Development of an Energy Harvesting Device using

Piezoceramic Materials

by

Vainatey Kulkarni

A thesis submitted in conformity with the requirements

for the degree of Doctor of Philosophy

Mechanical and Industrial Engineering

University of Toronto

© Copyright by Vainatey Kulkarni 2015

ii

Development of an Energy Harvesting Device using Piezoceramic

Materials

Vainatey Kulkarni

Doctor of Philosophy

Mechanical and Industrial Engineering

University of Toronto

2015

Abstract

Piezoelectric energy harvesters are increasingly being pursued for their potential to replace

finite-life batteries in wireless sensor modules and for their potential to create self-powered

devices. This work presents the development of a novel piezoelectric harvester that attempts to

improve upon the power output limitations of current piezoelectric harvesting technology. This

novel harvester uses the concept of torsion on a tube to produce shear stresses and hence uses

improved piezoelectric properties of the shear mode of piezoceramics to generate higher power

outputs. This concept is first presented in this work and a proof-of-concept prototype is utilized

to experimentally demonstrate the validity of this novel device. After this, the behaviour of the

novel harvester is explored through an investigation into three cross-section geometries of the

torsion tube and varying geometries of the eccentric mass using three different comparison

metrics. Through this, it is observed that configurations with higher torsional compliance and

high eccentric mass inertias have the potential for the highest power output and highest harvester

effectiveness. However, the mechanical damping in the system is also found to significantly

impact the harvester output resulting in prototypes of the various configurations not performing

as expected. As a result of this discrepancy, the factors affecting the performance of the harvester

are analyzed in greater detail through the development of a mathematical model that is then used

iii

to develop a set of guidelines to direct the design of a torsion harvester for a desired application.

These guidelines are then used to develop an improved torsion harvester with a demonstrated

ability to produce 1.2 mW of output power at its resonant frequency to power a wireless sensor

module. Finally, the use of alternative materials such as single crystals of PMN-PT in the torsion

harvester is also examined. Through finite element simulations and with material properties

reported in the literature, the torsion harvester used with the sensor module is found to

significantly benefit with the addition of the single crystal materials and ultimately generate

300% improvements in average output power while converting 11% of the input energy into

usable electrical energy.

iv

Acknowledgments

I would like to express my sincere thanks to Professor Ridha Ben Mrad and Dr. Eswar Prasad

for their insight, expertise and encouragement throughout the course of this project. This thesis

would not be possible without their constant support and their patience, especially during the

challenging aspects of the research process. I am also grateful to them for their positive influence

on my educational journey. I would also like to thank Professor Hani Naguib and Professor Jean

Zu for the comments and constructive feedback they have provided on this project during the

yearly committee meetings.

I would like to acknowledge financial contributions made to this project by the Natural

Sciences and Engineering Research Council of Canada (NSERC), Ontario Graduate Scholarships

(OGS), the University of Toronto and Sensor Technology Ltd. I would particularly like to thank

Dr. Sailu Nemana at Sensor Technology Ltd., for his assistance with piezoceramic

manufacturing methods and for providing access to testing equipment at Sensor Technology. I

would also like to thank Frédéric Giraud, Michel Amberg, and Christophe Giraud-Audine at

IRCICA, Lille, France for inviting me to their research institution and helping me develop the

interface circuitry presented in this thesis.

I would like to thank my parents (Arvind and Vinata) and my sister Anvita for their constant

and unending support. I am also grateful to my friends and colleagues: Sergey, Jalal, Mike, Paul,

Khalil, Trish, Sadegh, Hirmand, Andrew, Jacky, James, Imran, Tae, Eu-Jin, Steffen, Chakameh,

Ali, Alaeddin, Donn, Faez, Amro, and Martha for their help and advice and for making this

journey an enjoyable one.

v

Contents

Acknowledgments .......................................................................................................................... iv

List of Tables ............................................................................................................................... viii

List of Figures ................................................................................................................................. x

List of Appendices ........................................................................................................................ xv

Introduction ................................................................................................................................ 1 1.

1.1 Overview of Piezoelectric Materials ................................................................................... 1

1.2 Piezoelectric Materials for Energy Harvesting ................................................................... 6

1.3 Energy Harvesting Devices ................................................................................................. 9

1.4 Circuits for Energy Harvesting ......................................................................................... 13

1.4.1 Energy Storage ...................................................................................................... 13

1.4.2 AC/DC Converter ................................................................................................. 13

1.4.3 DC-DC Step Down Converter .............................................................................. 14

1.4.4 Non-Linear Voltage Processing ............................................................................ 14

1.5 Limitations of Current Technology .................................................................................. 15

1.6 Thesis Objectives .............................................................................................................. 15

1.7 Thesis Outline ................................................................................................................... 16

Novel Harvester Design ........................................................................................................... 17 2.

2.1 Harvester Concept and Model .......................................................................................... 19

2.2 Prototype and Experimental Results ................................................................................. 23

2.3 Comparison with Cantilever Harvester ............................................................................. 27

2.3.1 Generalized Comparison ....................................................................................... 30

2.4 Summary ........................................................................................................................... 32

Comparison of Harvester Geometries ...................................................................................... 33 3.

3.1 Comparison Metrics .......................................................................................................... 33

vi

3.2 Harvester Configurations .................................................................................................. 37

3.3 Simulation Results ............................................................................................................ 40

3.4 Experimental Results ........................................................................................................ 43

3.5 Summary ........................................................................................................................... 50

Analytical Modelling and Design Guidelines .......................................................................... 51 4.

4.1 Analytical Expressions ...................................................................................................... 51

4.2 Influence of Various Parameters ....................................................................................... 57

4.3 Finite Element Model Results ........................................................................................... 60

4.4 Summary ........................................................................................................................... 64

Improved Torsion Harvester for a Wireless Sensor Module ................................................... 65 5.

5.1 Harvester Design and Performance .................................................................................. 65

5.2 Experimental Results ........................................................................................................ 68

5.3 Interface Circuits ............................................................................................................... 74

5.4 Sensor Electronics ............................................................................................................. 78

5.5 Summary ........................................................................................................................... 81

Evaluation of Single Crystal Materials .................................................................................... 82 6.

6.1 Overview of Single Crystal Materials ............................................................................... 82

6.2 Manufacturing Methods .................................................................................................... 84

6.3 Characterization Considerations ....................................................................................... 85

6.4 Single Crystals for Torsion Harvester Concept ................................................................ 85

6.4.1 d15 Shear Mode ..................................................................................................... 86

6.4.2 d36 Shear Mode ..................................................................................................... 87

6.5 Characterization of Commercial Single Crystals Elements .............................................. 89

6.6 Summary ........................................................................................................................... 91

Concluding Remarks ................................................................................................................ 92 7.

7.1 Contributions ..................................................................................................................... 94

vii

7.1.1 Torsion Harvester .................................................................................................. 94

7.1.2 Design Guidelines for Torsion Harvester ............................................................. 94

7.1.3 Investigation of Single Crystals for Torsion Harvester ........................................ 94

7.2 Future Work ...................................................................................................................... 95

7.2.1 Mechanical Damping ............................................................................................ 95

7.2.2 Interface Electronics ............................................................................................. 95

7.2.3 Single Crystals ...................................................................................................... 95

7.3 Conclusion ........................................................................................................................ 96

References ..................................................................................................................................... 97

Appendix A: Buffer Circuit for Voltage Measurements from Piezoelectric Harvesters ............ 103

viii

List of Tables

Table 2.A: Piezoelectric properties of various PZT compositions ............................................... 18

Table 2.B: Major dimensions of torsion harvester model. Units are in Inches ............................ 21

Table 2.C: Results of FEM Modal Analysis ................................................................................. 21

Table 2.D: Impedance measurements of bonded PZT plates with associated measurement

accuracy ........................................................................................................................................ 25

Table 2.E: Comparison of output power from simplified cantilever and torsion harvesters ........ 31

Table 3.A: Characteristics and geometry of various torsion harvester configurations explored .. 39

Table 3.B: Summary of simulation results ................................................................................... 40

Table 3.C: Stiffness and mass estimation for SDOF harvester model .......................................... 47

Table 3.D: Source impedance measurements of each harvester prototype with associated

measurement accuracy .................................................................................................................. 47

Table 3.E: Summary of simulation and prototype performance ................................................... 49

Table 5.A: Characteristics of torsion harvester for wireless sensor module ................................. 66

Table 5.B: Summary of simulation results ................................................................................... 68

Table 5.C: Stiffness and mass estimation for SDOF model ......................................................... 73

