simultaneous energy harvesting and vibration control via ... · simultaneous energy harvesting and...

Simultaneous Energy Harvesting and Vibration

Control via Piezoelectric Materials

Ya Wang

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

Dwight Viehland

Mary Kasarda

Shashank Priya

Alexander Leonessa

January 27, 2012

Blacksburg, VA

Keywords: Energy Harvesting, Vibration Control, Multi-functionalities, Composite

Structures, Piezoelectric Materials, Gust Alleviation

Copyright © 2012 Ya Wang

Simultaneous Energy Harvesting and Vibration

Control via Piezoelectric Materials

Ya Wang

ABSTRACT

This work examines a novel concept and design of simultaneous energy harvesting and

vibration control on the same host structure. The motivating application is a

multifunctional composite sandwich wing spar for a small Unmanned Aerial Vehicle

(UAV) with the goal of providing self-contained gust alleviation. The basic idea is that

the wing itself is able to harvest energy from the ambient vibrations along with available

sunlight during normal flight. If the wing experiences any strong wind gust, it will sense

the increased vibration levels and provide vibration control to maintain its stability. This

work holds promise for improving performance of small UAVs in wind gusts.

The proposed multifunctional wing spar integrates a flexible solar cell array, flexible

piezoelectric wafers, a thin film battery and an electronic module into a composite

sandwich structure. The basic design factors are discussed for a beam-like

multifunctional wing spar with energy harvesting, strain sensing and self-controlling

functions. The investigated design factors for optimal power generation include the

configuration, location and actuation type of each piezoelectric transducer. The

equivalent electromechanical representations of a multifunctional wing spar is derived

theoretically, simulated numerically and validated experimentally.

Special attention is given to the development of a reduced energy control (REC) law,

aiming to minimize the actuation energy and the dissipated heat. The REC law integrates

a nonlinear switching algorithm with a positive strain feedback controller, and is

iii

represented by a positive feedback operation amplifier (op-amp) and a voltage buffer op-

amp for each mode. Experimental results exhibit that the use of nonlinear REC law

requires 67.3 % less power than a conventional nonlinear controller to have the same

settling time under free vibrations.

Nonlinearity in the electromechanical coupling coefficient of the piezoelectric transducer

is also observed, arising from the piezoelectric hysteresis in the constitutive equations

coupling the strain field and the electric field. If a constant and voltage-independent

electromechanical coupling coefficient is assumed, this nonlinearity results in

considerable discrepancies between experimental measurements and simulation results.

The voltage-dependent coupling coefficient function is identified experimentally, and a

real time adaptive control algorithm is developed to account for the nonlinear coupling

behavior, allowing for more accurate numerical simulations.

Experimental validations build upon recent advances in harvester, sensor and actuator

technology that have resulted in thin, light-weight multilayered composite sandwich wing

spars. These multifunctional wing spars are designed and validated to able to alleviate

wind gust of small UAVs using the harvested energy. Experimental results are presented

for cantilever wing spars with micro-fiber composite transducers controlled by reduced

energy controllers with a focus on two vibration modes. This work demonstrates the use

of reduced energy control laws for solving gust alleviation problems in small UAVs,

provides the experimental verification details, and focuses on applications to autonomous

light-weight aerospace systems.

iv

DEDICATION

To the memory of my grandma Ms. Ren

To my parents, Qiudong Wang and Yuexia Shao,

for their unconditional love, understanding and moral support,

To my brother Wengang Wang and my sister Xi Wang,

for the fond memories of childhood.

v

ACKNOWLEDGEMENTS

First and foremost, I would like to express my heartfelt appreciation and gratitude to my

advisor Professor Daniel J. Inman, the director of the Center for Intelligent Material

Systems and Structures (CIMSS) for providing me with endless encouragement and

guide me through multitude of challenges. I still remember my first meeting with

Professor Inman in his office, when I have no clue where my PhD was heading. I still

remember his encouraging words during the first meeting, when I have no idea whether it

was possible to finish my PhD given my circumstances. Things were not looking so

bright at that time and Prof. Inman gave me hope and made me realized that with right

direction, hard work and persistence, pursuing my dream was not an impossible task.

Over these years, Professor Inman has shown be the excellent example he has provided

as a successful scientist, researcher, teacher and advisor, both consciously and

unconsciously. The enthusiasm, curiosity, sense of humor, joy and effort he has for his

research was very impressive, motivational, infusive and contagious. He was always full

of ideas and open for new ideas. He never doubted my capabilities, always preserved his

confidence in me and guided me through tough times. Even though he is one of the

busiest men on earth, he was always available, always listened and always cared.

Professor Inman was also incredibly patient and never ever got angry or upset over my

missteps. Instead, he spent countless hours proofreading my research papers, discussing

research problems, seeking new research topics. These are all one student can expect and

ask from an advisor. There is no way to pay back for his admirable, inspirational and

immeasurable dedication, but at least I can promise that I will keep this spirit and mentor

to my own students in the future.

I would also like to extend my great acknowledgement to Dr. Kasarda, Prof. Viehland,

Prof. Priya and Dr. Leonessa for serving on my Ph.D committee. It has been my great

pleasure to have the support of such knowledgeable professors. Of course, my learning

experience would not have been this joyful without their wise advices and unquestionable

expertise.

vi

I would like to express my immense appreciation to Mechanical Engineering Department

at Virginia Tech, particularly the CIMSS. Sincere thanks to Ms. Margaret E. Howell

(Beth), the program manager of CIMSS, for creating such a friendly office and research

environment, for being supportive for any project and administrative related problems. I

feel so grateful to have so many amazing colleagues in CIMSS for sharing their ideas and

expertise in theoretical, numerical and excremental subjects. They are willing to provide

help in any way possible and they have contributed immensely to my personal and

professional time at Virginia Tech. There are too many to list here completely. First,

thank Dr. Onur Bilgen for his invaluable help and sincere assistance with experimental

setup. Thank Justin Farmer for helping me around the lab with introducing our lab

equipment. Thank Dr. Andy Sarles for his help with many dSPACE related problems.

Thank Dr. Steve Anton for introducing his self-charging structure and providing me a

solid research platform. Thank Dr. Ha Dong, Dr. Na Kong and John Tuner for their help

with printed circuit board design. Thank Dr. Amin, Karami for the discussion in all

academic issues. I would like to thank Mona Afshari and Jacob Dodson for being

amazing office mates. I also enjoyed sharing the same lab with my colleagues Jared

Hobeck and Alexander Pankonien during my visit in Aerospace Engineering at The

University of Michigan.

I am so in debts to my late grandma and my parents for their sacrifice, unreserved support

and belief in me. They are amazing parents who always putting their children first

without question. I also owe thanks to my amazing brother and sister for being a strong

support for the whole family when I am away from home. I am deeply blessed to have

them as my family. I also feel extremely lucky to have some of the most interesting

Chinese students at Virginia Tech and The University of Michigan. I appreciate for their

sincere friendship: Dr. Weihua Su, Jie Wang, Qingzhao Wang and many others.

I gratefully acknowledge the funding source that made my Ph.D. work possible. I was

funded by the U.S. Air Force Office of Scientific Research under the grant F9550-09-1-

0625 ‘Simultaneous Vibration Suppression and Energy Harvesting’ monitored by Dr B.L.

Lee.

vii

TABLE OF CONTENTS

ABSTRACT ........................................................................................................................ ii

DEDICATION ................................................................................................................... iv

ACKNOWLEDGEMENTS ................................................................................................ v

TABLE OF CONTENTS .................................................................................................. vii

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

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

CHAPTER 1 INTRODUCTION ........................................................................................ 1

Objective of the Dissertation .......................................................................................... 1

Layout of the Dissertation ............................................................................................... 3

CHAPTER 2 LITERATURE REVIEW ............................................................................. 7

Nomenclature .................................................................................................................. 8

Introduction ..................................................................................................................... 9

Review of Existing Modeling of Vibration-based Cantilever Piezoelectric Energy

Harvesters ..................................................................................................................... 12

Equivalent Electromechanical Circuit for Vibration-based Cantilever Piezoelectric

Harvesters ................................................................................................................. 13

Dynamic Modeling for Vibration-based Cantilever Piezoelectric Harvester ........... 13

Power Conditioning Circuitry and Power Optimization: ......................................... 16

Damping Effect due to Energy Dissipation Resulting from Energy Harvesting ...... 19

The State of Art of Vibration Control Laws via Piezoceramics ................................... 20

Purely Passive Shunt Damping ................................................................................. 21

Semi-passive Shunt Damping and the Switching Technology ................................. 22

Semi-active Control .................................................................................................. 29

Active Control ........................................................................................................... 32

Chapter Summary ......................................................................................................... 33

CHAPTER 3 REDUCED ENERGY CONTROL LAW .................................................. 35

Nomenclature ................................................................................................................ 36

Introduction ................................................................................................................... 37

Conventional Active Control Systems .......................................................................... 39

viii

PPF Control ............................................................................................................... 40

PID Control ............................................................................................................... 42

Nonlinear Control ..................................................................................................... 43

LQR Control ............................................................................................................. 44

Hybrid Bang-bang Control Systems ............................................................................. 45

Experimental Results ................................................................................................ 47

Numerical Simulations.................................................................................................. 53

Simulations with Voltage-independent Electromechanical Coupling ...................... 53

State Variable Simulation with Voltage-dependent Electromechanical Coupling ... 55

Chapter Summary ......................................................................................................... 60

CHAPTER 4 SIMULTANESOU ENERGY HARVESTING AND GUST

ALLEVIATION FOR A MULTIFUNCTIONAL WING SPAR ..................................... 61

Nomenclature ................................................................................................................ 62

Introduction ................................................................................................................... 63

Electromechanical Cantilever Beam Model of A Multifunctional Wing Spar using

Assumed Modes ............................................................................................................ 66

Electromechanical Energy Components Using Distributed-Parameter Method ...... 67

Solving Electromechanical Euler-Lagrange Equations for Piezoelectric Harvesters 69

Design Considerations for a Multifunctional Composite Wing Spar ....................... 73

Simultaneous Energy Harvesting and Gust Alleviation using REC ............................. 78

Equivalent Circuit Representation of a Piezoelectric Generator .............................. 78

Generation of Normal Wing Vibration and Wind Gust Signals ............................... 79

Gust Alleviation Using REC Control Laws: ............................................................. 81

Power Flow for Simultaneous Energy Harvesting and Gust Alleviation ................. 83

Chapter Summary ......................................................................................................... 87

CHAPTER 5 EXPERIMENTAL VALIDATION OF AN AUTONOMOUS GUST

ALLEVIATION SYSTEM ............................................................................................... 88

Nomenclature ................................................................................................................ 88

Introduction ................................................................................................................... 89

Experimental Validation of Reduced Energy Control on a Piezoelectric Layer Bonded

Aluminum Wing Spar ................................................................................................... 90

ix

Experimental Setup for Validation of Reduced Energy Control Law ...................... 90

Experimental Results ................................................................................................ 92

Experimental Characterization and Validation of an Autonomous Gust Alleviation

System on a Honeycomb Core Fiberglass Composite Sandwich Wing Spar ............... 98

Experimental Setup for Harvesting Ability Characterizations of Monolithic QP10n

and Micro Fiber Composite MFC 85281P1.............................................................. 99

Experimental Characterizations of Harvesting Abilities for Monolithic QP10n

Transducer............................................................................................................... 103

Experimental Characterizations of Harvesting Abilities for Micro Fiber Composite

MFC 8528 P1 .......................................................................................................... 106

Experimental Validations of the Autonomous Gust Alleviation System on the

Fiberglass Composite Multifunctional Wing Spar ................................................. 109

Chapter Summary ....................................................................................................... 112

CHAPTER 6 DISSERTATION SUMMARY ................................................................ 113

APPENDICES ................................................................................................................ 115

Appendix A Piezoelectric Constitutive Equations ...................................................... 115

Standard 3D Form of Constitutive Equations ......................................................... 115

Reduced Equations for 3-1 Actuation Modes ..................................................... 115

Reduced Equations for 3-3 Actuation Modes ..................................................... 116

Appendix B Euler-Lagrange Equations using Extended Hamilton’s Principle .......... 118

Appendix C Cross-section Transformation ................................................................ 119

Appendix D Energy Formulations of Electromechanical Cantilever Beam using

Distributed Parameter Method .................................................................................... 120

Appendix E Dryden Power Spectral Density Spectrum ............................................. 123

BIBLIOGRAPHY ........................................................................................................... 124

x

LIST OF FIGURES

Figure 1.1 Energy flow for simultaneous energy harvesting and vibration control. .......... 1

Figure 2.1 (a) A prototype of multifunctional structure with simultaneous energy

harvesting and vibration control abilities (b) Its schematic representation (c) Its feedback

control block. .................................................................................................................... 11

Figure 2.2 Equivalent circuit representation of the vibration-based piezoelectric harvester.

........................................................................................................................................... 13

Figure 2.3 A schematic diagram of (a) lumped-parameter (b) distributed-parameter

model................................................................................................................................. 15

Figure 2.4 (a) Standard energy harvesting (SEH) (b) Synchronous charge extraction

(SCE) (c) Synchronized switching harvesting on inductor (SSHI) .................................. 16

Figure 2.5 Purely passive shunted system using PZT-based transducers. ........................ 22

Figure 2.6 Schematic diagram of (a) state switch (b) SSDS (c) SSDI and (d) SSDV. ..... 24

Figure 2.7 (a) SSDS in Richard et al. (1999b) (b) SSDI in Richard et al. (2000). ........... 26

Figure 3.1 Block diagrams of the (a) conventional and (b) hybrid control system. ......... 46

Figure 3.2 (a) Picture and (b) schematic diagram of experimental setup. ....................... 47

Figure 3.3 Tip displacement measurements of the (a) Open-loop (b) PPF, Bang-bang-PPF

(c) PID, Bang-bang-PID (d) Nonlinear, Bang-bang-nonlinear (e) LQR, Bang-bang-LQR

control systems with identical initial conditions. .............................................................. 49

Figure 3.4 Experimental actuation voltage histories for the (a) PPF, Bang-bang-PPF (b)

PID, Bang-bang-PID (c) Nonlinear, Bang-bang-nonlinear (d) LQR, Bang-bang-LQR


Figure 3.5 Experimental actuation current histories for the (a) PPF, Bang-bang-PPF (b)

PID, Bang-bang-PID (c) Nonlinear, Bang-bang-nonlinear (d) LQR and Bang-bang-LQR


Figure 3.6 Experimental instantaneous power consumption for the (a) PPF, Bang-bang-

PPF (b) PID, Bang-bang-PID (c) Nonlinear, Bang-bang-nonlinear (d) LQR and Bang-

bang-LQR control systems with identical initial conditions............................................. 52

xi

Figure 3.7 Numerical and experimental comparisons of tip displacement, control voltage

and control current of (a) PPF and (b) Bang-bang-PPF control systems. ......................... 54

Figure 3.8 Numerical and experimental comparisons of tip displacement and control

voltage of (a) PID (b) Bang-bang-PID (c) Nonlinear (d) Bang-bang-nonlinear (e) LQR

and (f) Bang-bang-LQR control systems. ......................................................................... 55

Figure 3.9 Variation of the electromechanical coupling coefficient (feedback constant)

with actuation voltage. ...................................................................................................... 57

Figure 3.10 Block diagram of the state variable or adaptive control system. ................... 57

Figure 3.11 State variable numerical and experimental comparisons of tip displacement,

control voltage and control current of the (a) PPF (b) Bang-bang-PPF control systems. 58

Figure 3.12 State variable numerical and experimental comparisons of tip displacement

response and control voltage of the (a) PID (b) Bang-bang-PID (c) nonlinear (d) Bang-

bang-nonlinear (e) LQR and (f)Bang-bang-LQR control systems. .................................. 59

Figure 4.1 Multifunctional wing spar design showing various functionalities including

self-sensing, self-harvesting, self-storage and self-control. .............................................. 65

Figure 4.2 A composite spar for a small remote control aircraft( Anton et al. (2010)). ... 66

Figure 4.3 Relative tip frequency response function using both analytical and FEM

modeling. .......................................................................................................................... 73

Figure 4.4 Output power versus load resistance at mode 1 of 29Hz and mode 2 of 107 Hz.

........................................................................................................................................... 75

Figure 4.5 Output power of MFC 8528 P1versus distance from clamped end at mode 1 of

29 Hz and mode 2 of 107Hz. ............................................................................................ 76

Figure 4.6 Output power of QP10n versus distance from clamped end at mode 1 of 29 Hz

and mode 2 of 107Hz. ....................................................................................................... 77

Figure 4.7 The equivalent circuit for 1st mode piezoelectric generator with resistive

impedance. ........................................................................................................................ 78

Figure 4.8 The output voltage to base acceleration FRF for an 112Kohm Load

Resistance. ........................................................................................................................ 79

Figure 4.9 The harvested power spectrum for a 0.1 M ohm load resistance. ................... 79

xii

Figure 4.10 Block diagram of wind gust signal generation for open-loop and close-loop

tip displacement responses. ............................................................................................... 80

Figure 4.11Ambient wing vibration and wind gust acting on multifunctional wing spar

base, U0=15m/s, Lv =350m. .............................................................................................. 81

Figure 4.12 Schematic representations of gust alleviation using harvested energy. ........ 82

Figure 4.13The disturbed tip displacement spectrum of multifunctional wing spar before

and after REC control. ...................................................................................................... 83

Figure 4.14Winds disturbed multifunctional wing spar tip response in time domain before

and after REC control. ...................................................................................................... 83

Figure 4.15 Block Diagram of the 1st Mode PSF Control. ............................................... 84

Figure 4.16 Active and reactive power spectrum of 1st Mode and 2

nd Mode PSF and

buffer Op-amps. ................................................................................................................ 85

Figure 4.17Active and reactive power associated with the summing Op-amp and the

MFC 8528 P1actuator. ...................................................................................................... 86

Figure 5.1 (a) A photographic (b) A schematic representation of front view and back

view of the aluminum baseline multifunctional wing spar. .............................................. 91

Figure 5.2 Gust alleviation experimental setup using REC Laws. ................................... 92

Figure 5.3 Control performance of the PSF controllers for different control gains

(damping ratio of mode 1: ζ1= 0.15 and mode 2: ζ2 = 0.35). ........................................... 93

Figure 5.4 Vibration control performance using the PSF and REC Laws. ....................... 94

Figure 5.5 Control performance using PSF and REC laws (time history of relative tip

displacement response). .................................................................................................... 95

Figure 5.6 Actuation voltage measurements required by the PSF and REC laws. ........... 95

Figure 5.7 Actuation current measurements required by the PSF and REC laws. ........... 96

Figure 5.8 Instantaneous power required by PSF and REC laws. ................................... 96

Figure 5.9 Active and reactive power required by PSF and REC laws. ........................... 97

Figure 5.10 (a) A photographic representation (b) a schematic representation of the

autonomous gust alleviation system building on a honeycomb core fiberglass

multifunctional wing spar. ................................................................................................ 99

Figure 5.11 A prototype of (a) the MFC 8528P1 (b) the QP10n. ..................................... 99

xiii

Figure 5.12 A photographic representation of (a) the DP460 glue gun (b) the vacuum

process (c) the pressure meter panel. .............................................................................. 100

Figure 5.13 (a) Energy harvesting experimental setup for QP10n piezoelectric harvester

(b) dSPACE data acquisition system. ............................................................................. 102

Figure 5.14 Experimental and numerical simulation comparison of the first two mode

voltage-per-acceleration with a set of effective load resistance excited by clear sky

atmospheric turbulence for the QP10n piezoelectric harvester. ..................................... 103


current-per-acceleration with a set of effective load resistance excited by clear sky



power-per-acceleration with a set of effective load resistance excited by clear sky


Figure 5.17 Experimental and numerical simulation comparison of the voltage-to-base-

acceleration FRF at optimum load resistance excited by clear sky atmospheric turbulence

for the QP10n piezoelectric harvester. ............................................................................ 105

Figure 5.18 The measured voltage-to-base-acceleration FRF at a set of load resistance

excited by clear sky atmospheric turbulence for the QP10n piezoelectric harvester. .... 106



atmospheric turbulence for the MFC 8528P1 piezoelectric harvester. ........................... 106









for the QP10n piezoelectric harvester. ............................................................................ 108

Figure 5.23 A finished PCB prototype of Multimode REC Laws. ................................. 109

xiv

Figure 5.24 Experimental setup for the autonomous gust alleviation system. ............... 111

Figure 5.25 A comparison of relative tip displacement frequency spectrum response

predicted with numerical simulation for the first two modes, showing both open loop and

closed loop cases. ............................................................................................................ 112

Figure A.1Piezoelectric transducers with (a) 3-1 actuation mode (b) 3-3 actuation

mode. ............................................................................................................................... 116

Figure C.1 Cross section transformation of (a) original beam (b) transformed

homogeneous beam. ........................................................................................................ 119

Figure D.1 Representation of a Euler-Bernoulli cantilever beam with multiple PZT

layers. .............................................................................................................................. 120

xv

LIST OF TABLES

Table 2.1 Numerical and Experimental Shunt Parameters in Corr and Clark (2001a). .... 25

Table 2.2 Experimental Parameters and Damping Results for SSDI(Richard et al. (2000)).

........................................................................................................................................... 28

Table 2. 3 Summary of the Main Characteristics of SSDS and SSDI Systems in Free and

Forced Response in Ducarne et al. (2010). ....................................................................... 28

Table 3.1Properties of the Beam and the Piezoelectric Transducer (MFC). .................... 48

Table 3.2 Experimental comparisons of the PPF, Bang-bang-PPF, PID, Bang-bang-PID,

Nonlinear, Bang-bang-nonlinear, LQR and Bang-bang-LQR control systems. ............... 52

Table 4.1Selected properties of compared piezoelectric transducers. .............................. 74

Table 4.2 Selected properties of other components for multifunction wing spar design. 77

Table 4.3Power Associated With Each Electric Component............................................ 86

Table 5.1 Geometry and Material Properties for the Aluminum Baseline Multifunctional

Wing Spar. ........................................................................................................................ 91

Table 5.2 Control Performance versus PSF Control Gain. ............................................... 93

Table 5.3 Power and Energy Elements Associated with PSF and REC Laws. ................ 97

Table 5.4 Geometry and Material Properties for Two Unimorph Piezoelectric Harvesters.

......................................................................................................................................... 100

Table 5.5 Nominal Resistors and their Effective Values. ............................................... 101

Table 5.6 Experimentally Property Identification of Two Piezoelectric Harvesters. ..... 109

Table 5.7 Component Parameters of PCB Layout for Multimode Vibration Control. ... 110

Table 5.8 Experimentally Identified Properties for the PCB Device. ............................. 110

1

CHAPTER 1 INTRODUCTION

OBJECTIVE OF THE DISSERTATION

The goal of this dissertation is to demonstrate the feasibility, realization and

implementation of the concept and design of using harvested energy to directly control

the vibration response of flexible aerospace systems via piezoelectric materials.

Advanced techniques in aerospace systems usually require structures with low weight,

high strength, high damping and adaptive charging capabilities. Structural components of

satellites or unmanned aerial vehicles (UAVs) are often flexible and hence are easily

disturbed into vibration from a variety of sources. Repositioning maneuvers can cause

impulsive loads to the structure and hence excite broadband vibration, and rotating

components can cause persistent vibrations. Small, lightweight flexible UAVs provide

both harvesting opportunities and vibration suppression requirements. Hence the

motivation of this dissertation is to investigate the possibility that the aforementioned

ambient energy might be harvested and recycled to provide energy to mitigate the

vibrations through various control laws. Smart structure technology which incorporates

sensors, actuators, and real time control laws within composite sandwich substrates can

be implemented in such systems to achieve the required characteristics. The intention of

this dissertation is to develop an analytical basis for characterizing the feasibility of using

harvested ambient energy to suppress vibrations in aerospace structures. This intention

can also be illustrated by the basic scenario of energy flow shown in Figure 1.1. Ambient

sources from vibration, solar or thermal can be captured via piezoelectric transducers, and

then used for other purposes, such as vibration control or structural health monitoring.

Figure 1.1 Energy flow for simultaneous energy harvesting and vibration control.

ambient energy:

vibration, solar or thermal

Mechanical

Dissipation

Electrical

Dissipation

Vibration

Control

Thermal

Energy

Harvested

Energy

2

This research goal will be met by addressing the following objectives: 1) to develop a

model for piezoelectric and fiber composite materials integrated into flexible components

for a multifunctional cantilever beam; 2) to derive a predictive model for energy

conversion from embedded piezoelectric and fiber composite materials including the

associated electronics; 3) to explore a feedback control law based on minimum energy

constraints provided by the harvested ambient energy; 4) to experimentally validate the

theory produced in item 3; 5) to integrate the actuating, harvesting and sensing materials

into a composite sandwich structural element to form a multifunctional structure with

structural sensing , harvesting and control functionality; 6) to establish ambient vibration

levels for a typical small UAV to represent both normal fight condition and wind gust

disturbance; 7) to design frequency domain gust alleviation systems supplied by local

power sources harvested from ambient energy.

The following tasks outline approaches to achieve these proposed research objectives in

simultaneous energy harvesting and vibration control: 1) to develop and derive the

electromechanical governing equations for vibration and control of a multifunctional

composite structure consisting of embedded piezoelectric and fiber composite materials

in the general aerospace structures; 2) to explore the feedback control laws for vibration

suppression requiring the least amount of energy; 3) to design a multifunctional structure

with integrated piezoelectric and fiber composite materials, considering bending strength,

bending stiffness, and other optimal design factors; 4) to determine the feasibility of

using harvested energy to suppress vibration in the designed multifunctional structure; 5)

to examine the vibration suppression performance provided by a piezoelectric-based

harvesting device and the key parameters effecting maximum suppression; 6) to design a

proof of concept experiment to validate the results found in the first five tasks; 7) to apply

this concept and design for a composite sandwich wing spar with the goal of providing

self-contained gust alleviation; 8) to design a proof of concept experiment to validate the

design of task 7.

In summary, these research objectives address the question of whether or not harvested

ambient energy can be used to provide enough control effort to deliver a reasonable level

3

of vibration suppression and to quantify the degree to which such control can be

accomplished. The proposed effort focuses on the generic aerospace related systems. The

ambient energy sources considered are mechanical vibrations to be harvested by the

piezoelectric and fiber composite materials. Considering various control methodologies

explored for suppressing vibration, a reduced energy is developed to examine vibration

control performances with strong limits on the control input energy. A multifunctional

approach is applied to integrate the piezoelectric, fiber composite transducer materials

along with the control and harvesting electronics into the structure elements. One of the

promising applications of simultaneous energy harvesting and vibration control in aircraft

is in providing local power source to autonomous gust alleviation systems of a self-

contained small UAV. The research demonstrated in this dissertation integrates

piezoelectric energy harvester, smart materials, multifunctional structures and composite

sandwich structures into a UAV platform to perform simultaneously gust alleviation and

energy harvesting.

