than my last three years under the guidance of Prof. Inman at the Center for
Intelligent Material Systems and Structures (CIMSS).
Ms. Beth Howell, the program manager at CIMSS, has been very helpful since the first
day I started working at CIMSS. The friendly office environment and numerous beautiful aspects
of CIMSS have a lot to do with her presence and energy. She keeps so many things running
simultaneously with an amazing performance. I am indebted to Beth for several things she has
helped with.
The experience I have gained as a graduate student in the Department of Engineering
Science and Mechanics (ESM) is invaluable. I know that I am only at the beginning of my
academic career but my appreciation of Engineering Mechanics as a part of my life has been
enhanced dramatically after attending several classes by the pioneers of this field, such as Prof.
Liviu Librescu, Prof. Michael W. Hyer, Prof. Dean T. Mook, Prof. Raymond H. Plaut and Prof.
Ali H. Nayfeh, among others. I have been very proud of being an ESM student during my entire
graduate study.
I would like to thank Prof. Scott L. Hendricks, Prof. Michael W. Hyer, and Prof. Ishwar
K. Puri for serving on my PhD committee.
I would like to extend my appreciation to my Department, particularly the Graduate
Committee, once again, for honoring me with the “Liviu Librescu Memorial Scholarship.”
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Words are not enough to express the value of being the first recipient of a scholarship named in
memory of the late member of my PhD committee and my teacher, Prof. Librescu. I will have to
work very hard to honor Prof. Librescu’s memory…
I truly enjoyed sharing the same office and the lab with several colleagues and friends at
CIMSS over the last three years and I have had research collaborations with some of them (on
topics not discussed in this dissertation). I owe thanks to Dr. Pablo Tarazaga (presently with the
University of Bristol, England) for introducing me the lab equipment when I was a junior PhD
student. It was in those days when he helped me with conducting the experiments of the strain
nodes demonstration together with Justin Farmer. Justin was also a great colleague during the
short period of time we were at CIMSS together. Dr. Jamil Renno (presently with the University
of Southampton, England) was an outstanding colleague and a wonderful desk neighbor when
we shared the same office before his graduation. I could not count how many nights we spent in
lab Steve Anton while developing the concept of self-charging structures (a novel energy
harvesting concept that is not discussed in this dissertation). Steve has been a fantastic colleague
to collaborate with. It was also a great pleasure to work with Na Kong (a PhD student in the
VLSI for Telecommunications Group) and observe how a skillful electrical engineer can
improve that electrical domain (that we mechanical engineers often simplify) with efficient
regulator circuits. If one is interested in how brittle single crystals can be, he/she should find me
or Onur Bilgen. We have had long nights of research collaboration with Onur as well on macro-
fiber composites and single crystals. The gentlemen presently sitting close to me in the office,
Austin Creasy and Kahlil Detrich, are the best office neighbors one can find. I really enjoyed the
break chats with Austin every time and I am very glad that the tower of books and papers on my
desk has not yet collapsed onto Kahlil’s desk. I would like to thank also Andy Sarles, Andy
Duncan, Woon Kim, Zach Hills, Mana Afshari, Amin Karami, Brad Butrym and all the members
of the CIMSS community I could not mention here, for making CIMSS what it has been.
I met another great colleague during his visit to our lab for about a year: Prof. Carlos De
Marqui Junior, an aeroelasticity professor of the University of São Paulo in São Carlos, Brazil. I
have enjoyed collaborating with him on piezo-aero-elasticity (not covered here) during my PhD
study as much as I enjoyed the caipirinha when I visited his university to give a lecture last
summer.
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I would like to thank Julien Hoffmann, a student (presently an engineer) I advised during
his visit as a senior engineering student from the Institut Catholique d’Arts et Métiers in Lille,
France, for his help in planning and rapidly completing the manufacturing of the piezo-magneto-
elastic device with the ESM Machine Shop (which reminds me that I owe thanks to Mr. David
Simmons as well). If Julien had not undertaken the manufacturing process of the piezo-magneto-
elastic device, the idea of using the magneto-elastic structure for energy harvesting (dates back to
the summer of 2008 when I had taken Prof. Francis Moon’s book from Prof. Inman’s library at a
party) was going wait for at least one more year!