Table 5.D: Energy consumption of sensor and microcontroller circuit per active cycle of sensing

....................................................................................................................................................... 79

Table 6.A: Piezoelectric properties of various PZT compositions and PMN-PT single crystal ... 82

Table 6.B: Summary of simulation results with d15 PMN-PT single crystal compared to PZT-5A

results from chapter five ............................................................................................................... 86

ix

Table 6.C: Summary of simulation results with d36 PMN-PT single crystal compared to PZT-5A

results from chapter five ............................................................................................................... 88

Table 6.D: Results from impedance analyses of PMN-PT samples ............................................. 90

x

List of Figures

Figure 1.1: PZT unit cell with (a) cubic symmetry and no polarization above Curie temperature

(b) tetragonal symmetry and net polarization below Curie temperature ........................................ 3

Figure 1.2: Electric dipoles in a piezoceramic sample (a) before poling (b) during poling (c) after

poling. P represents the poling direction in the sample .................................................................. 3

Figure 1.3: Index convention for labelling anisotropic piezoelectric constants. P refers to the

poling direction in the piezoceramic sample .................................................................................. 4

Figure 1.4: Schematic of one cantilever beam harvester [10] ........................................................ 9

Figure 1.5: Schematic of the air-spaced cantilever beam. The bottom beam is the mechanical

beam while the top beam is piezoelectric [18] .............................................................................. 11

Figure 1.6: The frequency response of (a) a cantilever harvester compared to (b) a Smart-Sand

harvester [20] ................................................................................................................................ 12

Figure 2.1: Illustration of (a) transverse (d31) mode and (b) longitudinal (d33) mode of

piezoceramic materials with poling direction, P, and applied force directions, F, labeled. ......... 17

Figure 2.2: Poling and harvesting directions for PZT shear mode harvester ................................ 19

Figure 2.3: Design of a torsion based energy harvesting substructure ......................................... 20

Figure 2.4: Harvester deformation in torsional mode at approximately 600 Hz .......................... 21

Figure 2.5: ANSYS Model of harvester for piezoceramic analysis .............................................. 22

Figure 2.6: Voltage distribution in PZT elements of torsion harvester with optimal load

resistance for peak power dissipation. Units are in Volts. ............................................................ 23

Figure 2.7: Torsion harvester prototype ........................................................................................ 23

Figure 2.8: Mechanical response of torsion harvester prototype .................................................. 24

Figure 2.9: Open circuit voltage frequency response of harvester prototype ............................... 26

xi

Figure 2.10: Short circuit current frequency response of harvester prototype ............................. 26

Figure 2.11: Output power over varying load resistance values at 615 Hz .................................. 27

Figure 2.12: Output power frequency spectrum of harvester prototype ....................................... 27

Figure 2.13: ANSYS model of cantilever harvester ..................................................................... 28

Figure 2.14: Voltage distribution in cantilever harvester PZT elements. Units are in Volts. ....... 29

Figure 3.1: SDOF model of harvester with sinusoidal base excitation ......................................... 34

Figure 3.2: Energy dissipation from SDOF harvester model over time ....................................... 36

Figure 3.3: Torsion tube cross-sections examined in order of decreasing torsion constants: (a)

Solid (b) C-Channel (c) Slotted-C ................................................................................................ 38

Figure 3.4: Illustration of centimetre-scaled torsion harvester ..................................................... 38

Figure 3.5: Influence of modal damping ratio on peak power output at resonance through optimal

load resistance of each cross-section configuration ...................................................................... 42

Figure 3.6: Influence of modal damping ratio on harvester effectiveness at resonance through

optimal load resistance of each cross-section configuration ......................................................... 43

Figure 3.7: Prototypes of two centimetre-scaled torsion harvesters with a Canadian 10¢ coin for

scale: (a) solid cross-section and (b) slotted-C cross-section ....................................................... 44

Figure 3.8: Locations of interest in characterizing mechanical response of each torsion harvester

prototype ....................................................................................................................................... 44

Figure 3.9: Mechanical response of solid tube harvester .............................................................. 45

Figure 3.10: Mechanical response of slotted-C tube harvester ..................................................... 45

Figure 3.11: Open circuit voltage frequency response of torsion harvester prototypes ............... 48

Figure 3.12: Output power of harvester prototypes over varying load resistance values at

resonance ....................................................................................................................................... 48

xii

Figure 4.1: Simplified model of torsion harvester ........................................................................ 52

Figure 4.2: Cross-section of the simplified harvester. Domain I represents the substrate material

and domain II corresponds to the piezoelectric element ............................................................... 52

Figure 4.3: Influence of piezoelectric coefficients d15 and 𝜖11𝑆 on the power output of the

simplified torsion harvester ........................................................................................................... 59

Figure 4.4: Influence of mechanical damping on peak power output of original torsion harvester

....................................................................................................................................................... 61

Figure 4.5: Influence of base acceleration magnitude on peak power output of original torsion

harvester ........................................................................................................................................ 62

Figure 4.6: Influence of d15 piezoelectric constant on peak power output of original torsion

harvester ........................................................................................................................................ 62

Figure 4.7: Influence of dielectric permittivity (𝜖11𝑆) on peak power output of original torsion

harvester ........................................................................................................................................ 62

Figure 4.8: Influence of scaling factor on peak power output of original torsion harvester ......... 63

Figure 4.9: Influence of cross-section geometry and substrate Young’s modulus on peak power

output of original torsion harvester ............................................................................................... 63

Figure 5.1: (a) Torsion tube geometry and (b) cross section of improved torsion harvester ........ 66

Figure 5.2: Schematic of improved torsion tube harvester for the wireless sensor module showing

all components .............................................................................................................................. 66

Figure 5.3: Effect of torsion tube material’s elastic modulus on power output of torsion harvester

....................................................................................................................................................... 67

Figure 5.4: Voltage distribution in torsion harvester at resonance with optimal load resistance.

Units are in Volts RMS. Voltage at the output electrodes is 45.7 VRMS. ...................................... 68

Figure 5.5: Torsion harvester prototype ........................................................................................ 69

xiii

Figure 5.6: Locations of interest in characterizing mechanical response of each torsion harvester

prototype ....................................................................................................................................... 69

Figure 5.7: Mechanical frequency response of torsion harvester ................................................. 70

Figure 5.8: Open circuit voltage frequency response of torsion harvester under 0.25 gpk base

excitation ....................................................................................................................................... 71

Figure 5.9: Short circuit current frequency response of torsion harvester under 0.25 gpk base

excitation ....................................................................................................................................... 71

Figure 5.10: Output power of harvester prototypes over varying load resistance values at

resonance ....................................................................................................................................... 72

Figure 5.11: Output power from improved torsion harvester prototype under various excitation

accelerations at resonance ............................................................................................................. 73

Figure 5.12: Circuit layout of prototyped diode bridge rectifier .................................................. 74

Figure 5.13: Circuit layout of prototyped synchronous electrical charge extraction (SECE) circuit

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

Figure 5.14: Voltage waveforms in the SECE circuit ................................................................... 75

Figure 5.15: Output power of the two circuits considered with a variable resistive load ............. 76

Figure 5.16: Output power of the two circuits considered with a variable DC load .................... 76

Figure 5.17: Charging a 470 μF capacitor with both circuits under varying excitation conditions

....................................................................................................................................................... 78

Figure 5.18: Layout of wireless temperature sensor module with torsion harvester and interface

circuitry ......................................................................................................................................... 79

Figure 5.19: Wireless sensor module and interface circuitry ....................................................... 80

Figure 5.20: Temperature measurements as recorded on the RF memory by temperature sensor

and microcontroller system ........................................................................................................... 81

xiv

Figure 6.1: Example of domain engineering with (a) unpoled single crystal with possible

spontaneous polarization directions and (b) <001> poled crystal with possible polarization

directions [48] ............................................................................................................................... 83

Figure 6.2: Crystal orientations for PMN-PT with d36 response [50]. P is the poling direction and

T is the applied stress. ................................................................................................................... 84

Figure 6.3: Schematic of modified Bridgman technique for crystal growth [48] ......................... 85

Figure 6.4: Voltage distribution in torsion harvester with d15 PMN-PT single crystal at resonance

with optimal load resistance. Units are in Volts RMS. Voltage at the output electrodes is 67.9

VRMS. ............................................................................................................................................. 86

Figure 6.5: Piezoceramic poling directions to utilize d36 shear mode in the torsion harvester ..... 87

Figure 6.6: Voltage distribution in torsion harvester with d36 PMN-PT single crystal at resonance

with optimal load resistance. Units are in Volts RMS. Voltage at the output electrodes is 3.12

VRMS. ............................................................................................................................................. 88

Figure 6.7: Example impedance spectrum plot from one PMN-PT sample ................................. 90

xv

List of Appendices

Appendix A: Buffer Circuit for Voltage Measurements from Piezoelectric Harvesters ............ 103

1

Introduction 1.