The research issues to be addressed are: 1) the characterization of appropriate ambient

energy; 2) the electromechanical modeling of vibration control and collocated

piezoelectric harvesting and strain sensing; 3) the development of vibration control laws

with limited energy consumption; 4) the analysis of bending strength and beam stiffness

analysis for the composite sandwich substrates; 5) the integration of piezoelectric, fiber

composite harvesting materials into a load bearing composite sandwich structure

members to enable multi-functionality; 6) the experimental validation of the scenario of

using harvesting energy to perform control; 7) the incorporation of energy harvesting

devices and gust load alleviation systems into small UAVs, providing local power source

for low-power sensors and controllers in aircraft.

LAYOUT OF THE DISSERTATION

The problem statement of this dissertation is given in Chapter 1. Motivation behind the

proposed concept of simultaneous energy harvesting and vibration control is

demonstrated. The research objective, main tasks and scientific issues are also addressed.

4

A literature review is presented in Chapter 2, starting with mathematical and dynamical

modeling of vibration-based cantilevered energy harvesters ranging from lumped to

distributed parameter base excitation problem. A review of vibration control laws is

presented for schemes using harvested energy as the main source of energy to suppress

vibrations via piezoelectric materials. These control methods are reviewed along the lines

of purely passive, semi-passive, semi-active, and active control. The classification is built

on whether external power is supplied into piezoelectric transducers. Special attention is

paid to recent advances investigating semi-passive and semi-active control strategies

derived from synchronized switching damping (SSD). However, whether or not the

harvested energy is large enough to satisfy a vibration suppression requirement has

become an important topic of research but has not yet specifically been addressed in

previous studies. Hence this chapter also reviews the possible control methods aiming for

less control energy consumption and addresses the potential application for simultaneous

vibration control and energy harvesting.

Chapter 3 details the examination of four conventional vibration suppression control laws

and four hybrid modifications of these laws using a switching method, named Reduced

Energy Control (REC). A hybrid version of each controller is obtained by implementing a

bang-bang control law (on-off control). The bang-bang control algorithm switches the

control voltage between an external voltage supply and the feedback signal provided by

aforementioned four conventional controllers. The purpose of employing the bang-bang

control law is to reduce the power requirement for vibration suppression by providing an

active controller with limited voltage input. The motivation to consider REC is the idea

that in some applications very little energy is available for control, yet passive, semi-

passive or semi-active methods cannot meet performance demands. In particular the

eventual goal is to reduce transient vibrations of smart structures using energy obtained

from harvesting and/or low power storage devices (batteries or super capacitors) as is

often desirable in aerospace systems. Free vibrations of a thin cantilevered beam with a

piezoceramic transducer are controlled by these eight controllers with a focus on the

fundamental transverse vibration mode. Experimental results exhibit that the system with

hybrid bang-bang-nonlinear controller requires 67.3 % less power than its conventional

5

version. Experiments also reveal the presence of substantial piezoelectric nonlinearities in

the transducer. The voltage-dependent behavior of the electromechanical coupling

coefficient is identified empirically and represented by a curve-fit expression. A real-time

state variable control algorithm is developed to account for the voltage-dependent

behavior of the coupling coefficient, enabling good agreement between the simulation

and experimental results.

Chapter 4 presents the design of a multifunctional composite sandwich wing spar in order

to examine the gust alleviation problem of a small UAV. The basic idea is that the wing

itself is able to harvest energy from the ambient vibrations during normal flight along

with available sunlight. If the wing experiences any strong wind gust, it will sense the

increased vibration levels and provide vibration control to maintain its stability. The

multifunctional wing spar integrates a flexible solar cell array, piezoelectric wafers, a thin

film battery and an electronic module into a composite sandwich structure. The basic

design factors are discussed for a beam-like multifunctional wing spar with load-bearing

energy harvesting, strain sensing and self-controlling functions. In particular, the

configurations, locations and actuation types of piezoelectric transducers are discussed

for optimal power generation. The equivalent electromechanical representations of a

multifunctional wing spar are derived theoretically and simulated numerically. A reduced

energy control law is represented by a positive feedback operation amplifier (op-amp)

and a voltage buffer op-amp for each mode. This examines the concept of simultaneous

energy harvesting and vibration, and holds promise for improving UAV performance in

wind gusts.

Chapter 5 is dedicated to experimental characterization and validation of an autonomous

gust alleviation system building upon recent advances in harvester, sensor and actuator

technology that have resulted in thin, ultra-light weight multilayered composite wing

spars. These beam like multifunctional spars are designed to be capable of alleviating

wind gust of small UAVs using the harvested energy. Experimental results are presented

for cantilever wing spars with micro-fiber composite transducers controlled by reduced

energy controllers with a focus on two vibration modes. A reduction of 11dB and 7dB is

6

obtained for the first and the second mode using the harvested ambient energy. This work

demonstrates the use of reduced energy control laws for solving gust alleviation problems

in small UAVs, provides the experimental verification details, and focuses on

applications to autonomous light-weight aerospace systems.

Chapter 6 summarizes the results of this work, addresses the major contributions to the

research community and presents recommendations for future work.

7

CHAPTER 2 LITERATURE REVIEW

This chapter presents a summary of passive, semi-passive, semi-active and active control

methods for schemes using harvested energy as the main source of energy to suppress

vibrations via piezoelectric materials. This concept grew out of the fact that energy

dissipation effects resulting from energy harvesting can cause structural damping.

First, the existing equivalent electromechanical modeling methods are reviewed for

vibration-based cantilevered energy harvesters using piezoelectric transducers. Following

the literature on mathematical and dynamical modeling of these devices ranging from

lumped to distributed parameter base excitation problem, the commonly used electrical

power conditioning circuits and their optimization are presented and discussed.

The energy dissipation from harvesting induces structural damping and this leads to the

concept of purely passive shunt damping. Classification of previous results is built on

whether external power is supplied into piezoelectric transducers. This chapter reviews

the literature on vibration control laws along the lines of purely passive, semi-passive,

semi-active, and active control. The focus is placed on recent articles investigating semi-

passive and semi-active control strategies derived from synchronized switching damping

(SSD).

However, whether or not the harvested energy is large enough to satisfy a vibration

suppression requirement has become an important topic of research but has not yet

specifically been addressed in previous studies. Hence this chapter also reviews the

possible control methods aiming for less control energy consumption and addresses the

potential application for simultaneous vibration control and energy harvesting.

8

NOMENCLATURE

coupling coefficient of PZT-based transducers

feedback control gain

base excitation amplitude

equivalent damping of the beam structure for lumped-parameter moeq

b

A

c

deling

equivalent capacitance, the reciprocal of structural stiffness

inherent capacitance of the PZT-based transducers

damping matrix

base excitation force

s

p

s

C

C

f

f

C

equivalent structure force on PZT-based transducer due to vibration

base excitation forcein terms of a series of finite components

electrical current

equivaleneq

i

k

f

t stiffness of the beam structure for lumped parameter modeling

system coupling coefficient

stiffness matrix

equivalent inductance, the mass or inertia of the PZT-based

sys

s

k

L

K

generator

inductance of the inductive shunt circuit

equivalent mass of the beam structure for lumped parameter modeling

mass of the PZT-based generator

mass

eq

p

L

m

m

M matrix

turn ratio representing the piezoelectric coupling coefficient

equivalent resistance representing the structural damping

electrical voltage

absolutabs

n n

Rs

V

w

e displacement

base displacement

relative displacement

structural displacement at the position of interest

base excitation displacement at the position o

b

rel

w

w

x

y

f interest

external load impedance

structural damping ratio

device coupling coefficient vector

modal coordinates for mode

modal coordinate vec

L

th

r

Z

r

θ

η tor

9

modal loss factor

admissible trial function

base excitation frequency

shut time period for synchronized switching damping on inductance

l

sh

b

t

r

uT

INTRODUCTION

The large-scale and lightweight design trends in aerospace systems give rise to extremely

flexible structures with low-frequency vibration modes. Due to their mechanical

simplicity, light weight, small volume, and ability to be easily integrated into applications

with flexible structures, electromagnetic, electrostatic, and piezoelectric transducers have

been extensively utilized for both energy harvesting and vibration control purposes.

Among these three typical transducer mechanisms, piezoelectric harvesters are prominent

choice for mechanical to electric energy conversion, since the energy density is much

higher compared to other transduction materials for a given profile, see Roundy and

Wright (2004). Their popular utilization is attributed to their excellent sensing as well as

actuation abilities with relatively high electro-mechanical coupling coefficients.

Additionally, piezoelectric materials are easily integrated into applications with flexible

structures. The piezoelectric materials (PZT1) exhibit the piezoelectric effect, which is a

reversible effect that can be divided into two phenomena as the direct and the converse

piezoelectric effects. When a piezoelectric material is mechanically strained, it produces

an electric potential, which can be used for sensing and harvesting (direct piezoelectric

effect). Conversely, when an electric field is applied, it produces a mechanical strain

allowing actuation and shape control (converse piezoelectric effect).

Many ambient energy sources have been investigated for harvesting purposes, falling into

four typical types: solar energy, thermal gradients, acoustic and mechanical vibration.

Mechanical vibration-based energy harvesting is the most popular and practical one. Up

1 Although PZT commonly refers to monolithic piezoceramic material lead-zirconate-titanate, we refer to PZT for

brevity but any piezoelectric material is implied.

10

to now, the review of vibration-based energy harvesting in the past years have been

presented by the following researchers: Sodano, Inman and Park (2004a) reviewed topics

in PZT-based energy harvesting from ambient mechanical vibration. They evaluated the

harvesting efficiency, and discussed the power storage and circuitry. Beeby, Tudor and

White (2006) reviewed various harvesting sources subjected to mechanical vibration such

as household goods and structures, for the objective of removing the external battery

power supply for wireless sensor networks and self-powered micro-systems. The review

paper by Pereyma (2007) provides a performance comparison of different energy sources,

but mainly focuses on vibration energy harvesting devices. Anton and Sodano (2007)

reviewed efficient PZT-based harvesting design based on physical and geometrical

configurations and efficient circuitry through adaptive energy removal techniques, in the

years following the article by Sodano et al. (2004a). For the purpose of on-site real time

energy generation aiming to transfer ambient mechanical energy at the sensor location

into electrical energy, Priya (2007) provided a comprehensive review of PZT-based

energy converters using low profile transducers. He also investigated the energy

harvesting efficiency in literature and discussed the selection of PZT-based transducers

for both on and off resonance applications. Later, Chalasani and Conrad (2008) reviewed

and discussed the power density of various energy harvesting sources, not only from

mechanical vibration, but also photovoltaic cells and thermoelectric generators. Another

review paper given by Cook-Chennault, Thambi and Sastry (2008), addressed the recent

advances of PZT-based energy harvesting technology based on non-regenerative and

regenerative power supplies.

However, not much attention has been given to the important problem of simultaneously

harvesting energy and using that energy to perform control. Hence this chapter reviews

the control methods that consider vibration-based piezoelectric harvester as the main

source of power supply and aims for less control energy consumption. An application of

this concept is a multifunctional wing spar (Figure 2.1) for an Unmanned Aerial Vehicle

(UAV), which is designed to alleviate wind gust using the harvested energy from normal

vibration, proposed in Wang and Inman (2011c). The baseline/ host structure has to carry

out the multiple functions of energy harvesting, strain sensing and vibration control in

11

order to be considered as multifunctional. The multifunctional wing spar considered here

also includes a Printable Circuit Board (PCB) integrating the required electric circuitry,

as illustrated in Figure 2.1 (a). A schematic of the multifunctional wing spar is given in

Figure 2.1(b). Figure 2.1(c) is a representation of the feedback control loop contained

within the spar.

The organization of this chapter is presented in three major sections: section 1 presents

the literature on electromechanical modeling for vibration-based cantilever piezoelectric

energy harvesters. After introducing the equivalent electromechanical modeling, the

review covers modeling methods ranging from lumped parameter to distributed

parameter methods and power conditioning circuitries as well their optimization. Section

2 introduces the damping effect due to energy dissipated in the host structure resulting

from energy harvesting. Section 3 reviews existing vibration control laws via

piezoceramics, ranging from purely passive shunt damping, semi-passive shunt damping

to semi-active and active control methods.

(a) (b)

(c)

Figure 2.1 (a) A prototype of multifunctional structure with simultaneous energy

harvesting and vibration control abilities (b) Its schematic representation (c) Its feedback

control block.

AccelerometerQP10N Shaker MFC 8528P1

PCB

x

z

A. QP16N (Harvester, Sensor)D. Printable Circuit Board (PCB)

B. Honeycomb Core Fiberglass

C. MFC(Actuator)

L1=25 mm L2=84.6 mm

L3=110 mm

L4=552 mm

E. Epoxy DP 460, Kapton

L4=592 mm

PZT LayerSubstrate

PCB

Fixture

Wire

Plantu(t) x(t)

Controller

_r(t)

H

G

12

REVIEW OF EXISTING MODELING OF VIBRATION-BASED

CANTILEVER PIEZOELECTRIC ENERGY HARVESTERS

Researchers have sought ways to model the electromechanical behavior of PZT-based

energy harvesters for different design purposes. Roundy and Wright (2004) presented a

simplified lumped-parameter model for PZT-based generators under bending vibration.

Design considerations were provided for maximum power harvesting with respect to both

resistive and capacitive circuits. Another lumped-parameter model was presented by

duToit, Wardle and Kim (2005) along with spatially discretized approximate distributed-

parameter model based on the Rayleigh-Ritz solution given by Hagood, Chung and Von

Flotow (1990) for piezoelectric actuation. Two optima were identified for the maximum

power extraction, corresponding to short-circuit and open-circuit resonance frequencies

of the device. Badel et al. (2007) built a lumped model for a semi-passive shunt damping

system and concluded that it had very good agreement with finite element method in

ANSYS, but runs 100 times faster.

Later, Erturk and Inman (2008a) pointed out several oversimplified and incorrect

physical assumptions in the literature, addressed issues of incorrect base motion

modeling, and the use of static expressions in a fundamentally dynamic problem. They

corrected formulations for piezoelectric coupling, and derived improved analytical

solutions for distributed-parameter modeling of PZT-based energy harvesters, which

allows the single degree of freedom lumped model to be corrected and provides a more

accurate model prediction. Finite element simulations performed by Rupp et al. (2009),

De Marqui, Erturk and Inman (2009), Elvin and Elvin (2009a) and Yang and Tang (2009)

were also shown to agree with the analytical solutions.

In order to provide a high-level ‘road map’, this section starts the review with existing

equivalent electromechanical modeling for vibration-based cantilever piezoelectric

harvesters. Following the dynamic modeling of these devices (from the structure point of

view), the literature of harvesting conditioning circuits and their power optimization

(from electrical engineering’s interest) are reviewed respectively.

13

Equivalent Electromechanical Circuit for Vibration-based Cantilever Piezoelectric

Harvesters

The electromechanical piezoelectric coupling system is popularly modeled as a

piezoelectric transformer, originally reported by Katz (1959), discussed by Flynn and

Sanders (2002), and has been used widely. The typical circuit representation, shown in

Figure 2.2, has the following mechanical elements: an equivalent force fs, as a result of

vibration excitation, an equivalent inductance, Ls, representing the mass or inertia of the

generator; an equivalent resistance, Rs, representing the mechanical damping, an

equivalent capacitance, Cs, the reciprocal of stiffness. For the electrical elements, Cp is

the equivalent inherent capacitance of the PZT-based layers, and ZL denotes the load

impedance. Here the leakage resistance is usually ignored. The turn ratio n denotes the

piezoelectric coupling coefficient, which is important for providing valuable information

for harvesting/control performance and giving the designer of the system some intuition

about how to optimize design parameters.

Figure 2.2 Equivalent circuit representation of the vibration-based piezoelectric harvester.

Dynamic Modeling for Vibration-based Cantilever Piezoelectric Harvester

A comprehensive analytical model of the PZT-based harvester (the left side of Figure

2.2) is very important not only for optimizing the extracted power output, but also for

optimizing the geometric system design to improve its performance. Due to the diverse

nature of research objectives, there have been a number of approaches for modeling

electromechanical behavior of vibration-based cantilever piezoelectric harvesters. With

respect to various design goals, an accurate analytical model should be as simple as

possible but sophisticated enough to capture different important phenomena in order to

provide a reliable estimation of the physical system for the application of interest. The

literature on dynamic modeling for base excitation energy harvesters from ambient

vibration sources is reviewed next.

Ls Rs Cs

fs

n:1

ZL

Mechanical Electrical

Cp

14

From mechanical engineer’s point of view, since cantilevered PZT-based harvesters are

mostly excited under base motion, the well-known lumped-parameter modeling is enough

to estimate the fundamental behavior of the mechanical system. This method provides a

simple representation of PZT-based harvesters and only requires the lumped system

parameters of the point of interest (usually the free end of the beam). These parameters

are equivalent mass, stiffness, and the damping of the beam denoted by meq, keq, and ceq,

respectively, as shown in Figure 2.3 (a). The base excitation y(t) is mostly assumed to be

harmonic for simplicity. Taking the electromechanical system of Figure 2.1(a) for

example, the fundamental degree of freedom is generally modeled in the form of:

. ( ) ( )eq eq eq

f Vm x c x y k x y (2.1)

0.( )p

C V i x y (2.2)

Here, α denotes the coupling coefficient of PZT-based harvester, f is the external

excitation force, x is the structural displacement at the position of interest, y represents

the base excitation displacement, and V and i stand for electrical voltage and current,

respectively. Due to its simple nature, the lumped-parameter approach has been widely

employed in literature, such as duToit et al. (2005), and Daqaq et al. (2007).

As an alternative modeling approach, the distributed-parameter method (Euler-Bernoulli

model), originally derived by Hagood et al. (1990) has been employed by Sodano, Park

and Inman (2004b) and other researchers such as Erturk and Inman (2008b). This

modeling approach along with experimental validations implements the Rayleigh-Ritz

formulation to represent a discretized mechanical system by reducing its mechanical

degrees of freedom from an infinite dimension to a finite dimension (in Figure 2.3(b)).

The absolute displacement wabs (x, t) at any longitude point x and time t is the sum of base

motion wb (x, t) and relative motion wrel (x, t). The relative displacement can be

represented as a finite series expansion of admissible trial function ( )r

x and unknown

modal coordinates ( )r

t for the rth mode as:

15

1

,( , ) ( ) ( )

N

r

rel r rw x t x t

(2.3)

Also taking the electromechanical system of Figure 2.1(a) for example, the

electromechanical coupled governing equations derived by Erturk and Inman (2011) are

given by:

,V Mη Cη Kη f θ (2.4)

0.pVC i θη (2.5)

Here M, C and K denote the mass, damping, and stiffness matrices, respectively, the

vectors η and f stand for the modal coordinates and base excitation force, respectively,

and θ is the device coupling coefficient, which is the function of the size, location, elastic

modulus and other PZT-based properties of transducers as well as parameters of the

baseline structure. This method was also implemented by Anton, Erturk and Inman

(2011) for modeling a self-charging structure. This solution is more accurate and agrees

more precisely with experimental data. However, it tends to complicate design and

control efforts because of its potentially larger order.

(a) (b)

Figure 2.3 A schematic diagram of (a) lumped-parameter (b) distributed-parameter

model.

The harvesting power is proportional to the square of the base acceleration and reaches

maximum at optimal load resistance and optimal configurations of PZT-based

transducers. The efficiency of the conversion process at the resonance condition is also

dependent upon the coupling coefficient and mechanical quality factor of the PZT-based

transducers. Sodano, Inman and Park (2005b) compared three commonly used harvesting

Meq

Plantx(t)

PCB

u(t)keq ceq

Sensing/

Harvesting

f(t)

Actuating

y(t)

PZT Layer

Substrate

PCB

x1

z

x2 x3 x

EI, A, m, L

16

devices (PSI-5H4E piezoceramic, Quick Pack (QP) and Macro-Fiber Composite (MFC))

for recharging a specific capacity battery, using Standard Energy Harvesting (SEH).

Their experimental results showed that PSI-5H4E was the most effective device under

random vibration excitation, and MFC was not well suited for power harvesting.

However, the power extraction efficiency is also very sensitive to power conditioning

circuit. Therefore, with the consideration of optimal power output in mind, power

conditioning circuitries are summarized next.

Power Conditioning Circuitry and Power Optimization:

The generated alternate current (AC) power from a piezoelectric element (left side of

Figure 2.2) cannot be directly used by micro electric and electrochemical devices that

require a direct current (DC) power supply. Therefore, a power conditioning circuitry

(right side of Figure 2.2) to rectify and regulate the AC voltage extracted and converted

from a mechanically excited piezoelectric transducer to a stable DC voltage is very

necessary. Qiu et al. (2009b) compared four different conditioning circuits, and presented

their respective output power equations. While each type of conditioning circuit has its

benefits for certain kinds of applications, a brief comparison of various conditioning

circuitry is presented here by considering the likely power output of each type of

conditioning method. Figure 2.4 lists three common conditioning circuits: (a) Standard

Energy Harvesting (SEH), (b) Synchronized Charge Extraction (SCE) and (c)

Synchronized Switching Harvesting on Inductor (SSHI).

(a) (b) (c)

Figure 2.4 (a) Standard energy harvesting (SEH) (b) Synchronous charge extraction

(SCE) (c) Synchronized switching harvesting on inductor (SSHI)

PZT

ZL

Crect

ic(t)ip(t)

io(t)

DC AC

PZT

ZL

Crect

ic(t)ip(t)

io(t)

DC AC

PZT ZL

Crect

ic(t)

ip(t)io(t)

DC AC

17

Standard Energy Harvesting (SEH)

SEH circuit, shown in Figure 2.4 (a), the simplest AC to DC power conversion described

in Hambley (2000) and many others, such as Farmer (2007), includes a diode based full

bridge rectifier, a smoothing/filter capacitor Crect, and a load impedance ZL. The DC filter

capacitor Crect is usually added so that the output voltage is much smoother and

essentially constant (where we assume the capacitance of Crect is large enough). Shu and

Lien (2006) derived an analytical expression for the average harvested power PSEH per

unit generator mass that incorporates all of effected factors, which is given by:

2

2( , , , ).L

SEHb

p

ZP A

Pm

(2.6)

Here, the function is denoted by P , the frequency and acceleration magnitude of excited

vibration is denoted by ɷb and A, respectively, the damping ratio of the system is denoted

by ζ, the mass of the generator is represented by mp. Power output is optimized either by

tuning the electric resistance, selecting suitable excited vibration, or adjusting the system

coupling coefficient by optimal structural design.

Synchronous Charge Extraction (SCE)

Ottman et al. (2002) added a switching DC-DC step-down converter preceded by the

filter capacitance as shown in Figure 2.4 (b). Their experimental results showed that the

SCE method increased the power to the energy storage element (electrochemical battery)

by 400% as compared to SEH. In order to obtain optimized power flow, Ottman,

Hofmann and Lesieutre (2003) developed a simpler method to determine the optimal duty

cycle expression while operating a step-down DC-DC converter in discontinuous

conduction mode. Their experimental results revealed that, the harvested power from

SCE was increased to 30.66 mW from 9.45 mW by SEH. Lefeuvre et al. (2007)

improved their output power by tuning the mechanical acceleration and frequency. Wu et

al. (2009) studied the transient behavior of several energy harvesting circuit schemes

using PZT-based transducers, which included direct charging of a storage capacitor using

SEH in Figure 2.4 (a), synchronized switching and discharging to a storage capacitor

(similar to Figure 2.4 (b), but without an inductor), and synchronized switching and

18

discharging to a storage capacitor through an inductor (SCE). They developed and

compared analytical models of these circuits with a matched resistance to predict output

power and charging rate for various storage capacitances and quality factors. At the end,

they experimentally demonstrated that the most effective design SCE increased the

output power by about 200% over the direct charging case SEH and reduced the

charging time by about five times.

Synchronized Switching Harvesting on Inductor (SSHI)

Guyomar et al. (2005a) proposed to add a nonlinear circuit SSHI, to the SEH circuit in

Figure 2.4 (c). They experimentally validated that the electromechanical conversion

ability of PZT-based transducer was improved by adopting SSHI so that the output power

was increased by over 900% compared to the same PZT-based energy harvesting system

with SEH. However, this SSHI technique does not always satisfy the wide band multi-

modal cases. Guyomar et al. (2005b) implemented a novel multi-modal control law to

enhance the SSHI circuit. This probabilistic based control method produces optimal

energy dissipated in the nonlinear SSHI device connected to PZT-based transducers.

However, the frequency deviation from resonance of the SSHI circuit was ignored by

them, since they assumed that the periodic excitation and the speed of mass are in phase.

This deviation was considered and discussed in Shu, Lien and Wu (2007) for a more

accurate performance evaluation of the SSHI technique. Their analysis revealed that the

optimal results exist when SSHI circuit is used for systems in the mid-range of

electromechanical coupling, since the system has the least performance degradations in

these cases.

Badel et al. (2006a) investigated power optimization for three circuits: SCE in Figure

2.4(b), parallel SSHI in Figure 2.4 (c), and also series SSHI. The series SSHI technique is

very close to that of the parallel SSHI, but instead of connecting the voltage processing

device in parallel with the piezoelectric element and the rectifier input, the switching

device is connected in series. They proposed a nonlinear approach to shape the voltage

delivered by the PZT-based transducers so that the phase shift between the output voltage

and vibration velocity was reduced and the voltage amplitude was increased respectively.

19

Lallart, Anton and Inman (2010) experimentally demonstrated that SSHI increased the

effective bandwidth of the structure by a factor of 4 in terms of mechanical vibration and

had a 100% frequency band gain in terms of total power output of the device.

Besides the above three basic conditioning circuits, a hybrid circuit combining SCE and

SSHI was studied by Lallart et al. (2008b), named as double synchronized switching

harvesting (DSSH). Lallart et al. (2008b) showed DSSH allows a gain of more than 500%

in terms of harvested energy compared with the SEH circuit. Their experimental results

also demonstrated that DSSH harvest the same amount of energy as SHE circuit but use

one tenth the amount of piezoelectric material.