I owe special thanks to my parents, Selma and Hikmet, for their patience and support
during my PhD study. It was wonderful to visit them in my hometown (Eskisehir, Turkey) every
July to regain the energy I needed. This work is dedicated to them.
This work has been supported by the U.S. Air Force Office of Scientific Research MURI
under grant number F 9550-06-1-0326 “Energy Harvesting and Storage Systems for Future Air
Force Vehicles” monitored by Dr B.L. Lee and recently by the U.S. Department of Commerce,
National Institute of Standards and Technology, Technology Innovation Program, under the
Cooperative Agreement Number 70NANB9H9007: “Self-Powered Wireless Sensor Network for
Structural Health Prognosis.”

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“These oscillations arise freely, and I have determined various
conditions, and have performed a great many beautiful
experiments on the position of the knot points and the pitch
of the tone, which agree beautifully with the theory.” [1]
– Daniel Bernoulli (from a letter to Leonhard Euler)
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Table of Contents

Abstract ................................................................................................................................ ii
Acknowledgements .............................................................................................................. iv
Table of Contents .............................................................................................................. viii
List of Figures ................................................................................................................... xvi
List of Tables .................................................................................................................. xxvii
1 Introduction ....................................................................................................................... 1
1.1 Vibration Energy Harvesting Using Piezoelectric Transduction ...................................... 1
1.2 Review of the Existing Piezoelectric Energy Harvester Models ...................................... 4
1.3 Objectives of the Dissertation .......................................................................................... 6
1.4 Layout of the Dissertation ................................................................................................ 8
2 Correction of the Existing Lumped-parameter Piezoelectric Energy Harvester
Model .................................................................................................................................. 11
2.1 Base Excitation Problem for the Transverse Vibrations of a Cantilevered Beam ................. 12
2.1.1 Response to General Base Excitation ..................................................................... 12
2.1.2 Steady-state Response to Harmonic Base Excitation ............................................. 17
2.1.3 Lumped-parameter Model of the Harmonic Base Excitation Problem .................. 18
2.1.4 Comparison of the Distributed-parameter and the Lumped-parameter Model
Predictions........................................................................................................................ 21
2.2 Correction of the Lumped-parameter Model for Transverse Vibrations ............................... 24
2.2.1 Correction Factor for the Lumped-parameter Model .............................................. 24
2.2.2 Effect of a Tip Mass on the Correction Factor ....................................................... 26
2.3 Experimental Case Studies for Validation of the Correction Factor ..................................... 29
2.3.1 Cantilevered Beam without a Tip Mass under Base Excitation ............................. 29
2.3.2 Cantilevered Beam with a Tip Mass under Base Excitation ................................... 33
2.4 Base Excitation Problem for Longitudinal Vibrations and Correction of Its Lumped-
parameter Model .......................................................................................................................... 35
2.4.1 Analytical Modal Analysis and Steady-state Response to Harmonic Base
Excitation ......................................................................................................................... 35
2.4.2 Correction Factor for Longitudinal Vibrations ....................................................... 37
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2.5 Correction Factor in the Electromechanically Coupled Lumped-parameter Equations and a
Theoretical Case Study ................................................................................................................ 39
2.5.1 An Electromechanically Coupled Lumped-parameter Model for Piezoelectric
Energy Harvesting ........................................................................................................... 39
2.5.2 Correction Factor in the Electromechanically Coupled Lumped-parameter Model
and a Theoretical Case Study ........................................................................................... 40
2.6 Summary and Conclusions .................................................................................................... 42
3 Distributed-parameter Electromechanical Modeling of Bimorph Piezoelectric Energy
Harvesters – Analytical Solutions ..................................................................................... 43
3.1 Fundamentals of the Electromechanically Coupled Distributed-parameter Model ............... 44