With recent developments in personal electronics and micro-electronics, wireless sensor

networks are becoming increasingly ubiquitous in day-to-day applications. These devices allow

for the creation of adaptive and “smart” systems that are able to continuously monitor a dynamic

environment and autonomously respond to appropriate stimuli. As a result, wireless sensor

networks are widely utilized for structural health monitoring for civil infrastructure, aerospace

and automotive vehicles, and remote monitoring applications. However, despite the advances in

wireless sensor technology, such devices are still powered by finite-supply chemical batteries

that require frequent replacement. As a result of this maintenance demand, wireless sensor

modules cannot be placed in truly inaccessible locations, locations where they are most useful,

thus preventing them from achieving their full potential. To overcome this limitation, much

research has now been dedicated to the development of self-powered devices that can overcome

the current reliance and limitations of finite-supply batteries. The core of this research has

focused on utilizing ambient energy from a device’s surroundings and converting it into usable

electrical energy. While methods of harvesting ambient solar, magnetic and thermal energy have

all been proposed, one of the most common methods of power harvesting is to convert ambient

mechanical vibrations into electricity through the use of piezoelectric materials. These materials

exhibit the direct piezoelectric effect that produces an electrical charge across the material

corresponding to a mechanical deformation. This approach of using piezoelectric materials for

energy harvesting applications is the focus of this work and the development of a novel device to

power a wireless sensor module is described in subsequent chapters.

1.1 Overview of Piezoelectric Materials

Piezoelectric energy harvesting makes use of the material properties present in piezoelectric

materials to achieve the conversion of mechanical energy in the form of ambient vibrations into

usable electrical energy. This section provides a brief overview of the mechanisms in the

2

materials used to achieve this conversion along with material properties relevant for energy

harvesting.

Piezoelectric materials are characterized by the direct piezoelectric effect that was first

discovered by Jacques and Pierre Curie in 1880. In this, when certain materials are subjected to

mechanical strain, they become electrically polarized and the degree of polarization is found to

be proportional to the applied strain. This polarization is the result of uneven separation of

positive and negative charges in the unit cell of these materials, leading to net polarization at the

crystal surface. As a result of this uneven separation of charges, piezoelectric materials also

exhibit the inverse piezoelectric effect, where the application of an electric field results in

mechanical deformation of the material.

While many materials such as quartz, tourmaline, along with some polymers such as

polyvinylidene fluoride (PVDF) display the piezoelectric effect, the most commonly

piezoelectric materials are piezoelectric ceramics (“piezoceramics”) such as Pb(Zr1-xTix)O3, lead

zirconate titanate (PZT) due to its large electromechanical coupling coefficients [1]. PZT belongs

to a family of materials with a perovskite crystal structure that is characterized by a A2+

B4+

O2-

chemical formula. In this, A represents a large divalent metal ion such as lead (Pb), B denotes a

tetravalent metal ion such as zirconium (Zr) or titanium (Ti) and O represents an oxygen ion.

Above a temperature known as the Curie temperature, these perovskite crystals take on a cubic

crystal structure that is symmetrical and thus possesses no charge separation, resulting in a loss

of piezoelectricity. Below the Curie temperature, each unit cell of the material undergoes a

transformation and takes on a tetragonal symmetry that contains a net electric dipole as a result

of uneven distribution of charges within the unit cell. The transformation in the two unit cell

crystal structures is shown in Figure 1.1.

3

Figure 1.1: PZT unit cell with (a) cubic symmetry and no polarization above Curie temperature (b) tetragonal

symmetry and net polarization below Curie temperature

On the macroscopic scale of a piezoelectric sample containing a number of unit cells, the net

electric dipole of each perovskite unit cell below the Curie temperature is oriented randomly in

groups known as Weiss domains. When averaged over the size of the sample, this large number

of randomly oriented dipoles results in no overall polarization of a perovskite sample and hence

no piezoelectric behaviour. Obtaining piezoelectric response from a piezoelectric material

sample then requires a process known as poling, where the sample is exposed to a large electric

field near the Curie temperature of the material. This application of an external electric field and

high temperature causes the dipole domains in the piezoelectric sample to preferentially align

along the electric field. Cooling the sample to room temperature and removing the external

electric field then causes the dipole domains to remain in the same direction as the previously

applied electric field, thus creating remnant polarization in the sample and giving it overall

piezoelectric properties. This process of poling a piezoceramic is shown in Figure 1.2.

Figure 1.2: Electric dipoles in a piezoceramic sample (a) before poling (b) during poling (c) after poling. P

represents the poling direction in the sample

4

As a result of the poling process and the remnant polarization, the piezoceramic sample also

exhibits strong anisotropy in mechanical and electric behaviour. The physical constants

representing such behaviour are tensor quantities and hence relate to the directions of the input

quantities as well as the output. As a result, each constant of a piezoelectric material contains two

subscripts, each of which reflects the direction of the input and output quantities. By convention,

the direction of polarization of the piezoceramic sample is chosen to coincide with the “3”

direction of indices and the remaining directions are assigned the subscripts using the convention

shown in Figure 1.3. Thus, under this convention, the stiffness constant of a piezoelectric

material relating the stress along the “3” direction to the strain along the “2” direction would be

labelled 𝑐32 while the absolute permittivity of the material relating to the electric field and

dielectric displacement both along “1” direction would be labelled as 𝜖11. Occasionally, a

superscript is added to the material constants indicating the quantity that is held constant while

determining the material constant. For instance, 𝜖11𝑇 , then refers to 𝜖11 measured under constant

stress, T, conditions, while 𝜖11𝑆 refers to the same quantity measured under constant strain, S,

conditions.

Figure 1.3: Index convention for labelling anisotropic piezoelectric constants. P refers to the poling direction in

the piezoceramic sample

Finally, beyond common material properties such as stiffness and dielectric permittivity, the

presence of piezoelectric phenomenon also gives rise to two unique constants that couple the

mechanical domain to the electrical domain. The first of these is the piezoelectric charge

constant, d, that relates the charge generated in the piezoelectric material as a result of the

applied stress. Thus, for a direct piezoelectric effect application such as a piezoelectric sensor,

this constant can be used to estimate the amount of electrical charge produce by the sensor under

a given applied load. Alternatively, for an inverse piezoelectric effect application such as a

piezoelectric actuator, this same constant can be used to estimate the amount of stress generated

5

in a piezoceramic as a result of input charge. The exact expressions relating the charge and stress

in the piezoceramic for the direct piezoelectric effect can be expressed as:

𝑫 = [𝒅]𝑻 + [𝝐𝑻]𝑬 (1-1)

In this, 𝐷 is the charge displacement vector in the piezoceramic material, [𝑑] is the tensor of

d constants, 𝑇 is the applied stress vector, [𝜖𝑇] is the dielectric permittivity tensor under

constant stress conditions, and 𝐸 is the applied electric field vector. As with other piezoelectric

constants, the piezoelectric charge constant is a tensor and is anisotropic for a poled material and

thus carries two subscripts to identify the relevant directions. For this constant, the first subscript

refers to the direction of the electric field and the second subscript reflects the direction of stress

produced in the piezoceramic. Thus, during the direct piezoelectric effect, the constant d31

reflects the charge generated in the “3” direction as a result of stress in the “1” direction, while

for the inverse piezoelectric effect, the same constant refers to the stress generated in the “1”

direction as a result of charge applied in the “3” direction.

An alternative to the piezoelectric charge constant is the piezoelectric voltage constant, g.

Although commonly used in many transducer applications, this constant is not unique by itself

and relates to the d constant through the dielectric permittivity:

𝑔 =𝑑

𝜖 (1-2)

Like the piezoelectric charge constant, this constant relates the mechanical and electrical

domains in the piezoelectric material, but unlike the charge constant, this constant relates to the

electric field (i.e. voltage) produced in the material as a result of applied charge. However, this

constant uses the same subscript notation as the piezoelectric charge constant.