Damping Effect due to Energy Dissipation Resulting from Energy Harvesting

While power extraction methods have been widely investigated, the energy harvesting

community quickly noticed that the dissipation effect, resulting from energy harvesting

can also provide damping. The concept of simultaneous vibration suppression and energy

harvesting aims to use harvested energy via PZT-based transduction as the control power

source to directly suppress the vibration of flexible structures. The PZT-based transducers

play multiple roles in harvesting, sensing and actuation.

Lesieutre, Ottman and Hofmann (2004) addressed the damping effects in an extended

vibration energy harvesting circuit using SEH method in Figure 2.2(b). They derived the

modal loss factor ηl as a function of the coupling coefficient ksys of the system and the

voltage ratio. Here the voltage ratio is the operating rectifier output voltage (constant)

divided by the open-circuit rectifier output voltage (also the AC amplitude of the

harmonic piezoelectric voltage under open-circuit conditions). They demonstrated that

when the voltage ratio was maintained at the optimal value of 0.5, the effective loss

factor depended only on the system coupling coefficient, which is given by:

2

2

2 .

(2 )

sys

l

sys

k

k

(2.7)

20

This equation was experimentally validated on a base-driven piezoelectric cantilever

excited by a harmonic force, with ksys of 26%, resulting in a value of ηl for the first

vibration mode of 2.2%.

Liang and Liao (2009) discussed the energy dissipation effects on the structural damping

of PZT-based harvesters. They concluded that the SSHI would outperform the SEH in

terms of harvesting capability and would outperform the purely resistive shunt damping

in terms of vibration control.

However, simultaneous optimization for both harvested power and structural damping

has become an important topic of research but has not yet specifically been addressed.

The next section reviews the possible control methods that consider vibration-based

piezoelectric harvester as the main source of power supply and aims for vibration control

using harvested energy sources.

THE STATE OF ART OF VIBRATION CONTROL LAWS VIA

PIEZOCERAMICS

Vibration control has been a very comprehensive and active research area since century

ago, e.g. Frahm (1911). Different control concepts have been proposed and studied to

satisfy the diverse needs for newly developed algorithms and applications. Taking a few

recent review papers for example: Sun, Jolly and Norris (1995) reviewed tuned vibration

absorbers in terms of passive, adaptive and active methods. Housner et al. (1997), Yi and

Dyke (2000) and Yi et al. (2001), presented the similarities and differences of active,

passive, semi-active, and hybrid control laws for civil engineering structural control and

monitoring, e.g. earthquake hazard mitigation. Their investigation was comprised of

hybrid control, optimal control, stochastic control and adaptive control. Many of these

control laws have been summarized in Preumont (2002). However, this chapter presents a

review of control laws for use with PZT-based transducers concerned with the external

power supply, classified as purely passive control, semi-passive control, semi-active

control and active control.

21

Purely Passive Shunt Damping

The essential characteristic of passive shunt damping is the transfer of mechanical strain

energy into electrical energy via PZT-based transducers, whereas structural vibration is

damped through dissipating Joule heat in the shunt piezoelectric circuit. Note that the

definition of “passive” varies between authors. In some occasions, it means the

characteristics of autonomous shunt damping circuits without any external power supply.

For other researchers, such as Anderson and Sumeth (1973): an electric shunt impedance

is said to be passive if and only if it does not supply power to the system. In this chapter,

the preceding cases are classified as ‘purely passive’ and the succeeding ones ‘semi-

passive control’. Three typical purely passive shunt piezoelectric circuits are presented in

Figure 2.5. They do not need any external power or sensing sources, and thus they do not

introduce any instability. Figure 2.5 (a) represents resistive shunts, which adds structural

damping by dissipating the mechanical energy into heat. Its key features were

summarized by Johnson (1995). Figure 2.5 (b) represents an inductive shunt circuit,

where Hagood and von Flotow (1991) interpreted the resonant LC circuit in terms of an

analogy with a tuned mass damper. Figure 2.5 (c) is a capacitive shunt circuit

representation, where the vibration absorber is tuned by changing the structural

effectiveness. However, little research has been addressed about purely passive capacitive

shunts to date. Actively tuned capacitive shunts were investigated by Edberg and Bicos

(1991), and Davis and Lesieutre (2000). It is worth mentioning that, each shunt

component (resistor, inductor or capacitor) can be combined in series or parallel branches

with the intrinsic capacitance of the PZT-based transducers in a manner analogous to that

of a mechanical vibration absorber. The single-mode absorber can be applied for multi-

mode control with the use of many shunt branches. For example, dell'Isola, Maurini and

Porfiri (2004) distributed an array of PZT-based transducers adjacent to inductive shunt

branches on a host beam for multi-mode vibration suppression.

22

(a) Resistive Shunt (b) Inductive Shunt (c) Capacitive Shunt

Figure 2.5 Purely passive shunted system using PZT-based transducers.

Semi-passive Shunt Damping and the Switching Technology

The purely passive shunt is stable as no external power is supplied, however, it is not

very practical and effective, especially when large amplitude vibration suppression is

required, the structural modeling is not completely known, the structural resonant

frequencies are low, multimode wideband control is required, or if the system disturbance

is unknown. Therefore, a so-called semi-passive shunt was developed to remove these

drawbacks. Again, for semi-passive control, the shunt circuit does not supply power to

the system. A popular way is to design adjustable electric resonances shown in Figure 2.5

(b). Uchino and Ishii (1988) developed a mechanical damper with a controllable

damping factor. Hollkamp (1994) utilized two operational amplifiers to create a synthetic

inductor for multi-mode electric resonance tuning. Peak reduction of 19dB and 12dB in

the second and third bending modes were reached experimentally. Hollkamp and

Starchville (1994) designed a self-tuning shunt (PZT-based absorber) by adaptively

adjusting RMS values of electric resonances (i.e. tune the synthetic inductor and

motorized potentiometer for both inductance and resonance adjusting). Lesieutre (1998)

reviewed shunt circuits in terms of electric loads (resistive, inductive, capacitive and

switching circuits), which involves not only passive, semi-passive, but also semi-active,

active shunts.

However, for low frequency modes, the optimal value of the tuned inductance is always

too large to be practical (which is often the case for flexible structures). In order to solve

the problems associated with large inductance, the concept of suppressing vibration

L

R

PZT

VR

PZT

V C

R

PZT

V

23

nonlinearly using a switch leads to various semi-passive/ semi-active switching control

laws. Figure 2.6 shows four commonly used switching configurations: (a) state switch:

switching the shunt circuit between open circuit and short circuit; (b) synchronized

switched damping (SSD) on a resistive shunt (SSDS); (c) synchronized switched

damping on an inductive shunt (SSDI): only a very small inductance is required in this

case, instead of a huge one needed for inductive shunt shown in Figure 2.5 (b); (d)

synchronized switched damping on a small voltage source (SSDV). These semi-passive,

semi-active methods introduce nonlinearity by using a switch for vibration control, but

also deal with the energy transfer between the mechanical and electrical domains.

The concept of state switching was originally used by Larson (1996) to develop a high

stroke acoustic source at a wide frequency range. Using open- and short-circuit state

switching, the acoustic driver’s stiffness (and therefore its natural frequency) was tuned

to track high amplitude changing frequency signal. Clark (1999) applied the state

switching technique for vibration control by adjusting the stiffness between open- and

short-circuit states of the PZT, as shown in Figure 2.6 (a). He developed the state

switching control law using the following logic: when the system is moving away from

equilibrium (displacement times velocity is positive) the circuit is switched to the open-

circuit or high-stiffness state, and when the system is moving toward equilibrium

(displacement times velocity is negative) it switches back to short-circuit or low stiffness

state. The numerical simulation of a cantilever beam in bending showed that the vibration

suppression depended on the effective stiffness change. Depending on the operating

mode of the PZT, a layered beam can reach up to a factor of 2 in effective stiffness

change if the PZT is operated in d33 mode (factor of 1.18 for d31 mode). Much larger

stiffness changes can be produced by increasing the piezoelectric coupling coefficient.

Note that the author named the state switching as ‘adaptive passive’ or ‘semi-active’.

However, since external power is only supplied to drive the switch, not for the system,

we classified this as semi-passive shunt.

24

(a) (b) (c) (d)

Figure 2.6 Schematic diagram of (a) state switch (b) SSDS (c) SSDI and (d) SSDV.

Clark (2000b) numerically compared the damping effect of a state switch, against purely

passive resistive shunt in Figure 2.5 (a) under both impulse and harmonic excitation.

Note that, “state switch” were used by them to represent not only open/ short circuit

switching in Figure 2.6 (a), but also resistive switching (SSDS) in Figure 2.6 (b).

However, in this chapter, ‘state switch’ only denotes switching between open and short

circuit and SSDS represents for resistive shunt switching, which is the more widely used

term. Their simulation illustrated that for impulse response, the passive resistive shunt

provides almost the same performance for the optimized cases, but its control

performance dropped significantly when the resistors are no longer optimized, compared

to state switch and SSDS. For harmonic response, they also found that the passive

resistive shunt works well near resonance, and in a higher frequency range. However,

state-switch and SSDS performed better in lower frequency range where stiffness is a

more dominant factor in the response. Their conclusion was that the state-switch and

SSDS outperform the purely passive case, and may be an alternative to active control in

certain frequency ranges. They also presented a ratio of effective stiffness as a function of

baseline beam to PZT patch thickness ratio in both d31 and d33 modes. Both unimorph

and bimorph cases were studied, where an aluminum or steel baseline beam is entirely

covered by the PZT patch(s).

Corr and Clark (2001a) compared the performances of Figure 2.6 (a) state switching and

Figure 2.6 (b) SSDI against a tuned inductive shunt. For SSDI, the optimal shut time

interval (when the SSDI circuit stays at short-circuit) was derived by them as:

L

VSW

R

PZT VSW

V

PZT

V

PZT

V

R L

PZT

V

R

25

shut pT LC (2.8)

Here shutT is the shut time period, L is the inductance of the inductive shunt circuit and

Cp is the capacitance of the PZT-based harvester. Both numerical and experimental

comparisons (Table 2.1) showed that the SSDI and inductive shunt had better

performance than the state switching technique. It was hard to tell if the SSDI had better

performance than the inductive shunt. However, the use of inductance of 100 mH makes

the SSDI more practical than that with a 287H heavy inductor for the inductive shunt

(from Table 2.1).

Table 2.1 Numerical and Experimental Shunt Parameters in Corr and Clark (2001a).

Name Inductive shunt State Switching SSDI

Resistance _Numerical(Ω) 2900 1e-06 70

Inductance _Numerical (H) 287 0 0.10

Resistance _Experimental (Ω) 815 0.008 66

Inductance_ Experimental (H) 10.5 0 0.50

Their experimental comparison shown in Table 2.1 was conducted for the third vibration

mode reduction. The optimal resistance and inductance of the resistive shunt were

calculated first using the relations given by Hagood and von Flotow (1991). They were

then tuned (Table 2.1) to yield the smallest structural response at the third mode.

Experimental results showed that the state switching technique did not perform well (~2

dB reduction). The SSDI, however, did just as well as the inductive shunt (~12 dB

reduction), but with a much smaller shunt inductance (~20 times smaller). The SSDI

technique was also easier to tune and was less susceptible to system changes than the

inductive shunt. Note that, in order to keep the electrical natural frequency much larger

than the vibration mode of interest, an upper inductance bound was always set. To avoid

the chatter effect, a lower bound of inductance was set and introduced in Corr and Clark

(2001b). Cunefare (2002) implemented the concept of state switching for vibration

control under harmonic point force excitation. He integrated the switchable stiffness with

the spring element for a vibration absorber. In this way, the shift of stiffness

instantaneously 'retuned' the state-switched absorber to a new frequency. Holdhusen and

26

Cunefare (2003) analyzed the damping effect of this state-switched absorber compared to

classical absorbers. Corr and Clark (2003) expanded the multi-mode SSDI technique by

developing a power rate multi-modal control law for the switched PZT-based transducer,

which was numerically tested on a simple six-degree-of-freedom spring-mass system,

and also experimentally tested on an aluminum beam. The results of the simulations and

experiments showed that this method was able to dissipate energy in multiple structural

modes simultaneously and selectively. The experiments also demonstrated that 11 dB of

reduction in a single mode or 7 dB of simultaneous reduction in multiple modes was

achieved in a beam excited by a continuous random disturbance.

Originally, the SSD was proposed by another research group Richard, Guyomar and

Audigier (1999a) and Richard et al. (1999b) for resistive shunt switching from open-

circuit to short-circuit based on the variation of mechanical strain in Figure 2.7 (a).

However, in order to differentiate this SSD on a resistive shunt from SSD on other shunt

circuits, such as SSDI, we name the SSD on a simple resistive shunt as SSDS. Their

proposed switching circuit consisted a pair of N Channel MOSFET transistors 1T and

2T

and a pair of fast recovery diodes 1D and

2D wired as shown in Figure 2.7. Here gsV is

the drive signal for the transistors.

(a) (b)

Figure 2.7 (a) SSDS in Richard et al. (1999b) (b) SSDI in Richard et al. (2000).

This SSDS switching device introduced a phase shift and a distortion of the output

voltage V(t), which was modeled as:

D1

T1

D2T2

Driving Voltage Vgs

PZT

V

R

Opto

isolatorMC 68HC11

AC~DC

Converter

Switch Control Board

PZT

T1D2

T2

V

R

D1

L

27

cos( )( )V t j x y (2.9)

Here 1j , is the amplitude, and indicates the phase angle between the mass

velocity and the chopped voltage. Obviously, the damping performance will be enhanced

by increasing the factor cosb , by either decreasing the phase angle or increasing the

feedback control gain b or harmonic amplitude . The switching threshold value of the

displacement or voltage can be either predetermined or controllable. The shut time

interval of the short circuit was also tunable for the purpose of optimization.

Their experiments revealed that the shortest shut time interval resulted in the most

efficient damping. Experiments carried on by them using both harmonic and transient

excitation indicated that their proposed SSD technique increased both the output voltage

across PZT and the structural damping. The harmonic excitation experiments showed that

the switching circuit with a resistor of 54kΩ decreased 20% of the maximum amplitude

on the 1st mode resonant frequency of 10.5 Hz, and the damping appeared to be twice

what was usually obtained with a matched resistive shunt. The transient excitation

experiments revealed that the settling time using SSDS was nearly 30% faster than that

with the open-circuit one, and 15% faster than that with a matched resistive shunt of

54kΩ.

The SSDI (Figure 2.7(b)) developed by Richard et al. (2000), was achieved by building

the switching control board around a microcontroller, allowing it to digitize the PZT

voltage and to generate a controlled width pulse to drive the switch, as shown in Figure

2.7 (b). Both harmonic and transient excitation experiments were conducted by them on

three cantilever beams.

Table 2.2 gives global results including resonant frequency, transducer capacitance,

critical damping resistance and damping results.

28

Table 2.2 Experimental Parameters and Damping Results for SSDI(Richard et al. (2000)).

Name (Material) Beam 1

(Eposxy)

Beam 2

(Aluminum)

Beam 3

(Steel)

Short-Circuit Frequency(Hz) 10.33 13.09 12.75

Open-Circuit Frequency(Hz) 10.38 13.17 12.89

Coupling Coefficient 0.103 0.11 0.148

Capacitance 0C (nF) 280 190 90

Adapted Shunt Resistor(kΩ) 54 61 139

Max. Damping (dB) -Adapted Resistive -0.5 -2 -6

Max. Damping (dB) -Switch on Short-Circuit -1.3 -3.7 -8.4

Max. Damping (dB) -Switch on an Inductor -6 -10.5 -16.5

Time Constant (s) -Open-Circuit 0.8 1.5 6.5

Time Constant (s) Switch on Short-Circuit 0.6 0.7 1.3

Time Constant (s) Switch on an Inductor 0.4 0.3 0.5

The control performance of both SSDS and SSDI were compared by Ducarne, Thomas

and Deu (2010). Some theoretical results derived by them are shown in Table 2.3. Here

the added damping add depends only on the modal coupling coefficient ,rk which is

very close to the traditional effective coupling coefficient |kr| =

Keff, defined from

Thomas, Deu and Ducarne (2009). For a structure with a damping factor ζr = 0.1%, kr =

0.2, the added damping is 2.5% of with SSDS and 10 % with SSDI. An amplitude

attenuation denoted by AAB was defined in order to evaluate control performance for

forced response. It is expressed as the different between controlled system’s peak

amplitude and open-circuit amplitude (in dB). Additionally, the optimal value opt

e of the

electric damping factor using SSDI was obtained for free and forced responses.

Table 2. 3 Summary of the Main Characteristics of SSDS and SSDI Systems in Free and

Forced Response in Ducarne et al. (2010).

Name Free Response Forced Response

SSDS 2

2

11ln 2.5%

1

r

add

r

k

k

( , ) 30dB r rA f k dB

SSDI 11ln 10%

1

r

add

r

k

k

opt

e r rf k

( , ) 46dB r rA f k dB opt

e rf k

29

Semi-active Control

Qiu, Ji and Zhu (2009a) reviewed the semi-active control performance, compared with

passive and active control. However, in their paper, and many others in the literature,

both ‘semi-passive’ and ‘semi-active’ control, or ‘adaptive’ or ‘hybrid’ control have been

used to represent various shunt damping circuits with external power sources. In this

chapter, an electric shunt circuit, is said to be semi-passive if and only if it does not

supply power to the system, otherwise, it belongs to semi-active.

Fleming and Moheimani (2003), developed an online adaptive shunt damping circuit for

multi-mode vibration control in the use of only one PZT patch. A voltage controlled

current source and DSP system was implemented by them to provide the desired terminal

impedance of an arbitrary shunt network. Their experimental testing for a randomly

excited simply supported beam showed reliable estimation of the performance functions,

optimal tuning of the circuit parameters, and satisfactory maladjustment. In addition, the

second and third modes of this beam were reduced in magnitude by up to 22dB and

19dB.

Moheimani (2003) presented an overview and discussion of the feedback analogy of

PZT-based resistive shunt damping systems. He investigated the similarities and

differences between shunt damping systems and collocated active vibration controllers.

He also demonstrated that the shunted vibration control problem using PZT-based

transducers is a very specific type of feedback control problem.

Petit et al. (2004) investigated the damping performance of SSD on small voltage sources

VSW (SSDV) in using Figure 2.6 (c), compared with Figure 2.6 (a) SSDS, and Figure 2.6

(b) SSDI. A specific control box detecting the maximum and minimum voltage V is used

to drive the switches. For the SSDV configurations, the capacitance of PZT

charges/discharges through the inductor. They derived the damping coefficient as a

function of the system electromechanical coupling coefficient and this indicated that for

SSDS and SSDI circuits, the damping capability was strongly related to the coupling

coefficient. Experiments were conducted on a cantilever beam and validated their

30

theoretical derivation. A clamped steel plate with a variable amount and location of PZT

was tested using a SSDI circuit for multi-mode operation over a very large frequency

band, since the PZT position and volume also affect the damping. Experimental results

showed that the SSDI circuit damped vibration (between 2dB to 9dB) at various modes

for frequencies lying between 180 Hz and 280 Hz.

Guyomar et al. (2005b) developed a probabilistic multi-modal SSDI control law

maximizing the energy dissipated in the nonlinear processing device connected to the

PZT. The voltage was able to reach a statistically probable value before processing the

piezo-voltage inversion, utilizing a probabilistic method. The numerical simulations

revealed that this probabilistic approach can optimize both displacement-based and

energy-based vibration control criteria. The optimized displacement and energy damping

were -8.5 dB and -6.3 dB compared to -3.0 dB and -4.3 dB respectively of a conventional

SSDI which switches on voltage extremes.

Lefeuvre et al. (2006) added two voltage sources in the switch schemes from Figure 2.7

(d) to significantly reduce the required quantity of the PZT-based transducers. Their

experiments demonstrated an 83% reduction of the PZT volume by adding two voltage

sources of 10 V each in the switching circuit, while effecting the same vibration

attenuation of -24dB. Badel et al. (2006c) and Lallart, Badel and Guyomar (2008a) found

out if the external driving force is lower than the force induced by the PZT-based

transducer, the SSDV control becomes unstable. This problem was solved by them

through adjusting the voltage source to fit the piezoelectric generated control force to the

mechanical excitation. Guyomar, Lallart and Monnier (2008) implemented a SSDV with

single voltage source for structural stiffness control.

Guyomar, Richard and Mohammadi (2007) analyzed the voltage/displacement signal

during a given time period to statistically determine the probable voltage/displacement

thresholds using SSDI technique. The experiments revealed that either a probability or

statistical strain analysis could allow defining a criterion to identify the relevant switch

instants more accurately. The best results could be received using either the strain itself

31

or its square. In this way, almost 10 dB of global displacement damping can be reached,

nearly twice the achieved damping using classic SSDI techniques. Lallart et al. (2008c)

developed an adaptive SSDI circuit and found more than a 10% increase in both first and

second mode damping, compared with the classical SSDI circuit. Guyomar, Richard and

Mohammadi (2008) experimentally indicated that the PZT area and the excitation

frequency affect the damping results. While PZTs were in parallel, the larger surface size

of PZT was more efficient at low excitation frequencies and the smaller surface size was

more economical at high excitation frequencies. Harari, Richard and Gaudiller (2009)

developed an improved semi-active control method to satisfy low power supply

requirements for broad bandwidth excitation.

Ji et al. (2009b) improved the SSDV technique by adjusting the voltage coefficient for

effective vibration control. An improved switch control algorithm was developed for the

electronic switch drive instead of the conventional algorithm. They turned the switch to

an inactive state for a certain time period while no switching actions were executed

even if extremes were detected. This prevents the system from switching too frequently

and consequently increases the control stability, since classical SSDV became unstable

when the source voltage was too large. However, the adaptive SSDV was always stable.

Ji et al. (2009a) applied the adaptive SSDV technique on a composite beam for vibration

control, using the Least-Mean-Square (LMS) algorithm to adjust the voltage source. The

first mode control simulation indicated that the LMS-based adaptive SSDV consumed

less source voltage output but performed better control than the derivative-based adaptive

SSDV. This is because the phase of switching points also affects the control performance

in addition to the output voltage. Their application for multimode control was also

investigated in Ji et al. (2010). A simpler SSDI switching algorithm was developed by Ji

et al. (2009c) using a displacement threshold switch. This not only avoided overly

frequent switching, but also increased the converted energy and improved the control

performance. Experiments were carried on a cantilever composite beam under two

excited modes. It was revealed that this proposed method improved the control ability up

to 18.2 dB for the first mode compared to 3.7dB using classical SSDI. However, its

32

control performance dropped to 2.6 dB from 3.46 dB using classical SSDI for the second

mode when the first and the second modes were excited simultaneously.

Wilhelm and Rajamani (2009) presented a multimode vibration control law by using the

harvested and stored electrical charge from mechanical vibration. They proposed to

employ an array of one or more pre-charged capacitors to provide a selection of various

power supplies. The external capacitors can supply a control voltage to the PZT-based

actuators and can also collect current generated by the PZT-based actuators. In their

paper, the SSDI method was compared with the Single Switched Capacitor (SSC), and

the Multiple Switched Capacitor (MSC) method. The SSDI cannot harvest and store

energy continuously, which led to inferior performance as a self-powering semi-active

method. The SSC method could suppress vibration while harvesting and storing energy

continuously, but its single mode nature limited its control effectiveness. The MSC

method was more effective in both transient and continuous random excited vibration

excitation while harvesting over 20% of the ambient vibration energy.

Active Control

Considerable research on active control laws for the low frequency vibration reduction in

flexible structures has been conducted. Wagg and Neild (2010) investigated

interrelationships of nonlinear vibration identification, modeling and control in the

following separate, but related areas: nonlinear vibrations, nonlinear control, approximate

methods, cables, beams, plates and shells. The focus reported here is motivated by

searching for vibration suppression laws that use a minimum amount of energy, with the

idea that they might eventually be powered off of harvested energy and/or low power

storage devices. Bardou et al. (1997) analytically compared different active control

strategies of minimizing the total power supplied to a plate. They focused on physical

parameter optimization of the plate and locations of the excitation forces. Anthony and

Elliott (2000) investigated how the positions of the controlling actuators affect the control

performance in reducing the total vibration energy. Of four cost functions they studied,

two are energy-based, and the other two are based on velocity measurements. However,

the success of optimal active control is also dependent upon increasing demands on

33

design specifications in order to obtain improved performance and robustness

characteristics from structural control systems. Crawley and de Luis (1987) proposed to

build piezoelectric materials in laminated beams. Baz and Poh (1988) utilized a modified

independent modal space control for selecting optimal location, control gains and

excitation voltage of the PZT-based actuators, in order to reduce input control effort for

vibration suppression of large flexible structures. Morgan and Wang (1998) synthesized a

simple parametric control law involving an inductive shunt circuit to turn the resistor on

and off for reducing control power (a resonant inductive shunt circuit suppresses the

vibration equivalent to a tuned mass damper). Phan, Goo and Park (2009) developed a

genetic algorithm for parameter optimization of a Positive Position Feedback (PPF)

controller, in order to minimize the energy consumption in vibration suppression of a

flexible robot manipulator.

However, conventional optimal control methods calculate energy consumption only

indirectly by adjusting a weighting matrix on control effort, see Preumont (2002) and

Inman (2006) for instance. With minimizing actuation power in mind, motivated by

noting that the actuation voltage in most active vibration suppression control laws is

relatively high during the early control periods. Thus voltage saturation was found to be

very effective in reducing actuation power and obtaining the same performance. Wang

and Inman (2011a) introduced a reduced energy control law, employing a saturation

control logic to switch a control system from one state to another, which will be

introduced in next chapter. A reduced energy control is achieved by providing the

conventional active controllers with a limited voltage boundary.

CHAPTER SUMMARY

The main objective of this chapter is to review methods on how to suppress vibration

using harvested energy from ambient mechanical vibrations via PZT-based transduction.

First, the existing equivalent electromechanical modeling methods are reviewed for

vibration-based cantilevered energy harvesters using PZT-based transducers. Following

the literature of mathematical and dynamical modeling of these devices ranging from

lumped to distributed parameter base excitation problems, the commonly used electrical

34

power conditioning circuits and their optimization are presented and discussed. The

energy dissipation caused structural damping leads to the concept of purely passive shunt

damping. Classified by whether external power is supplied to the structures, this chapter

reviews the literature of vibration control laws along the lines of purely passive, semi-

passive, semi-active, and active control. Special attention is given on recent articles

investigating semi-passive and semi-active control strategies derived from synchronized

switching damping (SSD). This chapter also reviews the existing optimal active control

methods aiming for less control energy consumption. The aim of this review is to provide

background for researching the concept of using harvested energy as the main source of

powering control systems for the purpose of suppressing vibrations. The eventual goal is

to answer the question of whether or not harvested ambient energy can be used to provide

enough control effort to provide a reasonable level of vibration suppression and to

quantify the degree to which such control can be accomplished. One element of such

research is to develop minimum energy control laws as well as means of maximizing the

amount of energy harvested.