3.1.1 Modeling Assumptions and Bimorph Configurations ........................................... 44
3.1.2 Coupled Mechanical Equation and Modal Analysis of Bimorph Cantilevers ........ 46
3.1.3 Coupled Electrical Circuit Equation of a Thin Piezoceramic Layer under Dynamic
Bending ........................................................................................................................... 52
3.2 Series Connection of the Piezoceramic Layers ...................................................................... 55
3.2.1 Coupled Beam Equation in Modal Coordinates ..................................................... 56
3.2.2 Coupled Electrical Circuit Equation ....................................................................... 56
3.2.3 Closed-form Voltage Response and Vibration Response at Steady State .............. 57
3.3. Parallel Connection of the Piezoceramic Layers .................................................................. 59
3.3.1 Coupled Beam Equation in Modal Coordinates ..................................................... 59
3.3.2 Coupled Electrical Circuit Equation ....................................................................... 60
3.3.3 Closed-form Voltage Response and Vibration Response at Steady State .............. 60
3.4 Single-mode Electromechanical Expressions for Modal Excitations .................................... 62
3.4.1 Series Connection of the Piezoceramic Layers ....................................................... 62
3.4.2 Parallel Connection of the Piezoceramic Layers .................................................... 63
3.5 Multi-mode and Single-mode Electromechanical FRFs ........................................................ 63
3.5.1 Multi-mode Electromechanical FRFs ..................................................................... 64
3.5.1.1 Series Connection of the Piezoceramic Layers ........................................ 64
3.5.1.2 Parallel Connection of the Piezoceramic Layers ..................................... 65
3.5.2 Single-mode Electromechanical FRFs .................................................................... 66
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3.5.2.1 Series Connection of the Piezoceramic Layers ....................................... 66
3.5.2.2 Parallel Connection of the Piezoceramic Layers ..................................... 67
3.6 Equivalent Representation of the Series and Parallel Connection Expressions .................... 67
3.6.1 Modal Electromechanical Coupling Terms ............................................................ 68
3.6.2 Equivalent Capacitance for Series and Parallel Connections ................................. 68
3.6.3 Equivalent Representation of the Electromechanical Expressions ......................... 69
3.6.4 Equivalent Representation of the Multi-mode Electromechanical FRFs ............... 70
3.6.5 Equivalent Representation of the Single-mode Electromechanical FRFs .............. 71
3.7 Theoretical Case Study .......................................................................................................... 72
3.7.1 Properties of the Bimorph Cantilever .................................................................... 72
3.7.2 Frequency Response of the Voltage Output .......................................................... 74
3.7.3 Frequency Response of the Current Output ............................................................ 78
3.7.4 Frequency Response of the Power Output .............................................................. 80
3.7.5 Frequency Response of the Relative Tip Displacement ......................................... 84
3.7.6 Parallel Connection of the Piezoceramic Layers .................................................... 88
3.7.7 Single-mode FRFs .................................................................................................. 91
3.8 Summary and Conclusions .................................................................................................... 95
4 Experimental Validations of the Analytical Solutions for Cantilevered Bimorph
Piezoelectric Energy Harvesters ...................................................................................... 97
4.1 PZT-5H Bimorph Cantilever without a Tip Mass ................................................................. 98
4.1.1 Experimental Setup ................................................................................................. 98
4.1.2 Validation of the Electromechanical FRFs for a Set of Resistors ........................ 105
4.1.3 Electrical Performance Diagrams at the Fundamental Short-Circuit and Open-
Circuit Resonance Frequencies ...................................................................................... 111
4.1.4 Vibration Response Diagrams at the Fundamental Short-Circuit and Open-Circuit
Resonance Frequencies .................................................................................................. 113
4.2 PZT-5H Bimorph Cantilever with a Tip Mass .................................................................... 114
4.2.1 Experimental Setup ............................................................................................... 114
4.2.2 Validation of the Electromechanical FRFs for a Set of Resistors ........................ 117
4.2.3 Electrical Performance Diagrams at the Fundamental Short-Circuit and Open-
Circuit Resonance Frequencies ...................................................................................... 122
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