The second commonly used constant to describe piezoelectric phenomenon is the

electromechanical coupling coefficient for a piezoelectric material. This coefficient relates to the

ratio of the mechanical energy accumulated in the piezoelectric material as a result of an

electrical input or vice versa. As a result, it is defined as:

𝑘 = √𝑈𝑒(𝑠𝑡𝑜𝑟𝑒𝑑)

𝑈𝑚(𝑎𝑝𝑝𝑙𝑖𝑒𝑑) or 𝑘 = √

𝑈𝑚(𝑠𝑡𝑜𝑟𝑒𝑑)

𝑈𝑒(𝑎𝑝𝑝𝑙𝑖𝑒𝑑) (1-3)

6

In the above expression, Ue refers to electrical energy while Um refers mechanical energy. The

electromechanical coupling coefficients can be calculated for various piezoelectric modes of

vibrations and expressed using the same notation as the d-constant or the g-constant. In addition,

they can also be calculated for thickness and planar vibration modes and expressed as kt or kp,

respectively.

These piezoelectric constants along with mechanical and electric constants of the

piezoceramic are then used to evaluate piezoelectric materials for energy harvesting.

1.2 Piezoelectric Materials for Energy Harvesting

Efficient energy harvesting requires the use of appropriate materials to convert ambient

mechanical energy into usable electrical energy. In the case of piezoelectric materials, such

characteristics are represented by material properties such as the piezoelectric strain constant (d),

the piezoelectric voltage constant (g) and the dielectric constant (K or ϵ). For energy harvesting

applications, suitable materials are characterized by a large magnitude of the product d-g that

reflects the power generated by the material [2]. According to Inman and Priya, the condition for

obtaining this large product has been shown as |d| = ϵn, where n relates to the material properties

of the specific material and ϵ to the material dielectric [2]. In the tabulated values presented by

the authors, minimizing n has been shown to increase the energy density of the piezoceramic

material. Further considerations for energy harvesting applications include a high material

dielectric permittivity ϵ and a high strain capability. A high material dielectric corresponds to

low material impedance resulting in higher output current flow from the device while fracture

strains and flexibility reflect the durability of the material.

A large portion of existing piezoceramic energy harvesting research makes use of bulk

piezoceramics to capture ambient energy [2]. These are either polycrystalline or monolithic

materials that are available in the form of plates, cylinders or disks. Such materials provide the

advantage of a well-established research history and, as a result, their behaviour is fairly easy to

characterize. Bulk piezoceramics also demonstrate some of the most desirable properties for

energy harvesting (such as the d-g product) and have high stiffness making them suitable for

high load applications [2]. Furthermore, these materials have widespread use in actuators

7

creating well-established models for their behaviour. As a result, they are the most common

material used in energy harvesting devices. However, bulk piezoceramics suffer from poor

durability and robustness and are prone to cracking and fracturing. As a result, these materials

are limited to low excitation applications. As mentioned earlier, the most common bulk

piezoceramic used in literature is lead zirconate titanate (PZT), however other piezoceramics

such as BaTiO3 and PMN-PT are also being explored.

One method of overcoming the poor durability of bulk piezoceramics is through the use of

piezoceramic fibres composites (PFCs). PFCs are a type of piezoelectric material that contain

embedded piezoceramic fibres within a polymer matrix. Such piezoelectric fibres can have a

circular or square cross-section and can be assembled in the matrix using conventional injection

molding or by laminating the layers of polymer and fibrous components. The fibres are usually

aligned in the direction of stress and the electrode patterns are designed to utilize either the d31

piezoelectric mode or d33 piezoelectric mode depending on the material’s properties. In the case

of PFCs, the d33 mode can be exploited by using interdigitated electrodes that consist of

alternating positive and negative electrodes on the surface of the material. This electrode pattern

creates a capacitive field that is aligned with the axial direction of the fibres resulting in the d33

effect. The majority of PFCs make use of fibres on the order of 50-500 m [3].

Due to their improved mechanical properties, PFCs have been explored for energy harvesting

applications. For example, [4] compares the potential of conventional PFCs to two monolithic

ceramics with differing electrode patterns. In comparing the three different piezoelectric

materials, the authors attempt to determine the effect of the material and electrode patterning on

its capabilities to recharge a battery. Based on their experiments, it was observed that the power

developed by the PFC and the monolithic ceramic with interdigitated electrodes was much lower

than the bulk ceramic with a conventional electrode pattern. This was attributed to each electrode

pair in the interdigitated electrode pattern acting as a capacitor that was placed in series with its

neighbouring electrode pairs. This created a low overall capacitance of the device and

consequently a high material impedance limiting the amount of current required to recharge the

battery. However, it should be noted that the bulk PZT piezoceramics with interdigitated

electrode pattern performed similarly in the experiments to the tested PFCs indicating no

negative effects of the fibrous ceramic itself.

8

PFCs are also the subject of [5] that compares various composites with different fibre

diameters. In this, PZT fibres ranging from 15-250 µm in diameter were molded and laminated

in epoxy to create different PFCs with 40% of volume composed of piezofibres and 60%

composed of epoxy of varying thickness. The energy harvesting capabilities of the samples was

then tested by dropping a small steel ball and observing the material’s voltage and power

response. From this, it was observed that maximum power of 120 mW was obtained from the

thickest sample with the smallest fibre diameter (15 m). Upon analyzing the remaining data, the

authors were able to conclude that samples with larger thickness had greater capability for fibre

displacements resulting in larger power. Also, fibres with smaller diameters were observed to

have higher d33 coefficients resulting in greater power output.

Further research in PFCs was conducted by [6] that compared the energy harvesting

capabilities of piezoceramic fibres to PVDF piezoelectric polymers for wearable applications and

found PFCs to yield higher voltages than PVDF for the same excitations. However, the authors

note that PVDF yielded more repeatable results due to more stringent assembly controls in the

industry.

The behaviour of PFCs has also been the subject of much modelling and characterization.

Analytical methods of modelling PFC behaviour generally make use of the uniform fields

method [7], [8]. As the name implies, this model assumes that all fields are uniform within each

material (fibre or matrix) and only the material properties and volume fractions of each material

determine the magnitudes of this field. In essence, this model is the generalization of the rule of

mixtures applied to three-dimensions and provides accurate representation of the behaviour of

the composite structure. Finite element models have also been used to predict the variation of

electrical and mechanical fields within the piezofibres or the matrix and provide good agreement

with the analytical approximations.

However, despite the research interest and mechanical benefits of PFCs, their piezoelectric

performance remains significantly below bulk PZT piezoceramics, and consequently they are not

commonly utilized in energy harvesting devices.

Recently, single crystals of relaxor ferroelectric materials have been discovered with very

high piezoelectric charge coefficients of up to 2500 pC/N [2]. These single crystals are obtained

9

by controlling the crystal’s growth from a solution of raw materials resulting in very high

piezoelectric properties due to the uniformity in the crystal lattice. The most common

composition of single crystal utilizes a solid solution of lead magnesium niobate – lead titanate

(PMN-PT) [2]. Using this material, electromechanical conversion efficiencies as high as 90%

have been reported along with a d-g products ranging from 12,250 x 10-15

m2/N to 375,000 x

10-15

m2/N [9], several times larger than PZT. However, despite these improved material

properties, PMN-PT single crystals have not been utilized much for energy harvesting

applications due to challenges in synthesizing them in large volumes [2].

1.3 Energy Harvesting Devices

Due to significant research interest in piezoceramic energy harvesting, a wide variety of

harvesting devices have been proposed in recent years. To simplify categorization, these can be

divided into two broad groups: resonating structures and impulse driven generators.

Resonating structures are the most common type of energy harvesting device and make use of

a central structure that is excited by ambient vibrations approaching its resonant frequencies. The

oscillating structure then stresses a bonded piezoceramic element generating a time-varying

voltage. The most common example of such a resonating structure is a cantilevered beam

piezoelectric energy harvester that has piezoelectric elements bonded to the top and bottom

surfaces of the beam. For every deflection of the beam under ambient vibrations, the

piezoelectric elements are stressed in a transverse direction resulting in voltage generation across

electrodes placed at the top and bottom surfaces (see Figure 1.4). Often, a proof mass is also

placed near the tip of the beam to improve its displacement and control the frequency response of

the beam.

Figure 1.4: Schematic of one cantilever beam harvester [10]

V

M

10

Due to their simplistic design and predictable response, cantilevered harvesters are the subject

of extensive modelling and testing. One example of such work is by Roundy and Wright [10].

Using a volume constraint for a proposed device, the group developed an analytical model of the

system and subsequently validated it through two optimized prototypes. The prototypes were

then tested on a shaker at a frequency of 120 Hz producing a peak total of 375 µW and were able

to successfully power a custom-built radio.