35

CHAPTER 3 REDUCED ENERGY CONTROL LAW

The chapter presented here examines four conventional vibration suppression control

laws and four hybrid modifications of these laws using a switching method, named as

Reduced Energy Control (REC). The motivation is to determine which of these eight

controllers results in the least amount of power flow to the actuator to have the same

settling time under free vibrations. The motivation to consider REC is the idea that in

some applications very little energy is available for control, yet passive, semi-passive or

semi-active methods cannot meet performance demands. In particular the eventual goal is

to reduce transient vibrations of smart structures using energy obtained from harvesting

and/or low power storage devices (batteries or super capacitors) as is often desirable in

aerospace systems. The four conventional active control systems compared in this work

are Positive Position Feedback (PPF) control, Proportional Integral Derivative (PID)

control, nonlinear control, and Linear Quadratic Regulator (LQR) control. A hybrid

version of each controller is obtained by implementing a bang-bang control law (on-off

control). The bang-bang control algorithm switches the control voltage between an

external voltage supply and the feedback signal provided by the PPF, PID, nonlinear or

LQR controllers. The purpose of combining the bang-bang control law with the

aforementioned controllers is to reduce the power requirement for vibration suppression

by providing an active controller with limited voltage input. Free vibrations of a thin

cantilevered beam with a piezoceramic transducer are controlled by these eight

controllers with a focus on the fundamental transverse vibration mode. Experimental

results exhibit that the system with hybrid bang-bang-nonlinear controller requires 67.3

% less power than its conventional version. The hybrid versions require significantly less

power flow as compared to their conventional counterparts for the PPF, PID and LQR

controllers as well. Experiments also reveal the presence of substantial piezoelectric

nonlinearities in the transducer. The voltage-dependent behavior of the electromechanical

coupling coefficient is identified empirically and represented by a curve-fit expression. A

real-time state variable control algorithm is developed to account for the voltage-

dependent behavior of the coupling coefficient, enabling good agreement between the

simulation and experimental results.

36

NOMENCLATURE

, , , curve fit constants-describing the nonlinear piezoelectric behavior

state matrix in the state space representation

system constant determined by initial condi

f f f f

s

a b c d

A

A

tions and damping characteristics

nonlinear gain for the nonlinear controller

input matrix in the state space representation

electromechanical coupling coefficient f

nlA

B

b

33

eedback constant of the controller

output matrix in the state space representation

throughput matrix in the state space representation

longitudinal piezoelectric cons

C

D

d

tant of the piezoelectric material

state feedback gain for the LQR controller

control gain of the PPF controller

position of the MFC bottom layer

position

f

c

d

G

g

h

h

of the MFC top layer

total thickness of the MFC

total width of the MFC

actuation current

cost index function for the LQR controller

propor

p

t

p

h

h

i

J

k

tional gain of the PID controller

integral gain of the PID controller

derivative gain of the PID controller

active beam length

number of electrode pairs

i

d

e

k

k

l

N

over the active beam length

unique positive semi definite solution to the algebraic Riccati Equation

Quadratic matrices of the LQR controller

Regulator matrices of

P

Q

R

0

the LQR controller

average actuation power measured

instantaneous actuation power measured

width of each electrode of MFC in the longitude beam axis

w

avg

ins

e

P

P

r

r

idth of each non-electrode region of MFC in the longitude beam axis

Laplace transform complex argument

time

time constantc

s

t

t

37

settling time

control input actuation voltage

external voltage supply of the bang-bang-PPF controller

tip displacement of the beam

el

s

ext

p

T

u

V

X

Y

astic modulus of piezoelectric device MFC

generalized natural frequency of the PPF controller

fundamental un damped natural frequency of the beam with a tip mass

f

n

1

viscous damping ratio of the beam for the fundamental mode

generalized damping ratio of the PPF controller

mass normalized eigenfunction of the cantilevered beam for

f

the first mode

INTRODUCTION

Piezoelectric materials produce an electric field when strained, which is used for sensing

and energy harvesting, and produce a mechanical strain when an electric field is applied,

which is used for actuation and vibration control. Due to their mechanical simplicity,

light weight, small volume, and ability to be easily integrated into applications with

flexible structures, piezoelectric materials have found many applications in vibration

control. The increased research in energy harvesting offers promise in developing purely

passive, semi-passive and semi-active control systems that run using the harvested energy

(Wang and Inman (2010)). For instance, Makihara, Onoda and Minesugi (2005) proposed

a hybrid method integrating a bang-bang active method with an energy-recycling

technique (similar to SSDI reviewed in chapter 2), thus recycling the converted electrical

energy harvested via piezoelectric material from mechanical vibrations. Ji et al. (2011b)

analyzed converted energy numerically and experimentally to improve control

performance using SSD based on a displacement switching threshold. Ji et al. (2011c)

theoretically and numerically investigated modal coupling induced by energy conversion

using SSD. Liang and Liao (2009) studied the structural damping capacity by analyzing

the energy dissipated during the energy harvesting in the application of SSHI and

summarized that the damping capacity can be improved by increasing the

electromechanical coupling coefficient and/or enhancing the electrical quality factor. In

the experimental application of SSHI, Badel et al (2006) reported a 15dB amplitude

38

reduction of the beam tip displacement under harmonic force with the amplitude of

0.086N around its fundamental resonance frequency of 56Hz. However, due to the nature

of the low frequency of the fundamental mode, the light weight, typically for space

structures, as well as the limitation of electromechanical coupling of piezoelectric

materials, the maximum amplitude reduction for transient vibration case in the use of

SSHI is unable to reach 26dB (20lg(5%)), with reasonable inductance.

Our motivation is to develop energy-autonomous controllers powered by harvested

ambient energy using piezoceramic materials. The objective of this chapter is to compare

different active control laws to suppress low-frequency vibrations using reduced

actuation energy for the same system and under the same design constraint (identical

settling time Ts2

for free vibrations). We investigated the actuation power flow of four

conventional control systems as well as their hybrid versions employing a switching

technique. They are the PPF control (Fanson and Caughey (1990) and Goh (1983)). the

Proportional Integral Derivative (PID) control, see O'Dwyer (2009) for example, a

nonlinear control with a second-order nonlinear term of a product of the position and the

velocity feedback to create variable damping (Lewis (1953) and Lee and Castelazo

(1987)), the Linear Quadratic Regulator (LQR) control, summarized by Levine (1996)

and Dorf and Bishop (2008), as well as their hybrid versions integrating a bang-bang

control law (on-off control). Bang-bang control, usually employed to switch a control

system from one state to another, is a strategy typically used for optimal time problems

(Hermes and LaSalle (1969)). However, in this chapter, a bang-bang control algorithm is

implemented to switch a control input from an external voltage supply to the feedback

signal provided from the PPF, PID, nonlinear or LQR controllers. The purpose of

combining the bang-bang control law with the aforementioned controllers is not to obtain

minimum time but to reduce the actuation power requirement for vibration suppression.

It is known that the electromechanical coupling coefficient of a piezoelectric transducer

(feedback constant in the control problem) depends on the beam structure and the

2 The settling time is defined in this work as the time to reach 5% of the initial displacement in the

absence of any initial velocity.

39

piezoelectric actuator. Most researchers, such as Corr and Clark (2001a) and Onoda,

Makihara and Minesugi (2003), estimate this coefficient experimentally as a constant.

However, it is known that the electromechanical properties of piezoelectric materials

exhibit nonlinearity at high electric fields, see reference in Taylor (1985) and Uchino,

Negishi and Hirose (1989). This nonlinearity results in a considerable discrepancy

between the experimental measurements and simulation results if constant (voltage-

independent) coupling coefficient is assumed. Therefore, in this work, the

electromechanical coupling coefficient (feedback constant) is identified empirically by

using the results of the first experimental case. An exponential function of the voltage-

dependent coupling coefficient (feedback constant) is then obtained by curve fitting. A

state variable control algorithm is developed to account for the identified nonlinearity of

the coupling coefficient with control voltage in the simulations.

CONVENTIONAL ACTIVE CONTROL SYSTEMS

To compare control laws in terms of the required actuation power, it is emphasized that

they all designed to yield the same performance. In this case, we design each controller to

have the same setting time and each is given the same initial displacement and zero initial

velocity. Four conventional controllers and their hybrid bang-bang versions (REC) are

designed and compared in terms of the same settling time constraint. The free vibration

on the single degree of freedom mechanical system is considered here. The governing

equation of motion with small oscillations derived from equation (2.1) can be simplified

as:

22 0.n nx x x (3.1)

Here, x is the displacement response of the single degree of freedom system (the

transverse tip displacement of the cantilever in the experimental section introduced later

on), ζ and ωn are the viscous damping ratio and the undamped natural frequency of the

structural fundamental mode, respectively. The Laplace transform of this open-loop

system with the initial displacement x0 and zero initial velocity gives:

40

2 2

0 0( ) 2 ( ( ) )) ( ) 0.

n ns X s sx sX s x X s (3.2)

Here, X(s) denotes a Laplace transform of x (t), and s is the transformation complex of t.

The displacement response in Laplace domain then becomes:

0 0

2 2

2( ) ,

2

n

n n

x s xX s

s

(3.3)

and the time domain response becomes:

2

2 1 1( ) sin( 1 tan ).n

t

s nx t e tA

(3.4)

Here, As is the system constant determined by initial conditions and damping

characteristics. Thus the open-loop settling time can be derived as (examples can also be

found in Inman (2007)):

.

n

Open csT

t

(3.5)

Here, tc is the time constant. Its value is -ln5% (approximately 3) based on the definition

of settling time used here.

PPF Control

The first consideration is the PPF control methodology. PPF is popular for vibration

reduction since it offers rapid damping for a specific mode, is very stable, and is not

sensitive to spillover. It has been extensively applied in micro-vibration control of space

structures (Vaillon and Philippe (1999)), in suppressing thermally induced vibrations (

Friswell, Inman and Rietz (1997)), and for slewing flexible frames (Leo and Inman

(1994)). As illustrated in Figure 3.1(a), the PPF control algorithm introduces a second-

order filter G to the piezoelectrically coupled beam system H, which is fed back by the

41

sensed position signal. The position response of the filter is then fed back as a force input

to the structure. The PPF controller equation for the scalar case is given by:

2 22 .f f f f fu u u g x (3.6)

Here, u denotes the control input (actuation voltage), ζf and ωf are the generalized

damping ratio and the natural frequency of the controller respectively and gf is the control

gain. Note that the PPF controller is composed of a second order filter, the same form as

the structural equation but with much higher damping ratio (about 50 times higher in this

work to satisfy the settling time requirement). The transfer function of the PPF controller

in Laplace domain is given by:

2

2 2( ) .

2

f f

PPF

f f f

gG s

s s

(3.7)

The governing equations of the PPF closed-loop system, in free vibrations can be

expressed as:

2 2

2 2

2 0 0.

0 2 0

n n f f

f f f f f

gx x x

gu u u

(3.8)

In order to check system stability, Fanson and Caughey (1990) constructed a Lyapunov

function, and concluded that the PPF control system is stable if the determinant of

displacement coefficient matrix is positive, that is, if the following condition is satisfied:

2 2 2.f f ng (3.9)

The tip displacement response derived from equation (8) takes the following form:

1 12

1 1 1 21( ) sin( ) ( ).

t

fx t Ae t f

(3.10)

42

Here A1 and φ1 are system constants determined by initial conditions and damping

characteristics. ζ1 is the closed-loop damping ratio approximated to be ζf. The natural

frequencies of the PPF control system (ignoring damping effects) are the square roots of

the eigenvalues of characteristic polynomial of the mass normalized stiffness matrix:

2 2 2 2 2 4 2

2

1

2 2 2 2 2 4 2

2

2

( ) 4,

2

( ) 4

2.

n f n f f f

n f n f f f

g

g

(3.11)

Note that the natural frequency and damping ratio of the PPF control system 1 and ζ1 are

different from that of the open loop system n .The settling time and the relationship of

natural frequencies of PPF control system is of the form:

1 1

.PPF c

s

tT

(3.12)

PID Control

Next consider a PID control law, which remains an important control tool, due to its

simplicity and robust performance. The success of implementing the PID controller

depends on the appropriate choices of the PID gains. The transfer function of the PID

controller in Laplace domain is given by:

( ) ,i

PID p d

kG s k k s

s (3.13)

where, the proportional control gain kp is proportional to the size of the process error

signal, the integral control gain ki offers control correction by eliminating offset from a

constant reference signal value in the final state, and the derivative control gain kd uses

the rate of change of an error signal to predict the control action and tends to increase the

43

stability of the system. Since the effective damping coefficient of the PD control system

is thus: 2 ,n d

k rather than 2n

, and the settling time of the PID control system is thus

estimated to be:

.0.5

c

n d

PID

s

tT

k

(3.14)

However, the integral control gain ki may slowly decrease the amplitude. A MatLab code

is created and used to optimize these parameters in order to achieve the desired settling

time.

Nonlinear Control

The following consideration is a nonlinear control law. It has been shown that the system

response can be improved by constructing a nonlinear variable damping (using the

product of position and velocity), which eliminates overshoot (Castelazo and Lee

(1990)), improves the settling time, provides better system response than an optimal

linear controller (Rietz and Inman (2000)), and enhances robustness (Kuo and Wang

(1990)). In this chapter, this type of nonlinear controller is implemented by building a

position times velocity feedback and using a PD controller as the basis. The closed-loop

system is represented by:

2(2 | |) )( 0,n d nl n px k k x x k x (3.15)

where knl denotes the nonlinear gain. One should note that the position x in the damping

term could introduce negative damping to the system, which can lead to an unstable

response. Therefore the absolute value of position feedback is employed to avoid

negative damping. The effective damping coefficient of the nonlinear control is

approximately:( 0.5 )

2 n dk

n d nl nlk k A e

. The settling time is thus estimated by:

44

( 0.5 )/ ( 0.5 0.5 ).

n d nl nl

n dNL k

s cT k k At e

(3.16)

Here, Anl is the envelop amplitude determined by initial conditions and the system

damping characteristics. A MatLab code is created to validate and optimize these

estimated parameters in order to achieve the desired settling time.

LQR Control

The last conventional controller considered is the standard formulation of an LQR

optimal controller. The single input single output (SISO) control system from equation

(1) is:

22 ,n nx x x bu (3.17)

The state-space representation of this SISO system can be expressed as:

,

.

X AX BU

Y CX DU

(3.18)

Here, the state matrix A=[0,1;-ωn2, -2ζωn], the input matrix B=[0;b], b is the

electromechanical coupling coefficient (feedback constant) of the system. The output

matrix C = [1,0], and the throughput matrix D = [0]. The state vector X = [x; dx/dt], input

vector U = [u], and thus the output vector Y = [x].

The LQR approach entails the solution of an algebraic Riccati equation to obtain state

feedback gain G of the Linear Time Invariant system:

( ) ( ).Gu t x t (3.19)

The state feedback gain G is optimized to minimize the following cost (index) function:

45

0[ ( ) ( ) ( ) ( ) ( ) ( )] ,

s TT

TJ x Q x u R u d (3.20)

where the given constant penalty (weighting) matrices Q and R are all symmetric, with Q

positive semi definite and R positive definite. Choosing Q large relative to R means that

the response attenuation or minimal response will be more heavily penalized than the

control effort, and vice versa. In this chapter, the matrix Q is chosen as identity and the

scalar matrix R is tuned to be much larger to obtain the desired settling time with

reasonable control effort, while guaranteeing saturation-free control for a series of control

inputs. The minimized control input is:

1.

TG R B P

(3.21)

Here P is the unique positive semi-definite solution to the algebraic Riccati equation:

10.

T TA P PA PBR B P Q

(3.22)

An analytical expression for the settling time for a LQR controller was not found, but

numerical iteration is used in the design to satisfy the settling time constraint. Note that

each of the above controllers is designed to have the same settling time, but that each

design has a different closed loop frequency or/and damping ratio determined by the

controller dynamics. Equations (3.12), (3.14) and (3.16) are not used directly to design

for the same settling time as they are not exact, but rather they are presented to punctuate

the fact the different control laws can have the same settling time but with different

damping, and hence different actuation power dissipation because the controllers induce

different frequency shifts. Therefore each controller will consume a different amount of

actuation energy.

HYBRID BANG-BANG CONTROL SYSTEMS

Motivated by noting that the actuation voltage in most vibration suppression control laws

is relatively high during the early control periods, a bang-bang voltage clipping algorithm

which reduces the large amount of actuation power consumed in the early periods, is

examined to see if the total actuation power to obtain the same performance is possible.

Bang-bang control is usually employed to switch a control system from one state to

another. In this chapter, a bang-bang control algorithm is used to switch a control source

46

from the external voltage supply (constant) to a feedback signal from each conventional

controller. The purpose is not to obtain minimum time but minimum actuation power

requirement for vibration suppression. This objective is achieved by providing the

conventional controllers with a limited voltage boundary. The comparisons of the

conventional and hybrid single input single output feedback control systems are

schematically illustrated in Figure 3.1(a) and (b) respectively.

(a) (b)

Figure 3.1 Block diagrams of the (a) conventional and (b) hybrid control system.

Unlike a continuous feedback control law, the bang-bang method has a control threshold

value Vext. A switch position is changed between P1 and P2 to select the input control

signal, according to the following control logic:

When ( )u t > 0:

if ( )u t <Vext , connect to P1, input voltage u= ( )u t ; if ( )u t ≥Vext , connect to P2, input

voltage u=Vext ;

When ( )u t <0:

if ( )u t ≤Vext , connect to P2, input voltage u= Vext ;if ( )u t >Vext , connect to P1, input

voltage u= ( )u t .

This control logic sets up an upper and a lower bound on the voltage, in order to limit the

control input provided by the each conventional controller. The switch connects to the

external power supply if the input voltage reaches the limit value, and connects back to

the conventional controller once it drops in between the limits. Therefore, the logic is

Hu(t) x(t)

active

controllerG

+

r(t)beam w/ MFC

Hx(t)

active

controller

Vext

P1

P2

bang-bang

beam w/ MFC

G

r(t) u(t)

+

47

employed to control the switch in order to reduce the control input ( )u t when ( )u t is

positive, and increase it when ( )u t is negative. In this way, the electric current flows in

the desired circuit (P1 or P2 in Figure 3.1(b)) while vibration is suppressed more

effectively.

Experimental Results

The mechanical system of interest is a flexible cantilevered aluminum beam, controlled

in turn by each of the eight controllers studied in this work, with a focus on the

fundamental transverse vibration mode. Figures 3.2(a) and 3.2(b) show a picture and a

schematic diagram of the experimental setup, respectively. An aluminum beam, with an

MFC (Macro-Fiber Composite, originally developed by Wilkie et al. (2000) in NASA

Langley Research Center) patch attached to its root, is mounted vertically on an isolated

block. A magnetic tip mass is rigidly attached to the free end of the beam in order to

create an identical initial displacement condition for all the controllers, by means of an

electronically controlled electromagnet. This magnet (not shown in Figure 3.2(a) or (b))

is placed 4.8mm away from the tip mass allowing a repeatable release of the beam from a

fixed displacement with zero initial velocity.

(a) (b)

Figure 3.2 (a) Picture and (b) schematic diagram of experimental setup.

The tip displacement signal, measured with a MTI LTC-50-20 displacement laser sensor,

is low-pass filtered and fed into the controller. The control signal is low-pass filtered,

amplified 200 times with a TREK 2220 amplifier and fed back to the MFC actuator. The

dSPACE Board

Filter

Amplifier

Displacement Sensor

Tip Mass

Aluminum Beam

MFC

PC

Fixture

MFC

Beam

Fixture

Tip Mass

TREK Amp.

LPFilter

dSPACE Board

PC w/Simulink/Control Desk

Disp. SensorLP

Filter

48

control scheme in Matlab Simulink is implemented using the Control Desk software and

a dSPACE 1005 real time control board. The physical properties of the aluminum beam

are illustrated in Table 3.1. The MFC patch consists of rectangular piezoceramic rods

sandwiched between layers of adhesive and electrode polyimide film, with lateral

expanding motion and a collocated sensor Type S1. This film contains interdigitated

electrodes that transfer the applied voltage directly to and from the fibers of the

rectangular cross-section. Electrical impedance spectroscopy from 0.1 Hz to 100 kHz of

the MFC8528-P1 actuator by Bilgen, Wang and Inman (2011) showed that the

capacitance of MFC (4.05nF) is very low and remains constant during low frequency

band (0.1 Hz to 100 Hz). Their results also show that the MFC has the best control

effectiveness, compared with PSI PZT-5A, PSI PZT-5H, MIDE QP10N, due to its large

control authority (±500V voltage allowance) as a result of its larger d33 value.

Table 3.1Properties of the Beam and the Piezoelectric Transducer (MFC).

Physical property Value

Beam length x width 450mm x 28mm

Beam thickness 3.05mm

Beam mass M + tip mass Mt 139.4g+12g

MFC piezoelectric constant, strain/applied field d33 460(pm/V)

A single mechanical degree of freedom model (Equation (3.1)) is used here considering

only the fundamental vibration mode of the distributed-parameter model. The natural

frequency of the fundamental transverse vibration mode was experimentally identified as

12.0 Hz, using a Frequency Response Function (FRF) obtained from the Siglab. The

damping ratio was calculated as ζ=0.0037, using the quadrature peak picking on the FRF

plot.

Figure 3.3 shows the beam tip displacement for (a) the open-loop system with short-

circuit conditions (without feedback controllers), for (b) the PPF and the bang-bang-PPF

control system, for (c) the PID and the bang-bang-PID control system, for (d) the

nonlinear and bang-bang-nonlinear control system and for (e) the LQR and the bang-

bang-LQR control system. In all cases, the initial tip displacement is 4.8mm and the

49

initial tip velocity is zero.

Figure 3.3 Tip displacement measurements of the (a) Open-loop (b) PPF, Bang-bang-PPF

(c) PID, Bang-bang-PID (d) Nonlinear, Bang-bang-nonlinear (e) LQR, Bang-bang-LQR

control systems with identical initial conditions.

The settling time for the open-loop time response is 10.9s. The settling time for all the

control systems are all reduced to 0.85s as a design constraint (which is 92% of the open-

loop settling time). This design constraint is determined by the fastest response time for

all the conventional control systems under the largest actuation authority of MFC (-

500V~+500V). The parameters of each controller are tuned specifically to reach the same

settling time constraint. Figure 3 also illustrates that the conventional controls tend to

apply large forces to suppress vibration response to the final state (zero in this case), thus

using lots of energy early on. The hybrid control prevents this by limiting how much

control effort is expended in the early time intervals when the difference between the

initial state and final is large. Figure 3.4 compares the experimental time histories of the

control voltage of (a) PPF (solid line) and Bang-bang-PPF control system (dashed line);

(b) PID (solid line) and Bang-bang-PID (dashed line) control system; (c) Nonlinear (solid

line) and Bang-bang-nonlinear (dashed line) control system; (d) LQR (solid line) and

Bang-bang-LQR (dashed line) control system.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-5

0

5

Time(s)--(a)

Dis

p.(

mm

)

0 0.5 1 1.5 2-5

0

5

Time(s)--(b)

Dis

p.(

mm

)

0 0.5 1 1.5-5

0

5

Time(s)--(c)

Dis

p.(

mm

)

0 0.5 1 1.5 2-5

0

5

Time(s)--(d)

Dis

p.(

mm

)

0 0.5 1 1.5 2-5

0

5

Time(s)--(e)

Dis

p.(

mm

)

Open-loop

PPF

Bang-bang-PPF

PID

Bang-bang-PID

Nonlinear

Bang-bang-nonlinear

LQR

Bang-bang-LQR

50

Figure 3.4 Experimental actuation voltage histories for the (a) PPF, Bang-bang-PPF (b)

PID, Bang-bang-PID (c) Nonlinear, Bang-bang-nonlinear (d) LQR, Bang-bang-LQR


Figure 3.5 shows time-domain control current measurements of (a) PPF (solid line) and

Bang-bang-PPF control system (dashed line); (b) PID (solid line) and Bang-bang-PID

(dashed line) control system; (c) nonlinear (solid line) and Bang-bang-nonlinear (dashed

line) control system; (d) LQR (solid line) and Bang-bang-LQR (dashed line) control

system.

Figure 3.5 Experimental actuation current histories for the (a) PPF, Bang-bang-PPF (b)

PID, Bang-bang-PID (c) Nonlinear, Bang-bang-nonlinear (d) LQR and Bang-bang-LQR


0 0.2 0.4 0.6 0.8 1-500

0

500

Time(s)(a)

Vo

ltag

e(V

)

0 0.2 0.4 0.6 0.8 1-500

0

500

Time(s)(b)

Vo

ltag

e(V

)

0 0.2 0.4 0.6 0.8 1-500

0

500

Time(s)(c)

Vo

ltag

e(V

)

0 0.2 0.4 0.6 0.8 1-500

0

500

Time(s)(d)

Vo

ltag

e(V

)

PPF

Bang-bang-PPF

PID

Bang-bang-PID

Nonlinear

Bang-bang-nonlinear

LQR

Bang-bang-LQR

0 0.2 0.4 0.6 0.8 1

-0.5

0

0.5

Time(s)(a)

Cu

rre

nt(

mA

)

0 0.2 0.4 0.6 0.8 1

-0.5

0

0.5

Time(s)(b)

Cu

rre

nt(

mA

)

0 0.2 0.4 0.6 0.8 1

-0.5

0

0.5

Time(s)(c)

Cu

rre

nt(

mA

)

0 0.2 0.4 0.6 0.8 1

-0.5

0

0.5

Time(s)(d)

Cu

rre

nt(

mA

)

PPF

Bang-bang-PPF

PID

Bang-bang-PID

Nonlinear

Bang-bang-nonlinear

LQR

Bang-bang-LQR

51

The instantaneous electrical power Pins delivered to the MFC is calculated by multiplying

the measured control voltage ( )u t and current level ( )i t at each time step of the data (which

has a 1kHz sampling rate). The current is measured using the current monitor function

available on the TREK 2220 amplifier. The average power Pavg is defined as the time

average of the total electrical energy applied through the MFC until the settling time Ts is

reached. The expressions of Pins and Pavg are given respectively by:

( ) ( );insP u t i t (3.23)

0 0

1 1( ) ( ) .

s sT T

s s

avg insP P dt u t i t dtT T

(3.24)

Figure 3.6 compares instantaneous power of the (a) PPF (solid line) and Bang-bang-PPF

control systems (dashed line); (b) PID (solid line) and Bang-bang-PID (dashed line)

control systems; (c) Nonlinear (solid line) and Bang-bang-nonlinear (dashed line) control

systems; (d) LQR (solid line) and Bang-bang-LQR (dashed line) control systems.