Other research on straight cantilever beams includes the work by Sodano et al. in analytically

modelling cantilever beams ( [11], [12]) and predicting effects such as vibration damping in the

beam due to energy harvesting [13]. Sodano et al. have also investigated the amount of power

generated by a cantilevered plate and storing it in batteries or capacitors [14]. In this, it was

found that at resonant frequencies, the tested plate was able to produce 2 mW of power and was

capable of recharging batteries [15].

PMN-PT materials have also been used to develop energy harvesters with typically higher

outputs than conventional PZT materials. In [16], a PMN-PT cantilever beam utilizing the

longitudinal mode with interdigitated electrodes is presented. This device measures 7.4 mm x 2

mm x 110 µm in dimensions and is capable of producing 300 W of power at a resonant

frequency of 1.3 kHz. Another cantilever beam PMN-PT harvester is presented in [17] and is

capable of producing 19 mW under 0.2 gpk excitation with a power density of 0.73 mW/cm3.

This harvester was used to recharge a battery and power an accelerometer using harvested

power.

Alternatives to straight cantilever beams have also been explored. Zheng et al. have

developed an “air-spaced” cantilever beam designed with a tip mass to increase the amplitude of

the generated signal [18], and thus improve the efficiency of the AC-DC conversion for

subsequent storage. This design uses a fixed base and a proof mass that is attached using a

mechanical beam and a separate piezoelectric beam (see Figure 1.5). The resulting asymmetric

structure can undergo two modes of bending: pure bending and S-shaped bending. For energy

harvesting applications, the S-shaped bending mode must be avoided and the researchers use an

analytical model as well as a FEM model to develop criteria to determine the dominant mode of

bending. The model was successfully validated using a prototype that was able to produce 32.5

µW of power at a frequency of 150 Hz.

11

Figure 1.5: Schematic of the air-spaced cantilever beam. The bottom beam is the mechanical beam while the top

beam is piezoelectric [18]

Other alternative cantilever beam designs also include a tapered cantilever developed by

Glynne-Jones et al. to produce uniform strain over the length of the beam [19]. A tested

prototype using PZT piezoceramics was able to produce 3 µW at its fundamental frequency of 80

Hz with a load resistance of 333.1 kΩ. The device was made from thick film printing of the

piezoceramic material and as a result, suffered from reduced piezoelectric properties when

compared to bulk materials.

Despite their simplicity, one of the major drawbacks of the resonant energy harvesters is their

poor frequency response. For optimum performance, as implied by their name, resonant

structures must be excited close to resonant frequencies to maximize deflections and thus

maximize generated power. As a result, these devices must be placed in environments where the

ambient frequencies match the device’s operating frequencies. While the operating frequency of

an energy harvester can be tuned to a certain extent through variations of the proof mass or beam

properties, the bandwidth of the device remains more or less fixed. In order to address this issue,

some novel solutions have been proposed.

Marinkovich and Koser have proposed a concept for Smart-Sand, a wide bandwidth

piezoelectric energy harvester that makes use of four cantilever tethers with bonded

piezoceramic materials to support a central proof mass (see Figure 1.6(b)) [20]. The non-linear

dynamics of the device result in a large amount of bending as well as stretching of the tethers and

consequently, a 3D finite element model of the device was created to analyze its behaviour. This

model was then validated through a tested prototype that was able to successfully operate

between the frequencies of 160-400 Hz without the need for any tuning while producing a peak 1

µW of power. The authors note that the critical frequencies of the device could be decreased by

creating thinner and more compliant tethers or by increasing the mass of the proof mass.

Base Proof Mass

12

Figure 1.6: The frequency response of (a) a cantilever harvester compared to (b) a Smart-Sand harvester [20]

As can be observed from the above examples, most resonating energy harvesters operate at

frequencies greater than 100 Hz. For low frequency application, energy harvesting must usually

be conducted through impulse driven generators. Rastegar et al. have proposed one such energy

harvesting platform for very low frequency vibration operating conditions (on the order of 0.1 –

0.5 Hz) such as ships and trains [21]. This design consists of a primary travelling mass that

oscillates with the platform at very low frequencies moving back and forth within the device

enclosure. As it does so, it strikes a number of secondary pendulums with attached piezoelectric

elements that can then oscillate at their natural frequency. The pendulum frequency can be tuned

with a tip mass to optimize power output with respect to the overall platform frequency.

Prototypes of this device were still under construction at the time of the paper’s publication.

A similar approach of using a moving mass to strike secondary piezoelectric elements has

also been proposed by Renaud et al. in [22]. Using human motion to create mechanical

vibrations, the proposed device used a sliding mass within a rigid frame to strike piezoelectric

elements at either end of the frame. To improve the device’s power outputs, magnets were also

attached at either end of the device to increase the force of impact. Modelling results predicted

that the device would be able to produce a peak 40 µW of power at 1 Hz and 0.1 m/s2 excitation

amplitudes.

One of the main limitations of impulse driven harvesters is that the majority of the energy by

the impacting object on the piezoelectric harvester is returned to the object in the collisions.

Umeda et al. investigated the effects of such collisions between a steel ball and a piezoceramic

plate and developed an analytical model for the system in [23]. Through subsequent testing, they

found that the device was only 9.4 % efficient in extracting energy from the steel ball. Goldfarb

and Jones [24] found similar results with a stack piezoceramic configuration and observed that

13

the majority of the energy was returned back to the excitation source. Consequently, the stack

configuration was found to be most effective at low input frequencies around 5 Hz. Xu et al.

[25] compared the effects of impact loading to slowly applied compressive loading for a

piezoceramic material and found slow compressive loading to produce more energy than impact

loading due to the brittle nature of piezoceramics and poor energy transfer between the impacting

object and the material.

1.4 Circuits for Energy Harvesting

The previous sections describe the materials and configurations frequently utilized for

harvesting energy from ambient vibrations using piezoelectrics. However, the successful

integration of these devices with wireless sensor modules requires the use of specific circuitry to

rectify and optimize the output from the devices. These are characterized below.

1.4.1 Energy Storage

The power produced by piezoelectric elements from ambient energy is too low to directly

power most devices. As a result, the energy must first be stored using appropriate methods and

then be utilized for higher power applications. The two most common choices for energy storage

are capacitors and rechargeable batteries. In comparing the two, Umeda et al. [26] found that

capacitors were the more efficient choice for their tested device, but their storage capacity was

too low for many applications. Similarly, Starner [27] found batteries more suitable for high

power applications with capacitors more efficient for low-excitation piezoelectric elements.

Capacitors were also considered the storage means of choice for many in-vivo applications due

to their less intrusive nature.

1.4.2 AC/DC Converter

The nature of harvesting energy from mechanical vibrations means that the charge generated

by the piezoelectric element is oscillatory in nature and hence must be rectified through an

AC/DC converter before it can be stored. The most common implementation of this circuit

component is a standard interface diode bridge rectifier connected to the electrodes of the

piezoelectric element. However, to overcome the bias voltages of the diodes, the piezoelectric

14

element must often undergo significant excitation magnitudes [28]. Liu et al. [28] have proposed

an alternative active rectifier circuit that makes use of a MOSFET based full inverter circuit to

apply an average-value voltage across the piezoelectric element. The resultant circuit is

described as being 78% efficient and the tested device was able to produce 7 mW of power as

opposed to 5 mW obtained from a similar diode bridge rectifier.

1.4.3 DC-DC Step Down Converter

One method of optimizing the power flow from energy harvesting device to the battery is

through the use of a DC-DC step down converter. Since the voltages generated by a piezoelectric

element can be very high, a DC-DC converter can regulate the voltage to an acceptable level for

the battery or the load. To further optimize the circuit’s performance for varying voltages, the

converter can be controlled through a controller circuit. Ottman et al. [29] proposed one such

circuit that controls the DC-DC converter through the use of duty cycles to hold the optimal

voltage at the rectifier output. At high excitation levels of the piezoelectric element, the circuit

can be effectively implemented through fixed-duty PWM signals and a switching MOSFET. At

low excitation levels, however, the duty cycles vary over a wide range and this requires complex

controllers that are power inefficient for implementation.

1.4.4 Non-Linear Voltage Processing

Another method of improving the power flow from the piezoelectric element is through the

use of non-linear voltage processing. In this, the rectified voltage of the piezoelectric device is

subjected through a non-linear circuit component such as a switched inductor to improve power

flow. While many forms of non-linear voltage processing have been developed, the most

common of these are synchronous charge extraction and Synchronized Switching Harvesting on

Inductor (SSHI). Lefeuvre et al. compared these methods to a baseline-rectified energy

harvesting device to observe their performance in [30]. Based on their experiments, synchronous

charge extraction was found to be the most effective optimization technique and its performance

was independent of the circuit load. However, this method peaked at low electromechanical

coupling factors (k2 < 0.006) and its effectiveness decreased as k increased. The two SSHI

methods (parallel and series) delivered lower peak power outputs when compared to the

synchronous charge extraction but still offered improvements over the standard interface circuit.

15

1.5 Limitations of Current Technology

Despite the level of research interest, the biggest limitation preventing the widespread

implementation of piezoelectric harvester technology is the low power outputs of most

conventional harvesters. As the primary application of piezoelectric harvester technology, low-

power wireless sensor modules with transmission capabilities have been observed to require up

to 22 mW of power during transmission cycles [31]. Furthermore, sensors such as strain gages

and accelerometers used in structural health monitoring have been observed to consume 2-5 mW

of power during active cycles of sensing [31]. Consequently, powering such devices requires an

energy harvesting platform capable of producing a significant portion of this power requirement

to simultaneously operate the transmitter in the sleep mode as well as store any excess charge for

later use in the active cycles of transmission and sensing. With the power densities of 100-300

W/cm3 achieved with current designs, this would require several energy harvesters or one bulky

harvester to power each wireless sensor module, making them impractical for use. Despite the

modeling and optimization work with cantilever beam energy harvesters, the current state of

technology has been unable to meet the power demand of off-the-shelf wireless sensor modules

and hence an alternative solution is required.

1.6 Thesis Objectives

To address these shortcomings, this thesis will:

Design a novel energy harvesting structure using piezoceramic materials with the goal of

overcoming some limitations of the common cantilever beam harvester

Demonstrate this novel energy harvester’s capability by fabricating a prototype capable

of producing of 1 mW for base accelerations less than 1 gpk and frequencies below 300 Hz.

Investigate the potential for novel single crystal materials such as PMN-PT with the

potential for improved material properties over bulk PZT piezoceramics

Integrate harvester prototype with appropriate conditioning electronics and power

management modules to create a standalone energy harvesting device

16

1.7 Thesis Outline

This thesis is divided into seven chapters. Chapter two describes the conceptual design and the

design process used to develop a novel energy harvester with the goal of overcoming some of the

limitations of conventional cantilever beam harvesters. This conceptual design is developed

through the use of finite element models and is validated through experimental testing of a

prototype. Chapter three expands on this novel piezoelectric harvester concept by characterizing

the factors that influence the performance of the design. This is achieved by comparing multiple

harvester configurations that span the design space of the novel harvester and examining the

performance of each design. This characterization of influencing factors is expanded in chapter

four with the development of a mathematical model to create design guidelines for tailoring this

harvester concept to various operational environments. These guidelines are then utilized in

chapter five to create an improved prototype of the novel harvester with the goal of powering a

wireless sensor module. Interfacing electronics to achieve this are also discussed in this chapter

along with experimental results of testing the improved prototype and a sensor module. Chapter

six discusses the potential of using alternative materials such as PMN-PT single crystals and

considerations required to utilize these materials in the developed concept. Finally, chapter seven

summarizes the contributions made by this research and outlines future recommendations.

17

Novel Harvester Design1 2.

One method to address the goal of designing a novel piezoelectric energy harvester that

improves upon the current cantilever beam harvester technology is to examine piezoelectric

material properties most relevant to energy harvesting. As mentioned before, it is well known in

the field that the capability of a piezoelectric material for energy harvesting can be determined

by the product of two of its material properties: the piezoelectric strain constant, d¸ and the

piezoelectric stress constant, g [2]. While the previous chapter has outlined many of the common

examples of piezoelectric energy harvesters utilizing the longitudinal (d33) or transverse (d31)

piezoelectric modes as shown in Figure 2.1, a closer look at the material properties of

piezoceramics suggests that the shear mode of these materials has a higher d-g product than these

commonly used methods. Table 2.A presents a summary of the piezoelectric constants of

common piezoceramic compositions obtained from [32] and it can be noted that for all materials

listed, the shear mode d15-g15 product is several times higher than the d33-g33 product and almost

an order of magnitude higher than the d31-g31 product. This indicates that, if subjected to

identical excitation conditions as the other two modes, the shear mode can provide the highest

power outputs for a given PZT material.

Figure 2.1: Illustration of (a) transverse (d31) mode and (b) longitudinal (d33) mode of piezoceramic materials with

poling direction, P, and applied force directions, F, labeled.

1 The contents of this chapter were originally published: © 2014 IEEE. Reprinted, with permission, from V.

Kulkarni, R. Ben-Mrad, S. Eswar Prasad, and S. Nemana, “A shear-mode energy harvesting device based on

torsional stresses,” IEEE/ASME Transactions on Mechatronics, May 2014.

18

Table 2.A: Piezoelectric properties of various PZT compositions

Property

Material

PZT-8 PZT-4 PZT-5A PZT-5H

d33 [pC/N] 225 289 374 593

g33 [mV-m/N] 25.4 26.1 24.8 19.7

d33-g33 5,715 7,543 9,275 11,682

d31 [pC/N] -97 -123 -171 -274

g31 [mV-m/N] -10.9 -11.1 -11.4 -9.11

d31-g31 1,057 1,365 1,949 2,496

d15 [pC/N] 330 496 584 741

g15 [mV-m/N] 28.9 39.4 38.2 26.8

d15-g15 9,537 19,542 22,309 19,859

Some examples of shear mode harvesters can be found in the literature. A shear mode

harvester using a PMN-PT single crystal was presented in [33] and was designed to operate

under non-resonant vibrations. The device utilized a central mass supported by two PMN-PT

wafers that induced shear stress on the wafers under base excitation to produce voltage and

charge. A prototype of this design was found to produce a maximum power of 0.70 mW through

a matched load resistance of 91 kΩ at 500 Hz. The device was also able to maintain a high

output over a wide frequency range.

A resonant shear harvester was developed in [34] by using a PMN-PT single crystal mounted

on a cantilever beam with its poling direction oriented along the length of the beam. As a result,

the harvester was able to utilize shearing stresses occurring due to bending of the beam to

produce electric charge. A prototype of this design was able to produce a maximum of 4.16 mW

at a frequency of 60 Hz under action from a cyclic load.

Another resonant shear harvester using a cantilever beam was presented in [35]. This work

compared two d15 mode PZT elements connected in series on a cantilever beam with an

equivalent single d15 element. The series configuration was found to produce 8.7 µW of power

through a load resistance of 2.2 MΩ at its resonant frequency of 73 Hz and also generated higher

open circuit voltages when compared to the single element.

Finally, a novel shear mode energy harvesting diaphragm operating in pressurized water flow

was outlined in [36]. A prototype of this device was constructed and was found to produce an

19

open circuit voltage of 72 mVpp and peak power outputs of 0.45 nW under water pressures of

20.8 kPa varying at a frequency of 45 Hz.

The above devices all utilize shear stresses developed under bending for energy harvesting

purposes. An alternative concept using torsion was presented in [37], using a thin walled

piezoelectric cylinder excited by torsional vibrations at its base. An analytical model of the

concept was presented in the work with expressions for output voltage, current and power but no

experimental validation was presented.

The following work outlines the development of a novel energy harvesting device that uses

torsional shear stresses induced under non-rotational vibrations to generate electrical charge.

Upon investigation, this configuration is shown to have significant advantages over cantilever

designs.

2.1 Harvester Concept and Model

The application of torsion on a circular tube creates shearing stresses on its outer surfaces. If

piezoceramic elements are bonded to this surface, the shearing stresses can be used to generate

charge using the direct piezoelectric effect. In PZT piezoceramics, shear mode charge generation

occurs through the d15 mode that requires shear stresses to be oriented in the plane formed by the

“1” and “3” axes (the “5” direction; see Figure 1.3), resulting in charge displacement along the

“1” axis. Consequently, using this mode with torsion induced shear stresses on a circular tube

requires that the piezoceramic elements be poled along the length of the tube. This configuration

is shown in Figure 2.2.

Figure 2.2: Poling and harvesting directions for PZT shear mode harvester

20

Ambient vibrations are typically linear in motion and finding rotational vibrations to apply

torsion on a circular tube limits the potential applications of the harvester. Hence, for practical

use, the torsion based shear harvester must be able to harvest electricity from non-rotational

oscillations. To achieve this, the concept from Figure 2.2 was modified to the design shown in

Figure 2.3. This design consists of two primary components: the torsion harvester and its base.

The harvester itself is a tube as in Figure 2.2 with the addition of an eccentric mass placed at its

tip. As this device is excited by vibrations in the vertical direction, the eccentricity of the tip

mass creates torsional stresses in the tube. The harvester’s base supports the tube at its two ends

and prevents the tube from bending while allowing relative twisting, thus ensuring the primary

modes of vibration are torsion based.