Experimental comparisons show that the hybrid control systems reduce the actuation

voltage and actuation current, and thus the average actuation power while obtaining the

same vibration suppression performance. That is because the bang-bang control systems

cut off higher levels of actuation voltage during the early state of vibration control by

intermittent switching of the control sources. Even though, the current from the hybrid

controller becomes larger while the switch occurs, due to the sharp change in voltage, it

drops down rapidly after the switch finishes. Thus, the hybrid bang-bang controllers offer

more energy efficient control performance by reducing the average actuation power input

supplied to the piezoceramic transducer until the settling time is reached. The required

average actuation power of each controller is listed in Table 3.2, which is obtained by

measuring the time-domain current and voltage (for the interval 0 0.85st ). Table 3.2

also summarizes the maximum actuation voltage and the current of four conventional

controllers and their hybrid versions. It is observed that the proposed hybrid bang-bang

controllers consume much less power than their conventional versions. In particular, the

hybrid bang-bang-nonlinear control system requires 67.3% less power consumption

compared with its conventional counterpart.

52

Figure 3.6 Experimental instantaneous power consumption for the (a) PPF, Bang-bang-

PPF (b) PID, Bang-bang-PID (c) Nonlinear, Bang-bang-nonlinear (d) LQR and Bang-

bang-LQR control systems with identical initial conditions.

Table 3.2 Experimental comparisons of the PPF, Bang-bang-PPF, PID, Bang-bang-PID,

Nonlinear, Bang-bang-nonlinear, LQR and Bang-bang-LQR control systems.

Open-

Loop

PPF Bang-

bang-

PPF

PID Bang-

bang-

PID

Nonlinear Bang-

bang-

nonlinear

LQR Bang-

bang-

LQR

Initial disp.,

velocity(mm,

mm/s)

(4.8,0) (4.8,0) (4.8,0) (4.8,0) (4.8,0) (4.8,0) (4.8,0) (4.8,0) (4.8,0)

Settling time

Ts (s)

10.8 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85

Maximum

voltage (V)

N/A 450 130 450 130 450 130 450 130

Maximum

current (mA)

N/A 0.5 0.7 0.5 0.7 0.5 0.7 0.5 0.7

Average

power (mW)

N/A 10.6 5.73 15.5 6.47 16.7 5.46 15.52 6.20

0 0.2 0.4 0.6 0.80

50

100

150

200

Time(s)(a)

Inst.

Po

we

r(m

W)

0 0.2 0.4 0.6 0.80

50

100

150

200

Time(s)(b)

Inst.

Po

we

r(m

W)

0 0.2 0.4 0.6 0.80

50

100

150

200

Time(s)(c)

Inst.

Po

we

r(m

W)

0 0.2 0.4 0.6 0.80

50

100

150

200

Time(s)(d)

Inst.

Po

we

r(m

W)

PPF

Bang-bang-PPF

PID

Bang-bang-PID

Nonlinear

Bang-bang-nonlinear

LQR

Bang-bang-LQR

53

NUMERICAL SIMULATIONS

Numerical simulations of these eight controllers for the given system parameters are

performed using MATLAB Simulink. During the experiments, it was noted that the

piezoelectric constitutive equations coupling the strain field and the electric field behave

nonlinearly. This has also been reported in literature. Viehland (2006) experimentally

investigates the behavior of longitudinal piezoelectric coefficient d33 of various

ferroelectric ceramics and single crystals and indicates that the d33 performs uniaxial

stress-dependent behavior. The electromechanical coupling coefficient of MFC 8528 P1

is identified as actuation voltage dependent experimentally, which is understood as the

stress dependent behavior of d33 of MFC P1. Bilgen (2010) presents the constitutive

nonlinear effects for MFC 8507 P1 bimorphs (that arise from piezoceramic hysteresis),

and identifies the camber vs. voltage from open-loop responses. Note that MFC 8507 P1

is also d33 effected but is of different size than the 8528 P1 used in this chapter. His

curved shape is as same as our identification. To account for this voltage-dependent

electromechanical coupling coefficient in real time, a corresponding state variable

algorithm is developed correspondingly.

Simulations with Voltage-independent Electromechanical Coupling

The numerical simulations of these eight controllers exhibit a quantitative discrepancy

from the experiments when the system electromechanical coupling coefficient (feedback

constant) b is empirically identified as a constant of 2mm/Vs2 by matching the simulation

settling time of the PPF control system the same as the experimental measurements.

Figure 3.7 compares numerical simulations with experimental measurements (solid line)

of tip displacement responses, control voltage and control current for the (a) PPF and (b)

Bang-bang-PPF control systems. The comparisons demonstrate that discrepancies of the

tip displacement, control voltage and control current of both control systems arise at high

actuation voltage. The discrepancies also exist for the PID, the nonlinear, the LQR

controllers and their hybrid control versions.

54

(a) PPF (b) Bang-bang-PPF

Figure 3.7 Numerical and experimental comparisons of tip displacement, control voltage

and control current of (a) PPF and (b) Bang-bang-PPF control systems.

For simplicity, we display only the tip displacement and control voltage comparisons for

the other controllers, which are shown in Figure 3.8. In these graphs, the solid line

denotes experimental tip displacement and control voltage for (a) the PID (b) the Bang-

bang-PID (c) the Nonlinear (d) the Bang-bang-nonlinear (e) the LQR and (f) the Bang-

bang-LQR control systems. The discrepancy occurs because the longitudinal

piezoelectric coefficient d33 of MFC 8528 P1 from the datasheet (see Table 3.1) is voltage

dependent.

(a) PID (b) Bang-bang-PID (c) Nonlinear

0 0.2 0.4 0.6 0.8-5

0

5

Time(s)Dis

pla

cem

ent (

mm

)

0 0.2 0.4 0.6 0.8

-500

0

500

Time(s)

Voltage

(V)

0 0.2 0.4 0.6 0.8

-0.5

0

0.5

Time(s)

Curr

ent (

mA

)

0 0.2 0.4 0.6 0.8-5

0

5

Time(s)Dis

pla

cem

ent (

mm

)

0 0.2 0.4 0.6 0.8

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0

200

Time(s)

Voltage

(V)

0 0.2 0.4 0.6 0.8

-0.5

0

0.5

Time(s)

Curr

ent (

mA

)

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Experimental

Numerical

Experimental

Numerical

Experimental

Numerical

Experimental

Numerical

Experimental

Numerical

Experimental

0 0.2 0.4 0.6 0.8-5

0

5

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Dis

p.(m

m)

0 0.2 0.4 0.6 0.8

-500

0

500

Time(s)

Vol

tage

(V)

0 0.2 0.4 0.6 0.8-5

0

5

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Dis

p.(m

m)

0 0.2 0.4 0.6 0.8

-200

-100

0

100

200

Time(s)

Vol

tage

(V)

0 0.2 0.4 0.6 0.8-5

0

5

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Dis

p.(m

m)

0 0.2 0.4 0.6 0.8

-500

0

500

Time(s)

Vol

tage

(V)

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Experimental

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Experimental

Numerical

Experimental

Numerical

Experimental

Numerical

Experimental

Numerical

Experimental

55

(d) Bang-bang-nonlinear (e) LQR (f) Bang-bang-LQR

Figure 3.8 Numerical and experimental comparisons of tip displacement and control

voltage of (a) PID (b) Bang-bang-PID (c) Nonlinear (d) Bang-bang-nonlinear (e) LQR

and (f) Bang-bang-LQR control systems.

State Variable Simulation with Voltage-dependent Electromechanical Coupling

The analytical and experimental characterization of a MFC 8507 P1 actuated cantilever

unimorph beam is investigated by Bilgen, Erturk and Inman, (2010). In their paper, the

electromechanical coupling coefficient (feedback constant) b of Equation (3.17) is found

as a linear function of longitudinal piezoelectric constant d33. Again, the MFC 8507 P1 is

also d33 effected but is of different size than the 8528 P1 used in this chapter. This linear

equation is cited and re-expressed in the following form:

2 2

33 1

0

[( ) ( ) ]( ) ( ) .

2

t d p c p

p

e

h h h h hb u Y d u

r r

(3.25)

Here, Yp is the Young’s modulus of the MFC, ht and hp are the total width and thickness

of the MFC, hd and hc are the position of the top and bottom of the MFC layer from the

neutral unimorph beam axis, re is the width of each electrode in the longitude beam axis,

r0 is the width of each non-electrode region in the longitude beam axis, α is

experimentally identified as 0.20 in their chapter. The coefficient in the fundamental

0 0.2 0.4 0.6 0.8-5

0

5

Time(s)

Dis

p. (

mm

)

0 0.2 0.4 0.6 0.8

-200

0

200

Time(s)

Volta

ge

(V)

0 0.2 0.4 0.6 0.8-5

0

5

Time(s)

Dis

p. (

mm

)

0 0.2 0.4 0.6 0.8

-200

0

200

Time(s)

Volta

ge

(V)

0 0.2 0.4 0.6 0.8-5

0

5

Time(s)

Dis

p. (

mm

)

0 0.2 0.4 0.6 0.8

-500

0

500

Time(s)V

olta

ge

(V)

Numerical

Experimental

Numerical

Experimental

Numerical

Experimental

Numerical

Experimental

Numerical

Experimental

Numerical

Experimental

56

mode Γ1 is derived from:

1 0 1 1 1 2 1 3 1

1 0 1 3 2

3

0

( ) ( ) ( ) ( ) ( )( ).|

Ne

i i i i

i i i i i

i

i

x

x

x x x x d x

x x x x dx

(3.26)

Where Ne is the number of electrode pairs over the active beam length l, which is

approximate to be l/2(re+r0) and Φ1(x) is the mass normalized eigenfunction of the

cantilevered beam for the first mode in longitude beam coordinate x. For each beam

length period i, xi0, xi1, xi2 and xi3 are divided so that at an arbitrary instant of the motion,

the electric potential is assumed to be linearly decreasing in xi0 ≤ x ≤ xi1, whereas it is

assumed to be linearly increasing in xi2 ≤ x ≤ xi3. Due to this linear relationship, the

voltage-dependent behavior of d33(u) mentioned earlier in literature causes the coupling

coefficient, b(u) to vary nonlinearly. The experimental displacement responses of the PPF

controller are used to empirically identify the b(u) term as a function of actuation voltage

u(t). These empirically estimated values of b(u) for the PPF control system are compared

with the numerical simulation of the free vibration response of the fundamental mode.

This procedure is repeated until the simulation and experimental measurements converge

at this voltage level. The iterative process is performed for the voltage range of interest.

An approximate analytical function (using a curve fitting tool in MATLAB) is acquired

based on empirically identified data:

( ) ,f fb u d u

f fb u a e c e (3.27)

where the curve fit coefficients af, bf, cf and df are real valued constants. The fit function

agrees very well with experimental measurements (see Figure 9). The discrete data

denote experimentally identified values of the coupling coefficient b. The solid line is the

curve fit with a root mean squared error of 0.1431 and the coefficient of multiple

determination of 0.9965 (for the coefficients of af = 2.124, bf = 0.003, cf = 1.354e-15 and

df = 0.0775).

57

Figure 3.9 Variation of the electromechanical coupling coefficient (feedback constant)

with actuation voltage.

A state variable or adaptive control algorithm is then developed in order to account for

the voltage-dependent behavior of b. The general idea of state variable control is to create

a controller with parameters updated in real time to change the system response. In this

work, the parameters are updated based on actuation voltage, using the analytical curve

fit function given by Equation (3.27). As shown in Figure 3.10, the variation in the

coupling coefficient from the control system is adjusted by a state variable or adaptive

control algorithm GA with respect to the change of control voltage as demanded by GP

(the PPF, the PID, the nonlinear, the LQR conventional controller or their hybrid

versions). The state variable algorithm GA is used to determine an appropriate coupling

coefficient at any voltage level, and the voltage dependent variation Adj determined by GA

adjusts value of b by the variation due to the change of actuation voltage. The objective

of this state variable control system is to compensate for the voltage-dependent

electromechanical coupling coefficient in real time for better prediction of the

experimental results.

Figure 3.10 Block diagram of the state variable or adaptive control system.

0 100 200 300 400 5000

5

10

15

20

25

30

Votage(V)

Co

up

ling

Co

eff

icie

nt(

mm

/Vs

2)

Experimental

Curve fit

Hu(t) x(t)

Active/Hybrid

Controller

GP

Beam w/ MFC

GA Adj

Adaptive

Algorithm

Adjustment

58

Figure 3.11 compares the numerical simulations using state variable or adaptive

controllers of the tip displacement, actuation voltage and actuation current with the

experimental measurements (solid line) for (a) PPF and (b) Bang-bang-PPF control

systems. Note that the bang-bang switching algorithm of the hybrid controller is

employed before the state variable or adaptive control.

(a) PPF (b) Bang-bang-PPF

Figure 3.11 State variable numerical and experimental comparisons of tip displacement,

control voltage and control current of the (a) PPF (b) Bang-bang-PPF control systems.

As can be seen in Figure 3.11, after state variable compensation, numerical simulations of

tip displacement response, control voltage and control current histories of both PPF and

hybrid bang-bang-PPF control systems can predict experimental results precisely. Note

that, although the voltage-dependent electromechanical coupling given by Equation

(3.27) is identified from the voltage history of the PPF control system, the function works

successfully for the hybrid bang-bang-PPF control system as well. This is also true for

the PID, the nonlinear, the LQR and their hybrid control versions.

0 0.2 0.4 0.6 0.8-5

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5

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)

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Experimental

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500

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ge

(V)

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)

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)

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ent (

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)

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Experimental

59

(a) PID (b) Bang/bang PID (c) Nonlinear

(d) Bang-bang-nonlinear (e) LQR (f) Bang-bang-LQR

Figure 3.12 State variable numerical and experimental comparisons of tip displacement

response and control voltage of the (a) PID (b) Bang-bang-PID (c) nonlinear (d) Bang-

bang-nonlinear (e) LQR and (f)Bang-bang-LQR control systems.

For simplicity, Figure 3.12 only compares tip displacement response and control voltage

of the state variable numerical simulations and experimental measurements (solid line)

for (a) the PID (b) the bang-bang-PID (c) the nonlinear (d) the bang-bang-nonlinear (e)

the LQR and (f) the bang-bang-LQR control systems. This section therefore shows the

need for considering piezoelectric nonlinearity for reliable simulations since the actuation

voltage levels can be high enough to require it.

0 0.2 0.4 0.6 0.8-5

0

5

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Dis

p.(m

m)

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tage

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)

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p. (

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)

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200

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ge

(V)

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0

5

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Dis

p. (

mm

)

0 0.2 0.4 0.6 0.8-5

0

5

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isp. (

mm

)

0 0.2 0.4 0.6 0.8

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0

200

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ge

(V)

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Experimental

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60

CHAPTER SUMMARY

In this chapter, PPF, PID, nonlinear and LQR controllers and their respective hybrid

bang-bang control versions (REC) are used for vibration suppression and compared in

terms of their energy consumption. Experimental results for a thin cantilever with a

piezoceramic patch show that the REC laws require much less power than their

conventional versions. In particular, the hybrid nonlinear control system requires 67.3%

less power than its conventional counterpart (that uses the product of the displacement

and velocity feedback). This reduction is achieved by employing a bang-bang strategy to

control a switch between a fixed external voltage and the control input provided by the

PPF, the PID, the nonlinear or the LQR controllers. The switching strategy cuts down the

overall energy consumption by reducing maximum control voltage and control current.

The comparisons of numerical simulations and experiments indicate that the piezoelectric

constant of the MFC actuator is voltage dependent as a result of the piezoelectric

nonlinearity. A series of discrete values of electromechanical coupling coefficient

(feedback constant) are empirically identified and curve fit as an analytical exponential

function of the instantaneous control voltage. A state variable control algorithm is then

developed to compensate for the voltage-dependent behavior of electromechanical

coupling. The numerical simulation results of the tip displacement, actuation voltage, and

control current of all these eight controllers using state variable control and are then

successfully compared against the experimental results.

61

CHAPTER 4 SIMULTANESOU ENERGY HARVESTING

AND GUST ALLEVIATION FOR A MULTIFUNCTIONAL

WING SPAR

This chapter examines the concept and design of a multifunctional composite for

simultaneous energy harvesting and vibration control. The intention is to design a

composite wing spar for a small UAV which is able to harvest energy itself from ambient

vibrations during normal flight. If the wing experiences any strong wind gust, it will

sense the increased vibration levels and provide vibration control to maintain its stability.

The proposed multifunctional composite wing spar integrates a flexible solar cell array,

piezoelectric wafers, a thin film battery and an electronic module into a composite

structure. The piezoelectric wafers act as sensors, actuators, and harvesters. The basic

design factors are discussed for a beam-like multifunctional wing spar with energy

harvesting, strain sensing and self-controlling functions. The configurations, locations

and operating modes of piezoelectric transducers are discussed in detail for optimal

power generation.

The equivalent electromechanical representations of a multifunctional wing spar is

derived theoretically and simulated numerically. Special attention is given to the self-

contained gust alleviation with the goal of using available energy harvested from ambient

vibrations. A reduced energy control law developed recently in chapter 3 is implemented

to minimize the actuation energy and the dissipated heat. This law integrates saturation

control with a positive strain feedback controller, and is represented by a positive

feedback operation amplifier (op-amp) and a voltage buffer op-amp for each mode. This

work builds off of our previous research and holds promise for improving UAV

performance in wind gusts. Here we also include, but not use, a flexible solar panel in our

modeling. However, our focus is on using vibration based energy harvesting.

62

NOMENCLATURE

excited base acceleration

cross section area of PZT-based transducers

cross section area of beam like wing spar structure

width of specimens for bending tes

b

p

s

a

A

A

b

ts

thickness of specimens for bending tests

piezoelectric elastic stiffness constant for different operating mode

damping matrix in terms of a series of finite compone

kk

c

c

C nts

inherent capacitance of PZT-based transducers

elastic modulus

elastic modulus of specimens for bending tests

internal energy

base excited force

p

t

ie

C

E

E

E

f

2

p

base excited force matrix

acceleration gravity 9.81 m / s

wind gust transfer function

electrical current

second moment of inertia

couplin

v

g

G

i

I

J

f

g term for PZT-based transducers

stiffness for finite element component

stiffness matrix in terms of a series of finite components

wind gust gain

effective leng

g

k

k

l

K

th of specimens for bending tests

vertical scale of atmospheric turbulence

mass for finite element component

mass matrix in terms of a series of finite components

vL

m

P

M

transverse load for bending tests

failure load for bending tests

instantaneous power

structural strain component

time

kinetic energy

f

ins

xx

e

e

P

P

S

t

T

U

0

potential energy

aircraft trim velocityU

63

output gust signal

electrical voltage

absolute transverse displacement

base transverse displacement

relative transverse displacement

g

abs

b

rel

V

V

w

w

w

x

longitude coordinate

transverse coordinate

load impedance

damping ratio of the beam-like structure

generalized damping ratio of the PSF controller

f

y

Z

θ

device coupling coefficient in terms of a series of finite components

RMS vertical gust velocity

mass density

crosshead displacement specimen midpoint deflection

v

modal coordinates for mode

modal coordinates in terms of a series of finite components

admissible trial function of the cantilevered beam for the mode

th

r

th

r

r

r

η

natural frequency of the beam-like structure

generalized natural frequency of the PSF controllerf

INTRODUCTION

Gusts produce vibrations that can easily lead to stability and functional problems of small

UAVs. Gust response can detected directly and alleviated effectively to improve the

flight performance. Gust alleviation has been investigated years for large aircrafts, see

Fazelzadeh et. al (2008) for references. However, the current trend towards the increasing

use of small UAVs has renewed the interest in gust alleviation by virtue of the

differences between the effects of wind gusts on traditional aircraft and small UAVs. For

many of the applications, UAVs are designed relatively smaller and lighter, which

increases their susceptibility to variations in wind conditions. For many of the missions,

UAVs are required to fly at low levels and in urban or mountainous environments, which

expose them mostly in wind conditions. There are numerous active control methods of

64

gust alleviation. However, the energy constrain still remains the major issue for small

UAVs. A wing spar is used for fixed-wing aircraft, which includes some unmanned

designs. The increasing need for high resilience and flexibility lightweight structures in

aircraft wings raises structural vibration control issues induced by flutter mechanisms and

gust perturbations. This chapter focuses on a multifunctional wing spar design for an

Unmanned Aerial Vehicle (UAV), carrying on functions of energy harvesting from

normal vibration or sunlight, sensing ability for wind disturbances, and gust alleviation in

the application of REC laws developed in chapter 2 which uses available energy

harvested from ambient vibration.

Active control laws using external power supplies have been applied to the PZT-based

actuators for flutter suppression and gust alleviation, as reported by Suleman and Costa

(2004), Nam, Kim and Weisshaar (1996) and Fazelzadeh and Jafari (2008). However, the

use of batteries as power sources is a critical issue for sensing and control systems due to

their bulk size and limited lifetime, as discussed by Liao, Wang and Huang (2001).

Recent developments in microelectronics have made low power requirements possible

(on the order of several hundred microwatts) for signal conditioning electronics, as

reported by Chandrakasan et al. (1998). Therefore, current research mostly focuses on

harvesting energy from ambient environments instead of using batteries (Amirtharajah

and Chandrakasan (1998)). In the future, smart devices are expected to operate

autonomously, and facilitate self-sensing, self-evaluation, self-controlling and self-

powering from the ambient environment, such as vibrations and/or sunlight. Most recent

investigations on multifunctional structures have only focused on the design of self-

charging devices, such as Anton, Erturk and Inman (2010), Sodano, Inman and Gyuhae

(2005a), Yabin and Sodano (2008) and Yirong and Sodano (2009). Some others focused

on the sensitivity studies of self-powered sensors, such as Ng and Liao (2005) or the

collocated self-sensing piezoelectric actuators, such as Dosch and Inman (1992), Ji et al.

(2009d) and Ji et al. (2011a).

65

The objective of this chapter is to examine the concept and design of a multifunctional

composite sandwich structure for simultaneous energy harvesting and vibration control.

The motivating application is the multifunctional wing spar design of a UAV (as shown

in Figure 4.1) with the goal of providing self-contained gust alleviation. In particular, the

wing itself is able to harvest energy from normal vibration or sunlight, sense the wind

disturbances, and alleviate wind gust by the application of reduced energy control (REC)

laws developed in Chapter 3, which is supplied by available energy harvested from

ambient vibration.

Figure 4.1 Multifunctional wing spar design showing various functionalities including

self-sensing, self-harvesting, self-storage and self-control.

For weight reduction and strength purposes, the beam-like multifunctional wing spar in

Figure 4.1 is designed to fit on a fiberglass composite substrate (E) of 17.8g, with total

length of 735mm, width of 38mm and thickness of 2.38mm. The harvesting, sensing and

actuating PZT layers are placed at the root of the wing spar since this section will

experience the largest strain during normal wing vibration or wind gust disturbance. The

PZT-based harvester/sensor (B) layered on the top surface of the fiberglass substrate uses

monolithic PZT (QuickPack® QP10n). The Micro-Fiber Composite MFC 8528 P1 is the

PZT actuator (F) layer on the bottom surface of the fiberglass substrate. The MFC was

developed in the NASA Langley Research Center by Wilkie et al. (2000). The use of

interdigitated electrodes and piezofibers improves their actuation authority and

performance of MFCs, and therefore they become commonly used for vibration control,

as reported by Sodano, Park and Inman (2004c) and Bilgen et al. (2011). The thin film

battery (C) allows for power storage from harvesting and provides energy supply for

wind gust alleviation. The electronic module combines conditioning, sensing and control

x

z

A. Flexible Solar Panel B. QP16N (Harvester, Sensor)

C. Thinergy Thin Film Battery D. Printable Circuit Board (PCB)

E. Fiberglass Composite Substrate F. MFC(Actuator)

L1=25mm

L2=94.6mm

L3=110mm L4=735mm

G. Epoxy DP 460, Kapton

66

circuitry, on a single layer of Printable Circuit Board (PCB). These multifunctional

layers, together with the fiberglass composite substrate, form the multi-layer wing spar.

3M ScotchWeldTM

DP460 epoxy, bracketed by Kapton, is used in the individual layer

layup. Both of them are grouped together as layer G. No attempt is made here to study

energy storage, but rather composite beam model is used here to illustrate the concept of

multi-functionality. This multifunctional wing spar builds off an earlier self-charging

prototype and a remote control aircraft test platform developed in Anton et al. (2010), as

shown in Figure 4.2.

Figure 4.2 A composite spar for a small remote control aircraft( Anton et al. (2010)).

The remainder of this chapter is organized as follows: section 2 presents the

electromechanical cantilever beam model of a multifunctional wing spar using assumed

modes; section 3 discusses the design factors of the electromechanical composite

sandwich wing spar to obtain optimal harvested energy; section 4 investigates power

flows of simultaneous gust alleviation and energy harvesting using REC control laws;

section 5 summarizes and discusses the theoretical and numerical results.

ELECTROMECHANICAL CANTILEVER BEAM MODEL OF A

MULTIFUNCTIONAL WING SPAR USING ASSUMED MODES

A major research issue in energy harvesting theories is the accurate modeling of

piezoelectric generators, as discussed by Liang and Liao (2011). Experimental

measurements on an aluminum baseline bimorph cantilever thin beam show that the

distributed parameter model is more accurate than single degree freedom lumped model,

wing spar

67

as reported by Erturk and Inman (2011). Later, this distributed parameter method was

applied by Anton and Inman (2011) to a bimorph self-charging structure layered on an

aluminum beam. However, the electromechanical cantilever beam model with collocated

sensing/harvesting and self-controlling for multiple modes has yet to be investigated.