Figure 2.3: Design of a torsion based energy harvesting substructure

To observe the response of this design to base vibrations, a computer model of the harvester

was created with the dimensions summarized in Table 2.B. As an untested conceptual design at

this stage, the dimensions of this model were arbitrarily chosen to allow easy manufacturing and

easy assembly, should prototypes of this device become necessary. This model was then used in

a commercial finite element program (ANSYS, ANSYS Inc., Canonsburg, Pennsylvania, USA)

to identify the frequencies of the torsional modes of vibration. Resonant modes were searched

from 0 Hz to 5000 Hz and all the modes found are listed in Table 2.C. Among these, the first

three frequencies were observed to correspond to translations of the base and did not result in

significant shear stresses in the circular tube. The fourth mode occurring at approximately 600

Hz was found to be a torsional mode arising from the tube and generated significant shear

stresses along the outer surface of the tube. The displacements undergone by the base and the

harvester in this mode are shown in Figure 2.4.

21

Table 2.B: Major dimensions of torsion harvester model. Units are in Inches

Circular Tube Eccentric Mass Harvester Base

Length 4.250 Length 2.750 Length 4.600

Outer Dia. 1.000 Width 1.750 Width 4.000

Wall Thickness 0.045 Thickness 0.500 Height (Right) 3.750

Height (Left) 2.750

Wall Thickness 0.250

Table 2.C: Results of FEM Modal Analysis

Mode Frequency [Hz] Mode Frequency [Hz] Mode Frequency [Hz]

1 297.89 7 1216.2 13 3168.2

2 443.55 8 1465.1 14 3719.4

3 573.34 9 1651.3 15 3731.1

4 599.65 10 2007.6 16 4262.2

5 867.12 11 2226.6 17 4697.3

6 998.76 12 2817.9

Figure 2.4: Harvester deformation in torsional mode at approximately 600 Hz

After understanding the modal behaviour of the design, the energy harvesting potential of the

concept was explored with another finite element simulation in ANSYS. This simulation was

created as a coupled field analysis to quantify the piezoelectric response of any elements bonded

to the circular tube. To simplify the geometry and reduce computation time, only the harvester

tube was analyzed in this simulation and the constraints provided by the harvester base were

22

transferred to the relevant nodes of the finite element model. Four 5 mm x 5 mm x 0.5 mm

piezoceramic plates with generic PZT-5A properties listed in [38] were attached to the surface of

this tube to simulate energy harvesting elements. The torsion harvester was modeled with

SOLID187 elements and was assigned the material properties of aluminum while the PZT

material was modeled with SOLID227 coupled field elements. All PZT plates were connected

electrically in parallel. The geometry of the model is shown below in Figure 2.5 and this model

was excited at a frequency of 600 Hz using reaction forces calculated in the previous modal

analysis.

Figure 2.5: ANSYS Model of harvester for piezoceramic analysis

The open circuit response of the piezoceramic was first observed and then a CIRCU94

resistive element was placed across the PZT plates to model power harvested from the device.

The load resistance of this element was varied until peak power dissipation was observed. Using

this analysis, it was found that the peak power dissipation occurred with a load resistance of 6.03

MΩ, which was equal to the magnitude of the source impedance of the four piezoceramic

elements at resonance. At this load resistance, the ceramics were able to generate a peak

sinusoidal voltage of 73.9 V across the resistor, resulting in a peak instantaneous power

dissipation of 0.91 mW or, equivalently, 0.46 mW of average power dissipation. Under open

circuit conditions, the harvester produced a peak voltage of 101.2 V or 71.6 VRMS at a frequency

of 600 Hz. The voltage distribution across the piezoceramic with the optimal load resistance as

obtained from this analysis is shown in Figure 2.6.

1

11

12 X

Y

Z

AUG 8 2012

15:39:16

VOLUMES

TYPE NUM

23

Figure 2.6: Voltage distribution in PZT elements of torsion harvester with optimal load resistance for peak power

dissipation. Units are in Volts.

2.2 Prototype and Experimental Results

After estimating the performance of the torsion harvester concept with the ANSYS models, the

results were verified with the fabrication of a prototype using the dimensions listed in Table 2.B.

Off-the-shelf PZT-5A shear mode piezoceramic plates with dimensions of 5 mm x 5 mm x 0.5

mm were used with the harvester. The ceramics were polarized along one of their major

dimensions and electrodes were then applied across the other major dimension of these plates

perpendicular to the poling direction with Pelco conductive silver paint to obtain the

configuration shown in Figure 2.2. Wires were attached to these electrodes with silver epoxy

(MG Chemicals) and the piezoceramics were bonded to the harvester tube with electrically

insulating epoxy (AremcoBond 860) while ensuring that the electrodes on the sides of the

ceramic did not come in contact with the aluminum tube. The four PZT elements were then

connected electrically in parallel. The completed prototype is shown below in Figure 2.7.

Figure 2.7: Torsion harvester prototype

1

R0 MN

MX

Torsion Harvester Prototype - PZT

08.21154

16.423124.6346

32.846141.0577

49.269257.4808

65.692373.9038

MAR 12 2013

14:35:45

NODAL SOLUTION

STEP=1

SUB =1

FREQ=605.124

IMAGINARY

VOLT (AVG)

RSYS=0

DMX =.292E-04

SMX =73.9038

24

The mechanical response of this prototype was first tested by mounting it on an

electrodynamic shaker (Modal Shop 2110E) and exciting it with a 1 gpk acceleration amplitude

from 10 Hz to 5000 Hz. The response of the prototype was monitored by placing accelerometers

at three locations: at the tip of the eccentric mass, the connection of the circular tube with the

eccentric mass and the harvester base. The response of the assembly is shown in Figure 2.8.

Figure 2.8: Mechanical response of torsion harvester prototype

It can be observed that the harvester experiences its fundamental mode of vibration at a

frequency of approximately 620 Hz. This mode is characterized by high vertical accelerations of

the tip mass and smaller vertical accelerations of the circular tube, indicating rotation of the tube

and thus signifying a torsional mode. This result is in good agreement with the ANSYS torsional

mode predicted at 600 Hz. Several other modes can also be observed arising from vibrations of

the base and the tube at higher frequencies. These can also be observed to be in good agreement

with the higher modes listed in Table 2.C.

Next, the electrical response of the prototype was characterized. In this, the source impedance

of each of the four PZT plates was first measured for comparison with the ANSYS model. This

was done using an Agilent 4288A capacitance meter at 1 kHz with the source impedance of each

PZT plate modeled as an equivalent capacitor (Cp) in parallel with a resistance (Rp). The

measurement results are listed in Table 2.D.

0

2

4

6

8

10

12

14

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Ver

tica

l A

ccel

erat

ion

[g

pk]

Frequency [Hz]

Input Eccentric Mass Tip Circular Tube Harvester Base

25

Table 2.D: Impedance measurements of bonded PZT plates with associated measurement

accuracy

Element Cp [pF] Rp [MΩ]

1 16.799 ± 0.079 328.635 ±30.5

2 13.103 ± 0.079 421.743 ± 48.8

3 8.9554 ± 0.075 61.742 ± 0.89

4 14.177 ± 0.078 337.242 ± 40.0

When combined in parallel, the above four PZT plates have an equivalent impedance of 53.03

pF in parallel with a resistance of 40.7 MΩ. At the resonant frequency of the harvester of 620

Hz, this yields a combined source impedance of 4.3 MΩ. This value is lower than the ANSYS

predicted value of 6.03 MΩ at resonance but can be attributed to imperfect and non-uniform

bonding between the PZT plates and the tube structure. This non-uniform bonding creates

imperfect clamped boundary conditions on the PZT plates resulting in a different measured

capacitance than the ANSYS model. Imperfect bonding also results in leakage current between

the two electrodes of the PZT element through the conductive aluminum tube, decreasing source

impedance. This seems to be particularly the case with element 3 with its much lower equivalent

resistance and capacitance.

Further electrical characterizations of the harvester prototype were undertaken by measuring

its open circuit voltage and short circuit current outputs with respect to frequency. To do these

tests, the electrical connections were placed in the appropriate configuration and the harvester

was mounted on the electrodynamic shaker. A swept sinusoidal signal with a constant

acceleration amplitude of 1 gpk was fed to the shaker from 100 Hz to 5000 Hz. The open circuit

measurements were done with a National Instruments compactRio data acquisition system with a

buffer amplifier to account for the high source impedance of the energy harvester. The design

and characterization of this buffer amplifier is discussed in Appendix A. The short circuit current

measurements were taken with a Keithley 428 current amplifier connected in series with the data

acquisition system. The resulting open circuit voltage and short circuit frequency response are

shown below in Figure 2.9 and Figure 2.10.