This chapter employs a modeling method of assumed modes based on Rayleigh-Ritz

formulation for a multifunctional wing spar. The governing equations involve not only

the self-charging but also the self-sensing and the self-actuating functions of PZT

transducers. In addition to the fundamental mode, the second dominant mode of the

multifunctional wing spar is also predicted and controlled using REC laws.

Electromechanical Energy Components Using Distributed-Parameter Method

The motion of the wing spar is electromechanically coupled with the piezoelectric

transducers through strain. A closed-form electromechanical cantilever beam model is

developed based on Euler-Lagrange equations (Appendix D) that captures the basic

piezoelectric constitute equations (Appendix A). The distributed-parameter variable

relative vibration response wrel (x,t) at any longitude point x and time t can be represented as

a finite series of admissible trial functions Φr (x) and unknown modal coordinates ηr(t) for

rth

mode:

1

( , ) ( ) ( ),

N

r

rel r rw x t x t

(4.1)

where N is the number of terms required for convergence. The admissible trial function

Φr (x) has to satisfy the boundary conditions for cantilever beams. A advantageous trial

function (Meirovitch (2001)) to be used here is represented by:

(2 1)1 cos( ),

2r

r x

L

(4.2)

Substituting above assumed mode solutions into the kinetic, potential and internal energy

components introduced in Appendix D, one yields the energy expressions for

piezoelectric transducers. They are given separately as follows:

68

, , , ,

, 1

1[ ( ) ( ) ( ) ( ) ]

2.

N

o a o a o a o a

i l il i i

i l

U t t k t v t

(4.3)

Here U represents the potential energy, and v stands for the voltage across the PZT layer.

The superscript o stands for the output transducer (harvesting or sensing). The superscript

a denotes the actuating transducer. The stiffness component

,o a

ilk is defined as:

,

21 3

,

0 1

'' '' '' ''.

E o a

kk p

L L L

o a

il s s

L

i l i lk E I dx I dxc (4.4)

Here EI stands for the bending stiffness, (elastic modulus E multiply by second moment

of inertia I), which is taken as constant across each of individual longitudinal sections.

The bending stiffness of each longitudinal section is estimated using cross-section

transformation method(Vable (2002)) introduced in Appendix C, which is simple version

of modulus-weighted section method (Allen and Haisler (1985)), limited to rectangular

cross section Euler-Bernoulli beam like structures. The space-dependent bending stiffness

can be represented using Heaviside step functions, described in Appendix D.

The subscripts s and p represent the multifunctional structure and piezoelectric layers,

respectively. A prime denotes ordinary differentiation with respect to the spatial variable

x. Here, ckk is the piezoelectric elastic stiffness constant (k = 1 for 3-1 mode, 3 for 3-

3 mode). The superscript E means the corresponding parameters at constant electric

field. Note that the piezoelectric materials exhibit different piezoelectric constants due to

different poling direction of the material (Appendix A). For example, the monolithic d31

domain piezoelectric harvester and sensor QP10n satisfies the constitute equations given

in Equation (A.1). However, the MFC 8528P1 uses the d33 operating mode, which is

deformed and excited both in the 3-direction shown in Figure A.1 (b), the reduced

constitutive equations satisfy Equation (A.3).

The coupling coefficient ,o a

ie is defined by:

69

2, 3

, ,

1

''.

L L

o a o a

ie p

L

iJ dx (4.5)

Here the definition of coupling term Jp of the piezoelectric transducer over its cross

section area Ap is given in Appendix D.

If wb(x,t) denotes the base displacement of the wing spar in the absolute frame of the

reference at any longitude point x and time t, the total kinetic energy becomes

, , 2

, 1 0 0

1 1( ( ) ( ) 2 ( ) ( ) ) ( )( ) .

2 2

L LN

o a o a b b

i l il i s s

i l

e i

w wT t t m t A x dx A x dx

t t

(4.6)

Here ρ stands for the mass density and A is the cross section area of the beam like

structure in x direction. An over-dot stands for ordinary differentiation with respect to the

time variable t. The mass component, represented by ilm is defined as:

0

,( )

L

il s i lm A x dx (4.7)

If Cp denotes the piezoelectric capacitance, (see Appendix D for definition), the internal

energy Eie becomes

, , , , ,

1

1( ).

2

No a o a o a o a o a

ie i p

i

iE v C v

(4.8)

Solving Electromechanical Euler-Lagrange Equations for Piezoelectric Harvesters

After substituting each energy component back into the electromechanical Euler-

Lagrange Equations given by (B.4) in Appendix B, one can obtain the governing

equations for piezoelectric harvesters:

70

1

1

0,

( ) 0.

No o

il l il l i i i

l

o No o o o

p i ioi

m k v f

vC v

Z

(4.9)

Here Co and Z

o denote the piezoelectric capacitance and output circuit impedance. The

forcing term fi induced by base excitation is given by:

2

2

0

( ) ( ) .

L

b

i s i

wf A x x dx

t

(4.10)

Additionally, the distributed damping parameters are important for accurate model

prediction, as discussed by Erturk and Inman (2008a). Cudney and Inman (1989)

proposed a two parameter damping model which includes viscous and strain-rate

damping terms, and demonstrated that it provided the best fit to measure modal data

compared to other damping methods. However, the two-parameter damping model can

only reproduce the measured damping ratios to within 85%. Here the commonly used

Rayleigh damping is implemented, which assumes that the damping matrix is

proportional to the mass and stiffness matrices, given as follows:

. C M K (4.11)

The damping matrix must satisfy the following properties in order to uncouple the modal

equations:

22 .T

ni i i i ni C (4.12)

Here ɷni and ζi represent the ith

natural frequency and its associated damping ratio. The

damping ratios associated with the 1st and 2

nd modes, were measured as 0.37% and 0.1%

respectively, using quadrature peak picking on the frequency response function (FRF)

plot obtained from Siglab 20-42. Therefore, the coefficients κ and γ are determined as

71

1.34 and 1.89e-7

correspondingly. After introducing the damping components and

rewriting the Euler-Lagrange solutions (4.9) in matrix form, one obtains:

,o

o

Mη Cη Kη f v (4.13)

0.o o

p oC

Z

o

o ovv η (4.14)

Assuming the ambient normal vibration working on the clamped end of the wing spar

wb(t) has a harmonic form, the wing vibration base acceleration ab(t) and the forcing

vector fi(t) will have a harmonic form as well, given by:

( ) , ( ) , ( ) .j t j t j t

b b b b i i bw t w e a t a e f t m a e (4.15)

This harmonic forcing excitation leads to a harmonic solution for the generalized

coordinates and the voltage, as well, of the form:

( ) , ( ) . j t j tVt e t ev η η (4.16)

The steady-state forms of equations (4.13) and (4.14) then become:

2( ) ,Vj

o o oM + C + K η = f +θ

(4.17)

1( ) 0.o

p oVj C j

Z o o o

+ + θ η (4.18)

By solving steady-state equations, one can derive the output voltage, which is given by:

,1o

p o

Tr

Vj

j CZ

o oo θ η

+

(4.19)

The modal coordinate matrix ηo is represented by:

72

12( ) .1o

p o

Trjj

j CZ

o oo θ θη = M+ C+K + f

+

(4.20)

Here the superscript Tr stands for transpose. Substituting equation (4.20) into equation

(4.19), leads to the output voltage-to-base acceleration FRF:

12( )( ) .

1 1

o

o ofp po o

Tr Tr

j t

v j jj

j C j CZ Z

a e

o o oθ θ θ

= M + C + K + M

+ +

(4.21)

Substituting equation (4.20) into the assumed series solution given by equation (4.1),

leads to the relative displacement-to-base acceleration FRF:

12( , )( )( ) .

1Trel

obp o

Tr

j t

w x t jx j

j CZ

a e

o oθ θ

=Φ M+ C+K + M

+

(4.22)

As an alternative approach, a Finite Element Method (FEM) is implemented in Matlab to

capture three-dimensional behavior of the beam-like homogenous wing spar. The

geometry and material properties of each element are first solved for a convenient

coordinate system shown in Figure 4.1. The spar is discretized along the main longitude

axis using thin beam finite elements with a uniform cross-sectional shape but different

mass distribution within each element. These properties are then transformed with respect

to the centroid coordinate using the parallel axis theorem. Figure 4.3 compares the

relative displacement-to-base acceleration FRF using the analytical formulation and the

FEM modeling. The base acceleration is represented in terms of the acceleration of

gravity, g = 9.81 m/s2. The approximate analytical model is within 5% root mean square

(RMS) error of the FEM model.

73

Figure 4.3 Relative tip frequency response function using both analytical and FEM

modeling.

Design Considerations for a Multifunctional Composite Wing Spar

The increasing trend of lighter and stronger structures can be easily achieved by the use

of composite materials. Sandwich composites are one of popular examples which have

been tailored for many specific applications. They are manufactured with the central part

of the sandwich (core) bonded between two facing layers (skins), whose major advantage

is their high flexural stiffness to weigh ratio. The primary demands on facing materials

are high tensile and compressive strength, high stiffness and high flexural rigidity and so

on. Fiber reinforced composites with a low-density core find increasing use in both

civilian and military applications since they potentially provide more durable replacement

for aluminum or steels in primary aerospace, marine or automotive structures, see Gao et

al. (2009) and Hull and Clyne (1996) for examples. In sandwich structure composites, a

wide variety of materials are used as cores, such as honeycombs, low-density foams and

syntactic foams, which are required to possess a low specific weight and an adequate

shear stiffness, see Price and Nelson (1976) for references. In this project, a honeycomb

core fiberglass composite sandwich panel manufactured by Vopsaroiu et al. (2011)

Composite is used as the substrate of the multifunctional wing spar. This honeycomb core

fiberglass panels are sandwiched between two layers of uni-directional fiberglass pre-

101

102

-140

-130

-120

-110

-100

-90

-80

-70

Frequency [ Hz ]

Re

lative T

ip D

ispla

cem

ent F

RF

[ d

B r

ef 1 m

/g ]

Analytical

Finite Element Method

74

preg and an epoxy film adhesive. The fiberglass plies are layered up at 0° and 90° to

produce high strength-to weight and rigidity-to weight ratios. Recently, significant efforts

have been made for energy harvesting optimization, as reviewed in Chapter2. In

particular, Sodano et al. (2005b) have investigated the transduction performance of three

commercial piezoelectric devices: PSI-5H4E, QP, and MFC in order to recharge nickel

metal hydride batteries. Their experimental studies showed that QP and PSI- 5H4E are

more efficient in the random vibration environment. Bilgen (2010) identified harmonic

vibration-based harvesting characteristics of MFC, PZT-5H, PZT-5A and PMN-PZT with

respect to different configurations and material properties of the host structure. He

concluded that monolithic polycrystalline and single crystal piezoelectric offer better

harvesting capabilities than the MFC. But the MFC is more practical for large strain

aerodynamic applications, since it has much higher actuation voltage range.

This section focuses on ambient vibration transduction caused by atmospheric turbulence.

The ambient vibration source is simulated using Dryden’s PSD function (Appendix E).

The first comparison is given to two MFC devices under the 3-1 and the 3-3

operating modes. MFC 8528P1 and MFC 8528P2 are taken as examples for the 3-3

and the 3-1 operational modes, respectively. Their configurations and material

properties are listed in Table 4.1. One can see that the former has higher piezoelectric

constant, coupling coefficient, but much lower capacitance.

Table 4.1Selected properties of compared piezoelectric transducers.

Devices MFC 8528P1 MFC 8528P2 QP10n

active length x width 85mm x 28 mm 85mm x 28 mm 45mm x 25.4mm

thickness 0.18mm 0.18mm 0.38mm

mass 4.06g 4.06g 2.3g

elastic modulus 42Gpa 42GPa 51Gpa

operating mode 3-3 3-1 3-1

capacitance 5.7nF 172nF 117nF

piezoelectric constant d31/d33 400pC/N -170pC/ N -190pC/N

coupling coefficient Jp 8.9e-5 -3.8e-5 -4.4e-4

75

One can tell from equation (4.21) that load impedance also has an effect on optimal

power output. Pederson, Studer and Whinnery (1966) pointed out that a complex

conjugate matching load impedance delivers the maximum power output which has been

widely implemented among energy harvesting community for power optimization. In

order to make fair comparisons of harvesting abilities, the power output of the first two

modes using the above three types of piezoelectric transducers, are simulated along with

different resistive loads. A 1/8 inch thick honeycomb core fiberglass panel is cut using a

Tungsten Carbide saber saw into beam substrates with length of 24.375 inch and width of

1.5 inch. Figure 4.4 compares power output for MFC 8528 P1 and MFC 8528 P2 against

different load resistance for the first two modes. Both types of MFC are placed on the

clamped end. The plots in Figure 4.4 show that a P1 type MFC harvests much more

power at larger optimal load resistance compared to P2 type MFC. Here the acceleration

unit is g = 9.81 m/s2. It is worthy to mention that the P1 type MFC also has higher

actuation voltage range -500V to +1500V compared to the P2 type of -60V to +360V.

Another popular piezoelectric harvesting device is Quick Pack® (QP) group provided by

Midé () Technology, Corp. As shown in Table 4.1, consider a QP10n for example. It

operates in the 3-1 mode, has larger elastic modulus, capacitance, but smaller size,

mass, piezoelectric constant and coupling coefficient, compared to MFC 8528 P1.

Figure 4.4 Output power versus load resistance at mode 1 of 29Hz and mode 2 of 107

Hz.

102

104

106

108

10-8

10-6

10-4

10-2

100

Load Resistance (Ohm)

Ha

rve

ste

d P

ow

er

(mW

/g2)

MFC 8528-P1 - First Mode

MFC 8528-P1 - Second Mode

MFC 8528-P2 - First Mode

MFC 8528-P2 - Second Mode

76

The results presented in Figure 4.4 show that the optimal load resistance for MFC 8528

P1 is 1.98 Mohm for mode 1 and 72Kohm for mode 2. Using the same method, load

resistances of 112Kohm and 21Kohm are found to be the optimal values for QP10n. Plots

for QP10n are not shown here for brevity, but are compared next with MFC 8528P1, with

respect to another important design factor: the location of the piezoelectric transducers.

Figure 4.5 plots power output of MFC 8528 P1 at an optimal load resistance of 1.98

Mohm against the distance from clamped end (0.5 mm ~ 6.5mm) for the first two modes.

Figure 4.5 Output power of MFC 8528 P1versus distance from clamped end at mode 1 of

29 Hz and mode 2 of 107Hz.

Figure 4.6 presents the power output of QP10n at optimal load resistance of 112Kohm

against the distance from clamped end (0.5 mm ~ 6.5mm) for the first two modes. To

make a fair comparison, the normalized maximum power by piezoelectric active area

(length by width) is calculated as 42.1mW/g2/m

2 and 43.8mW/g

2/m

2, for MFC 8528P1

and QP10n, respectively. In order to obtain better power output, QP10n is selected as the

piezoelectric harvesting and sensing layer, and placed right next to the clamped end.

0 2 4 6 855

60

65

70

75

80

85

Distance From Clamped End [cm]

Ha

rve

ste

sd

Po

we

r [m

W/g

2]

MFC - 1st Mode - 1.98Mohm

0 2 4 6 80

2

4

6

8

10

12

14

16

18


Ha

rve

ste

sd

Po

we

r [m

W/g

2]

MFC - 2nd Mode - 1.98Mohm

77

Figure 4.6 Output power of QP10n versus distance from clamped end at mode 1 of 29 Hz

and mode 2 of 107Hz.

The results presented above provide a platform to build off when choosing an appropriate

piezoelectric transducer for optimal energy harvesting. In addition, for best actuation

authority concern, a 3-3 effected MFC 8528 P1 is selected as an actuator for gust

control, due to large actuation voltage range and high coupling coefficient, as reported by

Bilgen et al. (2011). Table 4.2 presented mechanical and material properties of other

components for multifunctional wing spar design.

Table 4.2 Selected properties of other components for multifunction wing spar design.

Devices Solar panel Battery PCB Epoxy, Kapton

length x

width

85mm x 28 mm 25.4mm x 25.4 mm 25.4mm x 25.4mm 1580mm x 28mm

thickness 0.2mm 0.18mm 0.2mm 0.00755mm

mass 0.69g 0.46g 0.23g 0.76

elastic

modulus

52GPa 55GPa 60GPa 3.35GPa

0 2 4 6 815

20

25

30

35

40

45


Harv

este

d P

ow

er

[mW

/g2]

QP - 1st Mode - 112Kohm

0 2 4 6 80

1

2

3

4

5

6

7

8

9


Harv

este

d P

ow

er

[mW

/g2]

QP - 2nd Mode - 112KMohm

78

SIMULTANEOUS ENERGY HARVESTING AND GUST

ALLEVIATION USING REC

This section discusses the possibility of a two mode wind gust control using REC for

small UAVs using the harvested energy from ambient wing vibration based on the

multifunctional composite concept developed in the previous sections. First, an

electromechanical equivalent circuit is described for representing a multifunctional wing

spar. After that, an estimation of harvested power and control power is analyzed.

Equivalent Circuit Representation of a Piezoelectric Generator

The equivalent circuit of piezoelectric generator has been derived by Liang and Liao

(2009) using the single degree freedom mode method. It has also been derived by Elvin

and Elvin (2009b) using the distributed parameter method. This chapter creates an

equivalent circuit in Matlab Simulink using distributed solutions presented by equation

(4.13) and (4.14), and aims to work for multi-mode harvesting. Figure 4.7 shows an

equivalent circuit for the first mode, where the piezoelectric generator is modeled as a

current source. Here, Cps is the piezoelectric capacitance of QP10n. The induced current and

voltage are generated from electromechanical coupled governing equations (4.13) and (4.14).

Figure 4.7 The equivalent circuit for 1st mode piezoelectric generator with resistive

impedance.

A resistance impedance of 112Kohm is implemented to maximize the harvested power

efficiency of the wing spar. The harvested voltage-to-base acceleration FRF for the first

two modes is presented in Figure 4.8 at this optimal load resistance. The assumed modes

79

formulation is truncated at order fifty to ensure the convergence of the first two modes.

The ambient normal wing vibration is simulated with 0.02g RMS acceleration over a

frequency bandwidth of 5Hz ~ 300Hz.

Figure 4.8 The output voltage to base acceleration FRF for an 112Kohm Load

Resistance.

The spectrum of the instantaneous harvested power for an 112Kohm resistive load is

plotted in Figure 4.9.

Figure 4.9 The harvested power spectrum for a 0.1 M ohm load resistance.

Generation of Normal Wing Vibration and Wind Gust Signals

The air through which an aircraft flies is never still. Therefore, the induced ambient

vibration provides an energy harvesting source during UAVs normal flight conditions.

The nature of atmospheric turbulence is influenced by many factors, which has been

101

102

-60

-40

-20

0

20

40

60

Frequency [ Hz ]

Vo

lta

ge

FR

F [ d

B r

ef 1

V/g

]

101

102

10-10

10-5

100

Frequency [ Hz ]

Ha

rve

ste

d P

ow

er

[ m

W ]

80

studied by Etkin (1972) and many others. The power spectral density (PSD) provides the

designer information of how the mean squared values of the argument is distributed with

frequency ɷ. Two classical representations for the PSD function of atmospheric

turbulence exist: The Von Karman and the Dryden spectrum, see McLean (1969) for

reference. Due to its simple form, the Dryden PSD spectrum in Appendix E is adopted

here for modeling ambience vibration source.

In order to generate the clear sky wing vibration and cumulus cloud gust signals with the

required intensity and scale lengths for a given flight velocity and height, a Gaussian

white noise source n(t) ~ N(0,1) with PSD function of 1, is amplified by a gust gain Kg

and filtered by a wind gust transfer function Gv(s). The schematic representation of the

wind gust generation for open loop and closed loop tip displacement response of the wing

spar is shown in Figure 4.10.

Figure 4.10 Block diagram of wind gust signal generation for open-loop and close-loop

tip displacement responses.

The relationship between the PSD of the output signal vg(t) and the input signal n(t) is

given by:

2 2( ) | ( ) | ( ).

v g v NK G (4.23)

If the power spectrum of the noise source is chosen as a Gaussian White noise, i.e. ΦN (ɷ)

= 1, then the wind gust transfer function becomes:

81

2 2| ( ) | ( ) / .

v v gG K

(4.24)

The solution of equation (4.23) yields:

0

2

0

20

3 3, ( )

( )

.v v

g v

v

v

Us

U LK G s

ULs

L

(4.25)

Here U0 is the aircraft trim velocity, Lv is the vertical scale of turbulence and σv is the

RMS vertical gust velocity. Figure 4.11 shows an example of simulated clear sky wing

vibration (dotted line) and cumulus cloud wind gust (solid line) signals, in acceleration

units g = 9.81 m/s2.

Figure 4.11Ambient wing vibration and wind gust acting on multifunctional wing spar

base, U0=15m/s, Lv =350m.

Gust Alleviation Using REC Control Laws:

Conventional active control laws usually suppress vibration very efficiently, but consume

large amount of external power supply. However, the REC control law developed was

reported to be able to suppress vibration efficiently but reduce the required power

significantly, see Wang and Inman (2011b) for reference. Our previous experimental

results in Wang and Inman (2011a), show a 67% energy reduction when using on-off

switching REC laws for transient vibration control compared to their conventional active

0 0.05 0.1 0.15 0.2 0.25 0.3

-3

-2

-1

0

1

2

3

Time [ s ]

Ba

se

acce

lera

tio

n [ g

]

Cumulus Cloud Wind Gust

Clear Sky Normal Vibration

82

control counterparts. Here, REC laws are implemented for gust alleviation over the

frequency band covering the first two modes, using the energy harvested from ambient

normal wing vibrations. The governing equations of the multifunctional wing spar can be

represented by:

,V V s s a a

Mη Cη Kη f θ θ (4.26)

0.s

p

sVC i

s s sθ η (4.27)

0.a

pVC i

a a a aθ η (4.28)

Here is and i

a denote the current flow through the sensor, and the actuator, respectively.

The circuit representation for gust alleviation using harvested energy is schematically

demonstrated in Figure 4.12. Here, the induced sensing current matrix [B] and sensing

voltage matrix [D] are derived from electromechanical coupled governing equations

(4.26) and (4.27). The induced actuating current matrix [A] and voltage matrix [C] are

derived from electromechanical coupled governing equations (4.26) and (4.28).

Figure 4.12 Schematic representations of gust alleviation using harvested energy.

Upon encountering the cumulus cloud disturbance shown in Figure 4.11, the REC law

yield 28 dB reduction of its 1st mode (29Hz) and 37dB reduction of its 2

nd mode (107Hz),

as seen in Figure 4.13. Here the logarithm relative tip displacement (dB reference 1

meter) is shown in linear coordinate.

83

Figure 4.13The disturbed tip displacement spectrum of multifunctional wing spar before

and after REC control.

In time domain, the wing spar tip displacement RMS value is reduced from 1.6 mm (open

loop in solid line) down to 0.35 mm (closed loop in dot line). The maximum tip

displacement is reduced from ±4.5 mm down to ±1.0 mm, as seen in Figure 4.14.

Figure 4.14Winds disturbed multifunctional wing spar tip response in time domain before

and after REC control.

Power Flow for Simultaneous Energy Harvesting and Gust Alleviation

The REC laws produce a nonlinear switching logic by saturating the actuating voltage

provided by the positive strain feedback (PSF) controller. The PSF control algorithm

101

102

-140

-120

-100

-80

-60

Frequency (Hz)

Re

lative

Tip

Dis

pla

ce

me

nt (d

B r

ef 1

m)

Open Loop - FRF

Reduced Energy Control - FRF

1.5 2 2.5 3

-6

-4

-2

0

2

4

6

Time [ s ]

Re

lative

Tip

Dis

pla

ce

me

nt [ m

m ]

Open Loop

Reduced Energy Control

84

takes the similar transfer function form as the positive position feedback control law, see

Wang and Inman (2011d) for reference. However, the feedback signals act from the PZT

strain sensor instead of a position sensor. This collocate function removes external

sensing sources and/or external power supply. As seen from Figure 4.15, the Laplace

transfer function of PSF for the 1st mode, driven by a PSF op-amp and a voltage buffer

op-amp, is of the following form:

2

22.

2

f

f

o

i f f

kv

v s s

(4.29)

Here the PSF filter parameters: gain k, damping ratio ζf , filter natural frequency ɷf are

achieved by:

1 2 3

1 2 2 3

252

1 4

1, ,

( ).

2(1 )

f

f f

C R R

C C R R

RR

R Rk

(4.30)

Here, C1 and C2 are capacitance components, R1, R2, R3, R4 and R5 are resistive

components.

Figure 4.15 Block Diagram of the 1st Mode PSF Control.

85

In Figure 4.15, the voltage buffer op-amp is interposed to reduce actuation current flow.

Two parallel PSF controllers are designed for gust alleviation for two specific modes.

The PSF feedback signals are periodically saturated to satisfy the REC switching logic

(not discussed here but demonstrated in Chapter 3. The instantaneous power associated

with the actuator is defined as the product of actuation voltage and actuation current:

( ) ( ) .( )ins a aP t v t i t (4.31)

Note that the piezoelectric actuation voltage associates with the REC laws, and the

actuation current depends on the electrical impedance of the control circuits. The real part

and the imaginary part of the cross-spectrum between instantaneous voltage and current

represent the active (real) power and reactive power associated with a given component,

respectively. The active power (thick line) and reactive power (thin line) associated with

each PSF and Buffer op-amp of the 1st mode and the 2

nd mode is presented in Figure

4.16. The unit of active and reactive power is watt (w) and volt-ampere reactive (var),

respectively. It shows that the 1st mode control Op-amps require 90 times more active

power than that of the 2nd

mode.

Figure 4.16 Active and reactive power spectrum of 1st Mode and 2

nd Mode PSF and

buffer Op-amps.

The active power (thick line) and reactive power (thin line) associated with the summing

Op-amp, and the MFC 8528P1 actuator is shown in Figure 4.17. The active and reactive

10 20 30 40

0

0.18

Active [ m

W ]

Frequency [ Hz ]

a. 1st Mode PSF Op-amp

10 20 30 400.08

0

Reactive [ m

VA

r ]Active

Reactive

10 20 30 40

0

1.1

Active [ m

W ]

Frequency [ Hz ]

b. 1st Mode Buffer Op-amp

10 20 30 400.2

0

Reactive [ m

VA

r ]Active

Reactive

170 175 180 185 190

0

3e-3

Active [ m

W ]

Frequency [ Hz ]

c. 2nd Mode PSF Op-amp

170 175 180 185 1902e-3

0

Reactive [ m

VA

r ]Active

Reactive

170 175 180 185 190

0

11e-3

Active [ m

W ]

Frequency [ Hz ]

d. 2nd Mode Buffer Op-amp

170 175 180 185 190

1e-30

Reactive[ m

VA

r ]

Active

Reactive

86

power associated with each PSF and buffer op-amp and each transducer are detailed in

Table 4.3.

Figure 4.17Active and reactive power associated with the summing Op-amp and the

MFC 8528 P1actuator.

Table 4.3Power Associated With Each Electric Component.

Electric Element Active Power

(mW)

Reactive Power

(mVAr)

Apparent Power

(mVA)

Power

Factor

1st PSF Op-Amp 0.15 0.09 0.18 0.83

1st Buffer Op-Amp 3.46 0.00 3.46 1

2nd

PSF Op-Amp 0.10 0.03 0.10 1

2nd

Buffer Op-Amp 0.30 0.00 0.30 1

Summing Op-Amp 3.65 4.94 6.78 0.54

Actuator 6.42 4.94 8.63 0.74

Harvester 0.16 0.00 0.16 1

Known from Table 4.3, the active power required for 28dB reduction of the 1st mode, and

37dB reduction of the 2nd

mode in tip displacement is 6.42mW, which is 40 times higher

than the harvesting power of 0.16mW. That is, in order to control a 1 second time span

cumulus wind gust, 40 seconds of harvesting are needed. Note that practical loads deliver

both active and reactive power and also waste energy into heat. Therefore, the apparent

power, which is the product of the RMS of voltage and current, is also calculated and

presented in Table 4.3, with the unit of Volt-Ampere (VA). The power factor here is

defined as the active power divided by apparent power.

10 20 30 40

0

1

Active [ m

W ]

Frequency [ Hz ]

a. Summing Op-amp

170 175 180 185 190

0

0.35

Active [ m

W ]

Frequency [ Hz ]

10 20 30 40

0

Active [ m

W ]

Frequency [ Hz ]

b. Actuation Power

170 175 180 185 190

0

0.35

Active [ m

W ]

Frequency [ Hz ]

10 20 30 402

0

Reactive [ m

VA

r ]

Active

Reactive

170 175 180 185 1900.2

0

Reactive [ m

VA

r ]

Active

Reactive

10 20 30 402

0

Reactive[ m

VA

r ]

Active

Reactive

170 175 180 185 1900.2

0

Reactive[ m

VA

r ]

Active

Reactive

87

CHAPTER SUMMARY

A novel concept of simultaneous energy harvesting and gust alleviation is presented in

this chapter. The motivating application is a multifunctional wing spar of a UAV with the

goal of providing self-contained gust alleviation. A wing spar is designed to be

compatible with collocated self-harvesting, self-sensing and self-controlling

functionalities using piezoelectric materials. The basic design factors are discussed for a

beam-like multifunctional wing spar with load-bearing energy harvesting, strain sensing

and self-controlling functions. The configurations, locations and operating modes of

piezoelectric transducers are also discussed for optimal power generation. The reduced

energy control law is implemented to reduce control power while preserving control

performance. Theoretical modeling and numerical simulations show that the tip

displacement due to a wind gust disturbance can be reduced by 28dB and 37dB for the 1st

and the 2nd

mode, respectively, using energy harvested from ambient wing vibrations. A

flexible solar panel is also modeled but not used in the analysis. Future work will focus

on solar energy harvesting in order to reduce the recharging time substantially. Another

consideration for future work is finite element modeling for optimized multifunctional

structures.

88

CHAPTER 5 EXPERIMENTAL VALIDATION OF AN

AUTONOMOUS GUST ALLEVIATION SYSTEM

This chapter details experimental characterization and validation of an autonomous gust

alleviation system building upon recent advances in harvester, sensor and actuator

technology that have resulted in thin, ultra-light weight multilayered composite wing

spars. This multifunctional spar is considered an autonomous gust alleviation system for

small UAVs powered by the harvested energy from ambient vibration during their normal

flight conditions. Experimental characterization and validation are performed on

cantilever wing spars with micro-fiber composite transducers controlled by reduced

energy controllers. Normal flight vibration and wind gust signal is simulated using

Simulink and Control desk and then generated for experimental validation analysis for

gust alleviation. Considering the aluminum multifunctional wing spar, a reduction of

11dB and 7dB is obtained respectively for the first and the second mode. Considering the

fiberglass composite multifunctional wing spar, a reduction of 16dB is obtained for the

first vibration mode. Power evaluations associated with various electronic components

are also presented for both cases. Energy harvesting abilities of monolithic and micro

fiber composite transducers are also compared for the fiberglass composite wing spar.

This work demonstrates the use of reduced energy control laws for solving gust

alleviation problems in small UAVs, provides the experimental verification details, and

focuses on applications to autonomous light-weight aerospace systems.

NOMENCLATURE

ζ1 = first mode damping ratio of the controller

ζ2 = second mode damping ratio of the controller

g1 = first mode control gain of the controller

g2 = second mode control gain of the controller

Etot = total required control energy

Etr = transient energy

Pst = required power for steady state vibration control

ts = setting time

tg = wind gust duration

89

INTRODUCTION

The increasing demand for ultra-light weight materials in small UAVs results in

extremely flexible structures with low-frequency vibration modes. Suppression of

undesired vibrations in such flexible structures with limited energy is becoming an

important design problem to develop energy-autonomous controllers powered using the

harvested ambient energy. Reduced energy control laws developed in Chapter 3 address

the trend towards autonomous aerospace structures with limited energy supply. Small

UAVs are usually comprised of large ultra-light weight structures with multilayered

composite components. The multifunctional wing spar designed in Chapter 4, is one

typical example. Within these composite structures, the individual components

themselves may have complex characteristics, for example a micro-fiber composite

(MFC) actuator in the context of gust alleviation. This chapter is dedicated to

experimental characterizations building upon recent advances in harvester, sensor and

actuator technology that have resulted in thin, ultra-light weight multi-layered composite

wing spars. The multifunctional wing spar of small UAVs is considered an autonomous

gust alleviation system powered by harvested energy from ambient vibration during their

normal flight conditions.

Experiments are performed on cantilever wing spars with MFC transducers controlled by

reduced energy controllers. Considering the first two vibration modes, the control energy

requirements are compared and evaluated. This work demonstrates the use of reduced

energy control laws for solving gust alleviation problems in small UAVs, provides the

experimental setup and verification details, and focuses on applications to autonomous

aerospace systems. These successful experiments validate the autonomous gust

alleviation systems on multifunctional wing spars with both aluminum and honeycomb

core fiberglass substrates. The experimental validation will be carried out through the

analysis of these two examples.

90

This chapter is organized as follows. In Section 2, experimental validation of reduced

energy control law is given for a piezoelectric layer bonded aluminum wing spar.

Experimental characterization of this control law under study is fully described. The

experimental results concerning the power flow associated with various electric elements

are also presented. Another experiment given in Section 3 is dedicated for the

autonomous gust alleviation system building on a honeycomb core fiberglass composite

sandwich wing spar, as presented in section 3. The first step characterizes and compares

energy harvesting abilities of monolithic and micro fiber composite transducers. The next

step addresses numerical predictions of Chapter 4, along with experimental data. Closing

remarks and future perspectives are briefly outlined in Section 4.

EXPERIMENTAL VALIDATION OF REDUCED ENERGY

CONTROL ON A PIEZOELECTRIC LAYER BONDED

ALUMINUM WING SPAR

This step of work is dedicated to experimental validation of the feasibility of reduced

energy control for autonomous gust alleviation system. In this part of study, an

autonomous gust alleviation system builds on a piezoelectric layered bonded aluminum

wing spar. This autonomous multifunctional wing spar is designed to evaluate control

performance of reduced energy control (REC) compared to positive strain feedback

control (PSF) developed in Chapter 4.

Experimental Setup for Validation of Reduced Energy Control Law

As stated before, this experimental study is conducted in order to evaluate control

performance of REC compared to PSF with a focus on the first two transverse vibration

modes. The experimental specimen is a flexible cantilevered multifunction wing spar. Its

photographic and schematic representation is given in Figure 5.1 (a) and (b). Both PZT

layers are positioned 54mm away from the root end of the beam, corresponding to the

beam length attached to the fuselage when the wing spar is fully inserted into the wing.

Therefore, both piezoelectric devices are placed at the root of the cantilever wings. The

MFC actuator is controlled by REC and PSF controllers, respectively, for control

performance comparison. The geometry and material properties of this multifunctional

91

wing spar are listed in Table 5.1. In the first step of this experiment, both controllers

build in Matlab Simulink using Control Desk software and a dSPACE real time control

board. Whichever controller that has the best performance is built and tested in a Printed

Circuit Board (PCB) for further experimental study.

(a)

(b)

Figure 5.1 (a) A photographic (b) A schematic representation of front view and back

view of the aluminum baseline multifunctional wing spar.

Table 5.1 Geometry and Material Properties for the Aluminum Baseline Multifunctional

Wing Spar.

Property/Component QP10n MFC8528P1 Aluminum Substrate

Overall Length 50.8mm 112mm 504mm

Overall Width 25.4mm 40mm 28mm

Overall Thickness 0.508mm 0.18mm 3.05mm

Overall Mass 2.835 gram 4.06gram 139.4gram

The overall experimental setup is shown in Figure 5.2, where the representative

multifunctional wing spar is clamped to an APS Dynamics ELECTRO-SEIS long stroke

shaker with frequency range 0.1 Hz ~ 200 Hz. The shaker is driven by an APS DUAL-

MFC8528-P1

QP10N

Shaker

Accelerometer

Charge Amplifier

x

z

B. QP16N (Harvester, Sensor)A. Aluminum Substrate

C. MFC(Actuator)

L1=54mm L2=113.6mm L4=504mm

D. Epoxy DP 460, Kapton

L3=400mm

92

MODE power amplifier. Considering the same nature of wing vibration signals (for

harvesting purpose) and wind gust signals (for control purpose), the Gaussian white noise

and Dryden PSD function in Appendix E are implemented in Simulink to produce input

signals in both cases. A 10 M ohms optimal resistive load is employed to provide

allowable operation voltage (5V for data acquisition system).

Figure 5.2 Gust alleviation experimental setup using REC Laws.

Experimental Results

The absolute tip displacement is measured using an MTI LTC-50-20 laser sensor. The

frequency spectrum of the measured relative tip displacement is calculated in order to

compare control performance for a range of control parameters. The investigated control

gains and corresponding vibration reduction levels are shown in Figure 5.3 and listed in

Table 5.2.

1. Multifunctional Wing Spar

2. Circuit Board3. PCw/Simulink4. dSPACE Board5. Low Pass Filter6. Displacement

Laser Sensor

7. Power Amplifier8. Siglab

1

8

4

65

2

3

7

93

Figure 5.3 Control performance of the PSF controllers for different control gains

(damping ratio of mode 1: ζ1= 0.15 and mode 2: ζ2 = 0.35).

Table 5.2 Control Performance versus PSF Control Gain.

1st Mode Control Gain 0.8 0.6 0.4

2nd

Mode Control Gain 0.8 0.6 0.4

1st Mode Reduction 13dB 11dB 9dB

2nd

Mode Reduction 8dB 7dB 6dB

In addition to above results, the experimental measurement also shows that when the

control gain is fixed (0.1 < g1, g2 < 1.2), changing the controller’s damping ratio (0.10 <

ζ1< 0.55, 0.30 < ζ2< 0.75) does not affect the control performance. Figure 5.4 shows that

the same control performance (11dB reduction for the 1st mode, 7dB reduction for the 2

nd

mode from open circuit control (dashed line)) is reached using the PSF controller (solid

line) with gains g1 = 0.6, g2 =0.6, damping ratios ζ1 = 0.15, ζ2 = 0.35; and using the

REC controller (dot line) with gains g1 = 0.8, g2 = 0.8, damping ratios ζ1 =0.15, ζ2 =

0.35, saturation voltage of 110 Volt. Again, this plots the relative tip displacement read

from laser sensor, not the tip-displacement-to-base-acceleration FRF but has the same

trend.

9 10 11 12 13 14-105

-100

-95

-90

-85

-80

Frequency [ Hz ]

Rela

tive T

ip D

ispla

cem

ent [ dB

ref 1m

]

Open-Circuit

g1 = 0.8 / g2 = 0.8

g1 = 0.6 / g2 = 0.6

g1 = 0.4 / g2 = 0.4

70 72 74 76-140

-135

-130

-125

-120

-115

Frequency [ Hz ]

Rela

tive T

ip D

ispla

cem

ent [ dB

ref 1m

]

Open-Circuit

g1 = 0.8 / g2 = 0.8

g1 = 0.6 / g2 = 0.6

g1 = 0.4 / g2 = 0.4

94

Figure 5.4 Vibration control performance using the PSF and REC Laws.

The time domain steady state responses of PSF and REC are shown in Figure 5.5. Both

the PSF controller (solid line) and the REC controller (dashed line) are designed to

reduce the RMS value of the relative tip displacement from 3.0 mm for the open circuit

case (dotted line) down to 1.2 mm for closed loop control. The two-mode PSF controller

has gains g1 = 0.6 (1st mode), g2 =0.6 (2

nd mode) and damping ratios ζ1= 0.15 (1

st mode),

ζ2 = 0.35 (2nd

mode). The two-mode REC has gains g1 = 0.8(1st mode), g2 = 0.8(2

nd

mode), and damping ratios ζ1 =0.15(1st mode), ζ2 = 0.35(2

nd mode), with saturation

voltage of 110 Volt.

9 10 11 12 13 14-105

-100

-95

-90

-85

-80

-75

Frequency [ Hz ]

Rela

tive T

ip D

ispla

cem

ent [ dB

ref 1m

]

70 72 74 76-140

-135

-130

-125

-120

-115

-110

Frequency [ Hz ]

Rela

tive T

ip D

ispla

cem

ent [ dB

ref 1m

]

Open-Circuit

PSF Control


Open-Circuit

PSF Control


95

Figure 5.5 Control performance using PSF and REC laws (time history of relative tip

displacement response).

The actuation voltage is amplified 200 times by a TREK 2220 power amplifier, and then

fed into the MFC actuator. Figure 5.6 presents the actuation voltage required by the PSF

(solid line) and the reduced energy (dotted line) control laws, satisfying identical control

performance.

Figure 5.6 Actuation voltage measurements required by the PSF and REC laws.

2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8-6

-4

-2

0

2

4

6

Time [ s ]

Rela

tive T

ip D

ispla

cem

ent [ m

m ]

Open-Circuit

PSF Control


0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8-300

-200

-100

0

100

200

300

Time [ s ]

Actu

ation V

oltage [

V ]

PSF Control g0.6

Reduced Energy Control v120

96

The TREK 2220 power amplifier also has a current monitor function available for

measuring actuation current. Figure 5.7 shows the actuation current required by the PSF

(Solid line) and the reduced energy (dotted line) control laws.

Figure 5.7 Actuation current measurements required by the PSF and REC laws.

The instantaneous power required by the PSF Controller (solid line) and the REC

Controller (dotted line) is shown in Figure 5.8.

Figure 5.8 Instantaneous power required by PSF and REC laws.

0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Time [ s ]

Actu

ation C

urr

ent

[ m

A ]

PSF Control g0.6

Reduced Energy Control v120

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5-10

-5

0

5

10

15

20

Time [ t ]

Insta

nta

neous P

ow

er

[ m

W ]

PSF Control Law

Reduced Engergy Control Law

97

Figure 5.9 compares the active and reactive power from experimental measurements in

the frequency domain. The data is recorded for duration of 60s.

Figure 5.9 Active and reactive power required by PSF and REC laws.

The average harvested power Phav

of the aluminum based multifunctional wing spar is

found to be 0.3 mW. The other power and energy elements associated with the PSF and

REC Laws are detailed in Table 5.3. All values are given to satisfy 11dB reduction of the

1st mode and 7dB reduction of the 2

nd mode, compared to open-circuit. Here Etr is the

energy consumed for transient vibration control and Pst is the power required for steady

state vibration control. The settling time ts is defined as the time needed for the controlled

response to reach 40 % of the amplitude of the open loop response, and is found to be 0.8

s in the present case.

Table 5.3 Power and Energy Elements Associated with PSF and REC Laws.

Element PSF REC Ratio of Reduction

Voltage RMS (V) 100 82.3 18 %

Current RMS (mA) 0.065 0.053 18 %

Required Energy Etr (mJ) 6.6 1.6 76 %

Average Power Pst (mW) 1.2 1.0 17 %

9 10 11 12 13 14

0

2

Active [ m

W ]

Frequency [ Hz ]

a. PSF Control Law

70 72 74 76

0

0.05

Active [m

W]

Frequency [ Hz ]

9 10 11 12 13 14

0

2

Active [m

W]

Frequency [ Hz ]

b. Reduced Energy Control Law

70 72 74 76

0

0.05

Active [m

W]

Frequency [ Hz ]

9 10 11 12 13 143

0

Reactive [ m

VA

r ]

Active

Reactive

70 72 74 760.05

0

Reactive [m

VA

r]

Active

Reactive

9 10 11 12 13 143

0R

eactive [m

VA

r]

Active

Reactive

70 72 74 760.05

0

Reactive [m

VA

r]

Active

Reactive

98

For any wind gust duration tg > ts, the total required control energy Etot can be calculated

by:

*( ).tot tr st g sE E P t t (5.1)

For PSF and REC control systems, the total required energy formula becomes:

6.6 1.2*( 0.8).

1.6 1.0*( 0.8).

PSF

tot g

REC

totg

E t

E t

(5.2)

The harvesting time th required to control a wind gust of duration tg is given by:

.tot

h harv

Et

P (5.3)

EXPERIMENTAL CHARACTERIZATION AND VALIDATION

OF AN AUTONOMOUS GUST ALLEVIATION SYSTEM ON A

HONEYCOMB CORE FIBERGLASS COMPOSITE SANDWICH

WING SPAR

The increasing trend of lighter and stronger aerospace structures can be easily achieved

by the use of composite materials. Sandwich composites are one of popular examples

which have been tailored for many specific applications, whose major advantage is their

high flexural stiffness to weigh ratio. Fiber reinforced composites with a low-density core

find increasing use in aerospace structures. A honeycomb core fiberglass composite

sandwich panel manufactured by ACP composites (ACP) is cut into a beam substrate to

build an autonomous gust alleviation wing spar for small UAVs. This honeycomb core

fiberglass beam substrate is sandwiched between two layers of uni-directional fiberglass

pre-preg and an epoxy film adhesive. The fiberglass plies are layered up at 0º and 90º to

produce high strength-to-weight and rigidity-to-weight ratios. The final experimental test

builds on recent advances on monolithic and micro fiber composite transducers, sandwich

composites, and printed circuit boards. The focus is experimental characterization and

validation of the autonomous gust alleviation system. Figure 5.10 illustrates an

experimental setup and schematic representation of the autonomous gust alleviation

system.

99

(a) (b)

Figure 5.10 (a) A photographic representation (b) a schematic representation of the

autonomous gust alleviation system building on a honeycomb core fiberglass

multifunctional wing spar.

Experimental Setup for Harvesting Ability Characterizations of Monolithic QP10n

and Micro Fiber Composite MFC 85281P1

The motivation of this step is to characterize harvesting abilities, and material properties

of two types of piezoelectric transducers. The considered vibration based cantilever

harvesters builds on two honeycomb core fiberglass composite sandwich wing spars, with

length 593.7mm, width 38mm and thickness 3.175mm. The hanging over length of the

substrate is 34.5mm. This results the effective length of 559.2 mm. The length to

thickness ratio of this design satisfies the required span wise rigidity for use in the small

UAVs. One of these rectangular wing spars is bonded by a monolithic piezoelectric layer

QP10n from QuickPack, as shown in Figure 5.11 (a). The other one is bonded by a micro

fiber composite piezoelectric layer MFC8528P1 from Smart Material Corp, as shown in

Figure 5.11 (b). The geometry and material properties of each piezoelectric transducer

are listed in Table 5.4.

(a) (b)

Figure 5.11 A prototype of (a) the MFC 8528P1 (b) the QP10n.

AccelerometerQP10N Shaker MFC 8528P1

PCB

x

z

A. QP16N (Harvester, Sensor)D. Printable Circuit Board (PCB)

B. Honeycomb Core Fiberglass

C. MFC(Actuator)

L1=25 mm L2=84.6 mm

L3=110 mm

L4=552 mm

E. Epoxy DP 460, Kapton

L4=592 mm

100

Table 5.4 Geometry and Material Properties for Two Unimorph Piezoelectric Harvesters.

Property/Component QP10n MFC8528P1 Composite Substrate

Overall Length 50.8mm 112mm 593.7mm

Overall Width 25.4mm 40mm 38mm

Overall Thickness 0.508mm 0.18mm 3.175mm

Overall Mass 2.835 gram 4.06gram 13.97gram

Both piezoelectric device layers are bonded with Kapton layer and 460 3D epoxy layers

under 0.8 atm vacuum for 6 hours to minimize variations in epoxy thickness and

eliminate air gaps. The processing procedure is 1) clean composite sandwich substrate

and the respective piezoelectric layer from Figure 5.10(a) and (b) using IPA and wipers;

2) prepare the vacuum bag, silk and cotton; 3) tape the silk layer on the bottom side of the

composite sandwich substrate; 4) clean the glue gun, as shown in Figure 5.12 (a); 5) mix

some 3D ScotchWeldTM

DP460 two part epoxy in a small try with the spatula; 6) spread a

thin layer of the mixed epoxy all over the respective device; 7) remove the surplus by

scrapping lines with the spatula; 8) glue together the respective device layer with the

composite sandwich substrate; 9) tape them together; 10) tape the silk layer over the

respective device layer; 11) put the bonded beam in the cotton layer and insert it in the

vacuum bag; 12) tape the vacuum bag tightly; 13) plug the vacuum pump 0523-101 from

Gast Manufacturing Inc, and vacuuming at 20kPa for about 6 hours, as shown in Figure

5.12 (b) and (c); 14) clean everything. After curing, turn off vacuum pump, take off silk

layer and tapes, and remove any excess epoxy from the edges of the device.

(a) (b) (c)

Figure 5.12 A photographic representation of (a) the DP460 glue gun (b) the vacuum

process (c) the pressure meter panel.

101

It is known that maximum power output occurs near the resonant frequency of each

unimorph cantilever harvester. Therefore, the first step focuses on output power

characterization of each harvester connected with a set of resistive load, (as shown in

resistive shunt in Figure 2.5(a)). Table 5.5 lists the nominal resistors used in the

experiment and their effective values due to the manufacturing error and the impedance

of the data acquisition system.

Table 5.5 Nominal Resistors and their Effective Values.

Nominal Resistor (Ω) Effective Load Resistance (Ω)

1k 1.8k

3k 3.3k

5k 4.9k

10k 9.9k

20k 21.7k

30k 32.5k

40k 40.6k

50k 50.3k

60k 62k

70k 71.3k

80k 81k

90k 92.8k

100k 98.8k

112k 112k

139k 139k

200k 199k

250k 250k

300k 300k

370k 377k

494k 494k

609k 609k

697k 697k

830k 839k

970k 977k

2M 1.985M

3M 2.724M

4M 3.76M

5M 4.770M

7M 7.01M

10M 9.44M

20M 22M

102

The purpose of this chapter is to study ambient vibration transduction caused by clear sky

atmospheric turbulence, therefore, the Dryden’s PSD spectrum function (Appendix E) is

used to represent atmospheric turbulence. Equations (4.23) to (4.25) are employed to

address ambient vibration, which is used to excite cantilever wing spar using the

dSPACE 1005 real time control board, Matlab Simulink and Control Desk software. In

consistent with Chapter 4, the small UVA trim velocity is 15m/s, and the vertical scale of

turbulence is 350m. The excitation acceleration is measured by a PCB Piezotronics

Model U352C67 shear accelerometer with sensitivity of 0.99 V/g and bandwidth from

0.5Hz to 10 kHz. A PCB charge amplifier 482A16 is implemented with a 10x gain. The

output voltage recorded using dSPACE 1005 and Control Desk software. The

photographic experimental setup is given in Figure 5.13.

(a) (b)

Figure 5.13 (a) Energy harvesting experimental setup for QP10n piezoelectric harvester

(b) dSPACE data acquisition system.

The objective is to compare the experimental measurement of voltage, current and power

output with analytical modeling and numerical simulation of Chapter 4. The focus is

placed on the first two resonant modes at 10.6Hz and 57.4Hz respectively. In order to

obtain better agreements of numerical simulation with experimental data, a tuning

procedure is required based on analytical solutions derived in Chapter 4. First, the modal

mechanical damping is experimentally identified with high accuracy. This is very

important, since it affects peak power amplitude significantly near resonance frequencies.

The mechanical damping ratio is identified graphically using the quadrature peak picking

on the open-loop FRF plot. In all these case studies, the mechanical damping ratio is kept

QP10n

Printed Circuit Board

Clamp

Shaker

103

at ζ1 = 0.009 for the first mode, and ζ2 = 0.0069 for the second mode. In addition, the

accurate distance from the clamped length to the harvester, the accurate stiffness, internal

capacitance and coupling coefficient of QP10n and MFC are also very important in order

to receive good agreements as shown from Figure 5.14-5.17 and Figure 5.19-5.22.

Details will be presented later with respect to each relevant plot.

Experimental Characterizations of Harvesting Abilities for Monolithic QP10n

Transducer

The experimental and numerical comparison of voltage-per-acceleration for the QP10n

harvester is plotted in Figure 5.14 along with a set of effective load resistance for both

resonant modes. The numerical curves agree very well with the experimental data points

for both cases. The peak voltage amplitude of both modes, as expected from the

theoretical modeling and simulation of Chapter 4, increases monotonically with

increasing load resistance. The peak voltage amplitude of both modes increases by two

orders of magnitude as the load resistance is increased from 1.8 kΩ to 22MΩ.



atmospheric turbulence for the QP10n piezoelectric harvester.

The experimental and numerical comparison of current-per-acceleration for the QP10n

harvester is plotted in Figure 5.15 along with a set of effective load resistance with

respect to the first two resonant modes. The simulation curves exhibit near perfect

agreements with the experimental data points for both modes.

103

104

105

106

107

100

101

102

103


Ha

rve

ste

d V

olta

ge

(V

/g)

QP10N - 1st Mode - Simulation

QP10N - 1st Mode - Experimental

QP10N - 2nd Mode - Simulation

QP10N - 2nd Mode - Experimental

104




The experimental and numerical comparison of power-per-acceleration for the QP10n

harvester is shown in Figure 5.16 along with a set of effective load resistance, with

respect to the first two resonant modes. Both simulation curves agree very well with the

experimental data points. It is observed from Figure 5.16 that, the first mode peak power

amplitude of 28.2mW/g2 is obtained for the optimum load resistance of 112kΩ, and the

second mode peak power amplitude of 9.8mW/g2 for the optimum load resistance of

20.2kΩ.




103

104

105

106

107

10-3

10-2

10-1

100

101


Ha

rve

ste

d C

urr

en

t (m

A/g

)





103

104

105

106

107

10-1

100

101

102


Ha

rve

ste

d P

ow

er

(mW

/g2)





105

Figure 5.17 shows numerical simulation of voltage-to-base-acceleration FRFs at optimal

load resistance of 112kΩ, along with experimental data. A few tuning procedures have

been performed to predict the dynamic response with very good accuracy. First, the

effective distance from the clamped root to the harvester is slightly tuned in order to

better match the peak amplitude ratio of the first mode and the second mode. Then, the

mass and stiffness of the substrate and harvester is tuned slightly to better match the

resonant frequency ratio of the first mode and the second mode. The slight tuning of the

composite substrate stiffness helps adjusting the resonant frequency of each mode, but

has no effect on the ratio of the frequencies. In the end, the internal capacitance of the

harvester is adjusted in order to better match the shape of the FRF curve. In addition, the

coupling coefficient term of the harvester is very easy to be identified, since it determines

the overall amplitude of the frequency response. All the characterized material properties

are presented in Table 5.5.



for the QP10n piezoelectric harvester.

In order to clearly see the change of peak-wise behavior resulted from variation of load

resistance, Figure 5.18 displays the output voltage-to-base-excitation FRFs for all these

resistive loads from Table 5.4. The trend exhibits consistency with derived analytical

models of Chapter 4.

0 10 20 30 40 50 60 7010

-2

10-1

100

101

102

103

Frequency (Hz)

Vo

lta

ge

to

Ba

se

Acce

lera

tio

n F

RF

(V

/g)

QP10N - Simulation - 139kohm

QP10N - Experimental - 139kohm

106

Figure 5.18 The measured voltage-to-base-acceleration FRF at a set of load resistance

excited by clear sky atmospheric turbulence for the QP10n piezoelectric harvester.

Experimental Characterizations of Harvesting Abilities for Micro Fiber Composite

MFC 8528 P1

The experimental and simulation comparison of voltage-per-acceleration for the MFC

8528P1 harvester is plotted in Figure 5.19 along with a set of effective load resistance for

both resonant modes. The numerical curves predict the experimental data points with

very good accuracy for both modes. The peak voltage amplitude of both modes, as

expected from the theoretical modeling and simulation of Chapter 4, increases

monotonically with increasing load resistance. The peak voltage amplitude of both modes

increases by three orders of magnitude as the load resistance is increased from 1.98 MΩ

to 0.35MΩ.



atmospheric turbulence for the MFC 8528P1 piezoelectric harvester.

0 50 100 150

10-4

10-3

10-2

10-1

100

101

102

Frequency [ Hz ]

Voltage to B

ase A

ccele

ration F

RF Increase R

103

104

105

106

107

108

10-1

100

101

102

103


Ha

rve

ste

d V

olta

ge

(V

/g)

MFC - 1st Mode - Simulation

MFC - 1st Mode - Experimental

MFC - 2nd Mode - Simulation

MFC - 2nd Mode - Experimental

107

The experimental and numerical comparison of current-per-acceleration for the MFC

8528P1 harvester is plotted in Figure 5.20 along with a set of effective load resistance

with respect to the first two resonant modes. Both simulation curves exhibit very good

agreements with the experimental data points for both cases.




The power-per-acceleration for the MFC 8528P1 harvester is shown in Figure 5.21 along

with a set of effective load resistance, with respect to the first two resonant modes. The

simulation curves agree very well with the experimental data points. It is observed from

Figure 5.21 that, the first mode peak power amplitude of 50.8mW/g2 is obtained for the

optimum load resistance of 1.98 MΩ, and the second mode peak power amplitude of

9.8mW/g2 for the optimum load resistance of 0.35 MΩ.




103

104

105

106

107

108

10-3

10-2

10-1

100


Ha

rve

ste

d C

urr

en

t (m

A/g

)





103

104

105

106

107

108

10-2

10-1

100

101

102


Ha

rve

ste

d P

ow

er

(mW

/g2)





108

Figure 5.22 displays numerical simulation of voltage-to-base-acceleration FRFs at

optimal load resistance of 1.98MΩ, along with experimental data. It can be observed that

numerical simulation agrees with experimental plots very well.



for the QP10n piezoelectric harvester.

As is the case for the QP 10n, a few tuning procedures have been performed to predict

dynamic response with very good accuracy. First, the effective distance from the clamped

root to the harvester is slightly tuned in order to better match the peak amplitude ratio of

the first mode and the second mode. Then, the mass and stiffness of the substrate and

harvester is tuned slightly to better match the resonant frequency ratio of the first mode

and the second mode. The slight tuning of the composite substrate stiffness helps

adjusting the resonant frequency of each mode, but has no effect on the ratio of the

frequencies. In the end, the internal capacitance of the harvester is adjusted in order to

better match the shape of the FRF curve. In addition, the coupling coefficient term of the

harvester is very easy to be identified, since it determines the overall amplitude of the

frequency response. All the experimental characterization of material properties of these

two types of piezoelectric harvesters are presented in Table 5.6.

0 10 20 30 40 50 60 7010

-2

10-1

100

101

102

103

Frequency (Hz)

Vo

lta

ge

to

Ba

se

Acce

lera

tio

n F

RF

(V

/g)

MFC - Simulation - R = 1.98Mohm

MFC - Experimental - R = 1.98Mohm

109

Table 5.6 Experimentally Property Identification of Two Piezoelectric Harvesters.

Property/Component QP10n MFC8528P1 Composite Substrate

Active Length 45mm 85mm 559.2mm

Active Width 20mm 28mm 38mm

Thickness 0.38mm 0.18mm 3.175mm

Mass 2.30gram 4.06gram 13.97gram

Young’s Modulus 51GPa 42GPa 10.29GPa

Internal Capacitance 117nF 7.9nF N/A

Piezoelectric Coefficient d33 -190e-12 400e-9 N/A

Effective Distance3 from Clamp 38.4mm 30.5mm N/A

Experimental Validations of the Autonomous Gust Alleviation System on the

Fiberglass Composite Multifunctional Wing Spar

The main contribution of this experiment on the autonomous gust alleviation system is

battery free and powerless. This experimental validation builds on recent development of

REC laws, PCB design and energy harvesting characterization. The first step of this

experiment is to characterize and validate PCB layout design, manufacturing and testing.

The PCB board is manufactured by PCB Universe using FR-4 (135ºC) Material. The

PCB layout contains a two mode REC law, provides device interface directly with the I/O

ports on the board. The finished PCB prototype is shown in Figure 5.23. This PCB

control performance is experimentally tested and verified by matching the REC controller

built using dSPACE 1005 hardware, Control Desk and MatLab Simulink software. Table

5.7 lists all the components contained in PCB layout and their nominal values. All these

surface mount resistors and ceramic capacitors use 0805 packaging. All the nominal

values are verified using digital meters.

Figure 5.23 A finished PCB prototype of Multimode REC Laws.

3 Distance from clamp to the start of active element of respective piezoelectric transducers.

110

Table 5.7 Component Parameters of PCB Layout for Multimode Vibration Control.

Component Symbol Nominal Values Component Purpose

C1 150pF C1_1st Mode

C2 5600pF C2_1st Mode

C3 150pF C1_2nd

Mode

C4 500pF C2_2nd

Mode

R1 7.5MΩ R1_1st Mode


R3 10MΩ R3_1st Mode



R6 7.5MΩ R1_2nd

Mode

R7 7.5MΩ R2_2nd

Mode

R8 7.5MΩ R3_2nd

Mode

R9 7.5MΩ R4_2nd

Mode

R10 15MΩ R5_2nd

Mode

R11 7.5MΩ Summing Resistor 1



J1 Inputs

J2 Outputs

U1 LT1179SW

U2 LT1782IS5

The mechanical and material properties of PCB are experimentally identified by slightly

tuning numerical simulation for better agreements with experimental data shown from

Figure 5.14 to 5.17 and Figure 5.19 to Figure 5.22. Their values are shown in Table 5.8.

Table 5.8 Experimentally Identified Properties for the PCB Device.

Property/Component PCB

Length 40 mm

Width 10 mm

Thickness 1.016 mm

Distance from clamp to start of PCB 114 mm

Mass 2.625gram

Young’s Modulus 30 GPa

111

The PCB control circuit requires a ±4V DC voltage supply, which is currently provided

from BK Precision 9130, as shown in Figure 5.24, the experimental setup for the

autonomous gust alleviation system. This ±4V DC voltage supply can be replaced

directly by two Thinergy MEC 101-7 Lithium batteries or other equivalent thin-film

storage batteries. Experimental characterizations of capture and storage abilities of thin-

film batteries have been investigated by Sodano et al. (2005a)and Anton (2011). In

Figure 5.24, the autonomous gust alleviation wing spar is clamped to an APS Dynamics

ELECTRO-SEIS long stroke shaker with frequency range 0.1 Hz ~ 200 Hz. The shaker is

driven by an APS DUAL-MODE power amplifier. Gaussian white noise and Dryden

PSD function in Appendix E are implemented in Simulink to produce input signals in

both cases. The absolute tip displacement is measured using an MTI LTC-50-20 laser

sensor. A Spectral Dynamics SigLab 20-42 data acquisition system is used for recording

the absolute tip displacement-to-base acceleration FRF, where a Piezotronics U352C67

accelerometer is employed for input acceleration measurements.

Figure 5.24 Experimental setup for the autonomous gust alleviation system.

Figure 5.25 displays multi-mode predictions of the relative-tip-displacement-to-

acceleration FRF of the autonomous gust alleviation wing spar. As in the case of Figure

5.4, frequency response plots of relative tip displacement FRF is valid strictly. A 16dB

reduction for the first mode agrees very well with experimental data. An 11dB reduction

of the second mode can be obtained by numerical simulation, but this also requires a 20

times higher voltage supply. In addition, the tip disturbance caused by the second mode is

TREK2220 SigLab 20-42BK9130

MTI LTC-50-20

PCB

Wing Spar

112

30dB less than the first mode. Therefore, no effort is taken in the experiment for second

mode control. However, this method still applies very well for other UVA applications,

whose second mode response is significant enough to be controlled.

Figure 5.25 A comparison of relative tip displacement frequency spectrum response

predicted with numerical simulation for the first two modes, showing both open loop and

closed loop cases.

CHAPTER SUMMARY

The analytical solutions and numerical simulation derived in the previous chapter are

validated for various experimental cases. The first experiment is given for the aluminum

based multifunctional wing spar in order to characterize and validate reduced energy

control law. An 11dB and 7dB reduction for the first and the second vibration mode is

experimentally validated. The second experiment is dedicated to the autonomous gust

alleviation system building on a fiberglass composite sandwich wing spar. The first step

characterizes and compares energy harvesting properties of monolithic and microfiber

composite transducers. These identified properties are then used as input data for the

numerical prediction of the dynamic properties of the multifunctional wing spar. In the

second step, the frequency spectrum responses of the wind spar are measured with a

16dB reduction of the first vibration mode. Finally, predicted and measured results are

compared. Results indicate that the analytical formulation and numerical simulation

presented in previous research paper yields acceptable accuracy within 5%.

113

CHAPTER 6 DISSERTATION SUMMARY

This dissertation has demonstrated the feasibility, realization and implementation of the

concept and design of using harvested energy to directly control the vibration response of

flexible aerospace systems via piezoelectric materials. Smart material technology which

incorporates sensors, actuators, and real time control laws with composite sandwich

substrates have been implemented in such systems to achieve the required characteristics.

This dissertation has developed a scientific basis for characterizing the feasibility of using

harvested ambient energy to suppress vibrations in aerospace structures.

This main contributions of this research are: 1) established ambient vibration levels in

time histories; 2) developed a model for piezoelectric and fiber composite materials

integrated into flexible components for a multifunctional cantilever beam; 3) derived a

predictive model for energy conversion from embedded piezoelectric and fiber composite

materials including the associated electronics; 4) developed and derived the

electromechanical governing equations for vibration and control of a multifunctional

composite structure consisting of embedded piezoelectric and fiber composite materials

in the general aerospace structures; 5) developed a feedback control law based on

minimum energy constraints provided by the harvested ambient energy; 6)

experimentally validated the theory produced in item 4; 7) integrated the actuation,

harvesting and sensing materials into a composite sandwich structural element to form a

multifunctional structure with structural, sensing , harvesting and control functionality; 8)

established ambient vibration levels for a typical small UAV to represent both normal

flight condition and wind gust disturbance; 9) designed frequency domain gust alleviation

systems supplied by local power sources harvested from ambient energy; 10) designed a

proof of concept experiment to validate the results found in the first five tasks; 11)

applied this concept and design for a composite sandwich wing spar with the goal of

providing self-contained gust alleviation.

The research issues have been addressed are: 1) the characterization of appropriate

ambient energy; 2) the electromechanical modeling of vibration control and collocated

piezoelectric harvesting and strain sensing; 3) the development of vibration control laws

114

with limited energy consumption; 4) the analysis of bending strength and beam stiffness

analysis for the composite sandwich substrates; 5) the integration of piezoelectric, fiber

composite harvesting materials into a load bearing composite sandwich structure

members to enable multi-functionality; 6) the experimental determination if the scenario

of using harvesting energy to perform control is feasible or not; 7) the incorporation of

energy harvesting devices and gust load alleviation systems into small UAVs, providing

local power source for low-power sensors and controllers in aircraft.

In summary, this research addressed the question of whether or not harvested ambient

energy can be used to provide enough control efforts to provide a reasonable level of

vibration suppression and to quantify the degree to which such control can be

accomplished. A reduced energy law has been developed to examine vibration control

performances with strong limits on the control input energy. A multifunctional approach

has been applied to integrate the piezoelectric, fiber composite transducer materials along

with the control and harvesting electronics into the structure elements. One of the

promising applications of simultaneous energy harvesting and vibration control in aircraft

is in providing local power source to autonomous gust alleviation systems of a self-

contained small UAV. The research has demonstrated the integration of piezoelectric

energy harvester, smart materials, multifunctional structures and composite sandwich

structures into a UAV platform to perform simultaneously gust alleviation and energy

harvesting.

115

APPENDICES

APPENDIX A PIEZOELECTRIC CONSTITUTIVE EQUATIONS

This appendix summarizes the standard three-dimension (3D) form of constitutive

equations on piezoelectricity. Reduced 1D forms are presented for piezoelectric

transducers working on both 3-1 and 3-3 operating mode.

Standard 3D Form of Constitutive Equations

According to Standards Committee of the IEEE Ultrasonics (1984), linear

piezoelectricity can be represented by four standard sets of constitutive equations, by

taking two of the four field variables as the independent variables. The field variables are

the mechanical strain Sij, the mechanical stress Tij, the dielectric charge displacement Dk,

and the electrical field strength Ek. All of these four sets describe the same piezoelectric

phenomena. For studies of damping effects, the mechanical stress and electric field

strength are more often taken as the independent variables, see Clark (2000a) for

reference. However, For investigations of energy harvesting, the mechanical strain and

electric field strength are more popular independent variables, see Badel et al. (2006b) for

reference. The corresponding constitutive equations become:

E

ij ijkl kl kij k

S

i ikl kl ik k

T c S e E

D e S E

. (A.1)

Here, c represents the elastic stiffness constants, e denotes the piezoelectric constants and

ε indicates the permittivity constants. The superscripts E and S denote the corresponding

parameters at constant electric field or mechanical strain, respectively. The subscripts i, j

and k are tensor notations.

Reduced Equations for 3-1 Actuation Modes

For cantilever configurations with piezoelectric transducers satisfying the Euler-Bernoulli

beam assumptions, the only non-zero stress component T1 is in the axial direction, see

Bent (1994) for reference. If the piezoelectric transducers are under a 3-1 operating

mode, the constitutive equations can be reduced to:

116

1 11 1 31 3

3 31 1 33 3

E

S

T c S e E

D e S E

. (A.2)

As illustrated in Figure A.1 (a), the 3-1 mode PZT-based transducers are excited under

force fp in the 3-direction and deformed by wp in the 1-direction. The monolithic d31

domain PZT-based harvester and sensor QP10N satisfies the constitute equations given in

Equation (A.2).

(a) (b)

Figure A.1Piezoelectric transducers with (a) 3-1 actuation mode (b) 3-3 actuation

mode.

The piezoelectric coefficients in reduced form in Equation A.2 are not equal to those in

standard 3-D form in Equation A.1. These reduced coefficients can be derived from the

elastic modulus and the dielectric permittivity of the piezoelectric material. The following

constitute relations for 3-1 operating mode, are given by duToit et al. (2005):

11

11

11 12

11 31 12 31

31

11 12

2

31 11 12

33 33

11 12

2 2

2 2

2 2

,

2

,( ) ( )

( ) ( )

( ).

( ) ( )

E

E

E E

E E

E E

E E

S T

E E

sc

s s

s d s de

s s

d s s

s s

. (A.3)

Here, s is the elastic compliance constant, which is the reciprocal of elastic/Young’s

modulus, d is the piezoelectric constant. The superscript T denotes the corresponding

parameters at constant mechanical stress.

Reduced Equations for 3-3 Actuation Modes

For the piezoelectric transducers operating in the 3-3 operating mode, such as Micro

Fiber Composites (MFC), their interdigitated electrode configurations are deformed and

+ + +

- - -f p ,wp

1

2

3

f p ,wp

+ - +

- + -

2

1

3

117

excited both in the 3-direction shown in Figure A.1 (b). Assuming that the electrode

region is electrically inactive, whereas the region between the electrodes utilizes the full

3-3 effect, the linear constitutive equations for cantilever configuration under 3-3

operating mode can be reduced to:

3 33 3 33 3

*

3 33 3 33 3

E

S

T c S e E

D e S E

. (A.4)

The piezoelectric coefficients in the 3-3 mode are given by:

11

33

11 33 13

11 33 13 31

33

11 33 13

2 2

* 33 31 13 31 33 11 33

33 33

11 33 13

2

2

2

2

,( )

,( )

( ),

( )

E

E

E E E

E E

E E E

E E E

S T

E E E

sc

s s s

s d s de

s s s

s d s d d s d

s s s

. (A.5)

118

APPENDIX B EULER-LAGRANGE EQUATIONS USING

EXTENDED HAMILTON’S PRINCIPLE

For a given structure, the total kinetic energy Te is a function of the generalized

coordinates and their time derivatives, denoted by:

1 2 1 2( , , , , ).e e n nT T q q q q q q (B.1)

The total potential energy Ue and internal electrical energy Eie are functions of the

generalized coordinates, denoted by:

1 2 1 2

1 2 1 2

,( , , , , )

( , , , , ).

e e n n

ie ie n n

U U

E E

q q q q q q

q q q q q q

(B.2)

If the non-conservative work is denoted by Enc, the extended Hamilton’s principle over a

given time period satisfies the following relation:

2

1

( ) 0.

t

e ie nc

t

T U E E dt (B.3)

The Euler-Lagrange Equations can be derived and represented by:

.e e ie ncT T E Ed U

dt q q q q q

(B.4)

119

APPENDIX C CROSS-SECTION TRANSFORMATION

If a multi-layer beam is made of layers with rectangular cross sections, the cross-section

transformation (CST) method is a simple way to compute an equivalent modulus, see

Vable (2002) for reference. Taking a four-layer beam as an example, shown in Figure

C.1, the dimensions of the cross section in the y direction are transformed using the

equation ( / ) .k k ref k

h E E h Here Eref is the modulus of elasticity of the reference material

(layer D) on the cross section, k

h and k

h are the original width and transformed width of

each layer (k=A, B, C, D).

Figure C.1 Cross section transformation of (a) original beam (b) transformed

homogeneous beam.

The total bending rigidity thus can be written as

( )yy k yy k ref yyEI E I E I (C.1)

The location of neutral axis y in composite beam is determined from the following

equation:

( ) / ( ).c k k k k kE A E A (C.2)

Here k

is the location of the centroid of the kth layer as measured from a common datum

line.

Eref =Ec y

z

EA

EB

EC

ED

y

z

EA

EB

EC

ED

hA

hc

(Ek/EC)hA

hc

120

APPENDIX D ENERGY FORMULATIONS OF

ELECTROMECHANICAL CANTILEVER BEAM USING

DISTRIBUTED PARAMETER METHOD

The multiple layer beam structure provides multifunctional abilities, such as energy

harvesting, strain sensing and vibration control, and also introduces complex static and

dynamic characteristics. The approximate distributed-parameter model was derived

analytically and validated experimentally by Erturk and Inman (2011) for cantilevered

piezoelectric energy harvesters. This appendix summarizes the energy formulations of the

electromechanical cantilever beam using a distributed-parameter approach, which will be

applied for equivalent electromechanical model prediction. This summary is for Euler-

Bernoulli cantilever beams with multiple PZT-based transducer layers in the general case,

as shown in Figure D.1.

Figure D.1 Representation of a Euler-Bernoulli cantilever beam with multiple PZT

layers.

The total potential energy U in the cantilever beam structure can be written as:

1.

2xx xx

V

U S T dV (D.1)

Here Sxx and Txx denote the stress and strain vectors of the host structure and the PZT-

based transducers over the whole structural volume V. If wrel(x,t) stands for the

displacement relative to the clamped end of the beam, at an arbitrary position in the x

direction along the neutral axis, and at a given time t, the non-zero strain component is

given by:

2

2

( , )( , , ) .

rel

xx

w x tS x z t z

x

(D.2)

PZT layer

substrate

tip mass

x1

z

x2 x3 x

EI, A, m, L

121

According to Hooke’s law, if the beam is linearly elastic, the structural stress Txx is the

product of the global elastic modulus Es and the strain vector Sxx. Thus, the structural

strain energy becomes:

2

2

0

( , )1( ) .

2

L

rel

s s sU

w x tE I dx

x

(D.3)

Here, Es Is stands for the bending stiffness of the structure, which is taken as constants

across each of the three individual longitudinal sections in Figure D.1. The bending

stiffness of each longitudinal section is estimated by cross-section transformation method

introduced in Appendix C. The space-dependant bending stiffness can therefore be

represented using Heaviside step functions:

1 1 1 2 2 2 1 3 3 3 2( ) [ ( ) ( )] ( ) ( ) ( ) ( ).

s sE I x E I H L x H x E I H L x H x L E I H L x H x L (D.4)

Again, here x is the longitude coordinate, as shown in Figure D.1. According to the

constitutive equations (A.2) and (A.4), the stress components in PZT transducers under

dji actuation mode are given by:

.E

i ii i ji jT c S e E (D.5)

Here, Si represents the strain vector Sxx in the x direction. Since the voltage v(t) across the

PZT layers is the product of the thickness of each PZT layer with the instantaneous

electric field Ej(t), the total strain energy in each PZT layer becomes:

2 2

2 2

2 2

1 ( )[ ( ) ( ) ] ,

2P

E rel rel

ii ji

V p

p

w wv tU c z ze t dv

x h x

(D.6)

The subscript p denotes the piezoelectric layer. The bending stiffness of each PZT layer

is derived using the parallel axis theorem. If the coupling term Jp of the piezoelectric

transducer over the cross section area Ap is defined as:

122

,

p

ji

p

A p

eJ z dydz

h (D.7)

where hp represents the thickness of the PZT layer, then Equation (D.6) becomes:

1 2 2

2

2 2

0

1[ ( ) ( ) ] ,

2

x

E rel rel

p ii p pU

w wc I J v t dx

x x

(D.8)

If w(x,t) denotes the transverse displacement of the multifunctional beam in the absolute

frame of the reference base, the total kinetic energy of the multi-layer beam structure Te

can be represented by:

2

0

1 ( , )( ) ( )( ) .

2

L

se

w x tT x A x dx

t

(D.9)

Where ρ(x) stands for the density and As(x) for the cross section area of the beam

structure in the x direction. The internal electric energy of the PZT layer can be expressed

by:

1.

2ie j j p

Vp

E E D dV (D.10)

After substituting the expressions for the electric field and the electric displacement from

the constitutive equations (A.2) to (A.4) in Appendix A into equation (D.10), the internal

electric energy becomes:

21

2

0

21 1( ) ( ) .

2 2

x

rel

ie p p

x

wCE J v t dx v t

(D.11)

Here the capacitance term Cp of the PZT transducer is defined as:

.

s

ji p

p

p

CA

h

(D.12)

Here Ap denotes the electrode surface area of the PZT transducer. Since the base

excitation from ambient vibration is accounted for in the kinetic energy, the only non-

conservative work done by the PZT transducer is:

( ) ( ).nc

E Q t v t (D.13)

Here Q denotes the electrical charge across the PZT transducer.

123

APPENDIX E DRYDEN POWER SPECTRAL DENSITY

SPECTRUM

At any particular frequency ɷ, the power spectral density (PSD) of any function x(t) is

the mean squared value of that part of x(t), whose frequency is within an infinitely narrow

band, centered on ɷ. It provides the designer information of how the mean squared

values of the argument are distributed with frequency ɷ. For a given function of x(t), its

PSD function is given by:

2

, 00

1( ) lim ( , , ) .

T

T

x t dtT

(E.1)

Here, Φ(ɷ) is the PSD function of x, T is the duration in seconds of the record of x(t), and

x(t, ɷ, Δ ɷ) is the component of x(t) which lies within the frequency band ɷ ± Δ ɷ/2.

The Dryden PSD spectrum is a popular and accepted way to represent the PSD function

of atmospheric turbulence, since it is simpler and more easily programmed compared to

other approaches. The definition of the Dryden PSD spectrum can be found from McLean

(1969), represented by:

2

2

0

2 20

0

1 3( )

( )

(1 ( ) )

.

v

v v

Dry

v

L

L U

LU

U

(E.2)

Here U0 is the aircraft trim velocity. Lv is the vertical scale of turbulence and σv is the

RMS vertical gust velocity. In arriving at these representations, the following

assumptions are made:

The atmospheric turbulence is a stationary random process.

The vertical gust velocity σv for clear air, cumulus cloud and severe storm are

0.5m/s, 2.0m/s and 4.0m/s, respectively.

The turbulence scale length varies with height, and follows the definition of

reference Chalk et al. (1969).

124

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