26

Figure 2.9: Open circuit voltage frequency response of harvester prototype

Figure 2.10: Short circuit current frequency response of harvester prototype

From these, the harvester prototype was found to have a maximum RMS open circuit voltage

of 59.6 VRMS at a frequency of 621.5 Hz and a maximum RMS short circuit current of 30.5

µARMS at 618 Hz. The open circuit voltage of the prototype was lower than the ANSYS

predicted result of 71.6 VRMS, but as mentioned before, this was caused due to imperfect bonding

between the piezoceramics and the circular tube that resulted in dissipation of stresses being

transferred to the piezoceramic, thus leading to lower voltages.

To quantify the peak power output of the prototype, varying load resistors were attached

across the PZT plates and the voltage across the resistor was measured. As expected, it was

found that near resonance, the peak power transfer from the harvester to the load resistor

occurred when the resistance matched the source impedance of the harvester. In this case, at a

load resistance of 4.35 MΩ, the harvester prototype was found to deliver a maximum average

power output of 0.403 mW at 620 Hz. The results of the resistance sweep undertaken near

resonance at 615 Hz are shown in Figure 2.11 and the frequency response of the harvester with

the optimal load of 4.35 MΩ is shown in Figure 2.12.

27

Figure 2.11: Output power over varying load resistance values at 615 Hz

Figure 2.12: Output power frequency spectrum of harvester prototype

It should be noted that although the torsion harvester prototype exhibits several significant

peaks beyond the fundamental torsion modes in the open circuit voltage spectrum (Figure 2.9) or

short circuit current spectrum (Figure 2.10), it is unable to deliver significant power to a load

through any of these other peaks. The majority of the power output from the harvester is

delivered through its fundamental torsional mode.

2.3 Comparison with Cantilever Harvester

After validating the ANSYS model of the torsion based shear harvester with experimental

results, its performance was compared with the commonly found cantilever beam configuration

of most energy harvesting devices. For this, an ANSYS model of a cantilever beam harvester

0.00

50.00

100.00

150.00

200.00

250.00

300.00

0.00 1.00 2.00 3.00 4.00 5.00 6.00

Aver

age

Pou

t [µ

W]

Load Resistance [MΩ]

28

was created with a length of 7.5 in. and a cross section area of 1 in. x 1 in. These dimensions

were chosen such that the volume occupied by this harvester was roughly equal to the volume

occupied by the torsion based harvester. Furthermore, when modeled from aluminum, these

dimensions were found to produce a fundamental bending mode of the harvester at 575 Hz,

approximately the same as the torsion harvester, thus allowing for direct comparison. As before,

four generic PZT-5A elements were bonded on the top surface of this harvester and were

connected electrically in parallel after being configured to operate in the d31 mode to utilize

bending vibrations. One end of the beam was then excited by accelerations with amplitude of 1 g

at the resonant frequency of the beam to simulate base excitation. The ANSYS model of this

cantilever harvester is shown in Figure 2.13.

Figure 2.13: ANSYS model of cantilever harvester

The results of the ANSYS cantilever model predicted that this cantilever harvester

configuration would produce an open circuit voltage of 6.3 VRMS at resonance and dissipate

0.295 mW of average electrical power through an optimal load of 67 kΩ. When compared to the

torsion harvester ANSYS model, the cantilever harvester had a lower predicted voltage but was

still expected to produce approximately 70% of the torsion harvester’s power output. This was a

result of the cantilever harvester’s lower source impedance, resulting in much higher current

outputs from the cantilever configuration.

However, despite similar predicted power outputs, a closer look at the PZT elements in the

cantilever harvester shows significant shortcomings of this design. A plot of the voltage

1

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Y

Z

AUG 8 2012

15:49:51

VOLUMES

TYPE NUM

29

distribution with the optimal load as shown in Figure 2.14 indicates that although the maximum

voltage produced in the harvester was 7.1 V, located in element closest to the base of the

cantilever, the overall voltage drop across the electrodes in the harvester was found to be 6.2 V.

Hence, only a portion of the electric potential generated by the harvester was utilized for

electrical work across the load. This phenomenon was due to the variation in bending stresses

along the length of the beam, resulting in lower voltage outputs from elements farther from the

base of the beam. As a result, when connected in parallel, a portion of the generated charge was

dissipated along PZT elements as opposed to being dissipated through the attached electrical

load in the form of useful work.

Figure 2.14: Voltage distribution in cantilever harvester PZT elements. Units are in Volts.

In comparison, all piezoceramic elements on the outer surface of the torsion harvester

experience the same amount of stress resulting in equal electric potential across all individual

elements (see Figure 2.6). Hence, as more elements are added in parallel in the torsion harvester

to harvest all available strain energy, the torsion harvester is capable of producing the same

output voltages while simultaneously reducing its source impedance, resulting in higher power

outputs. On the other hand, in the cantilever harvester, adding more elements can result in

increased losses due to charge cancellation.

To illustrate this, the finite element simulations of the cantilever beam and torsion harvester

were repeated at resonance with eight PZT elements bonded to each harvester. Despite the added

elements, it was observed that the cantilever harvester only experienced a slight increase in

output power and was able to produce 0.347 mW of average power through a matched load

resistance of 39 kΩ with a significantly lower output voltage of 3.68 VRMS, an increase in power

of only 18 % despite doubling the number of piezoelectric elements. On the other hand, with the

1

R0

MNMX

X

Y

Z

Cantilever Harvester

0.7959

1.59182.3877

3.18363.9795

4.77545.5713

6.36727.1631

MAR 13 2013

14:35:21

NODAL SOLUTION

STEP=1

SUB =1

FREQ=576.821

AMPLITUDE

VOLT (AVG)

RSYS=0

DMX =.591E-04

SMX =7.1631

30

additional elements, the torsion harvester nearly doubled its power output, producing 0.706 mW

of average power through its optimal load resistance and an output voltage of 39.2 VRMS,

indicating minimal losses between the PZT elements. Thus, the torsion based shear harvester has

the potential for much greater power outputs if the entirety of its energy harvesting surfaces is

utilized.

2.3.1 Generalized Comparison

While the example provided above using finite element simulations provides an insight into

the benefits of the novel torsion harvester, a more comprehensive comparison is necessary to

examine other aspects involved in the design. For instance, while it is shown that the shear mode

of piezoceramics offers better energy harvesting potential due to its higher d-g product, this

potential can only be realized when these piezoceramics are coupled with mechanical systems

that expose them to similar stress conditions as found in other harvesters. The criteria for

achieving this are examined here.

An established model of a piezoelectric harvester was presented by DuToit et al in [39] that

outlined a simplified single degree of freedom (SDOF) model of a general energy harvester

operating near resonance and examined its output power. In this work, the optimal power output

from an energy harvester was expressed as:

|𝑃𝑜𝑢𝑡|𝑜𝑝𝑡 =𝑚𝑤

2

16𝜁𝑚𝜔𝑛 (2-1)

In this, m represents the equivalent mass of the harvester in the SDOF model, 𝑤 represents the

input base acceleration, ζm is the mechanical damping ratio in the system and ωn is the resonant

frequency of the system. This equation can then be used to compare the power outputs of the

novel torsion based harvester to the common cantilever beam configuration.

To simplify the analysis for this comparison, any tip masses were neglected in each harvester

since these generally serve to change the frequency of operation of the harvester and the

vibration modes of the harvester but do not directly affect the output power when the harvester is

operated at resonance. Furthermore, the cantilever harvester was assumed to be a uniform

rectangular beam with length l, width b, and height h, while the torsion harvester was assumed to

be a uniform hollow circular tube with length l, outer radius ro, and inner radius, ri. Finally, to

31

ensure proper comparison, the constituent material for both geometries was assumed to be

identical and the volume of both harvesters was set to be equal, giving:

𝑙𝑏ℎ = 𝜋𝑙(𝑟𝑜2 − 𝑟𝑖

2) (2-2)

With these assumptions, the parameters for each harvester were developed for use in (2-1). These

are summarized in Table 2.E.

Table 2.E: Comparison of output power from simplified cantilever and torsion harvesters

Cantilever Torsion

Geometry

SDOF equivalent stiffness, k 𝑌𝑏ℎ3

4𝑙3 𝜋𝐺

2𝑙(𝑟𝑜

4 − 𝑟𝑖4)

SDOF equivalent mass, m 0.23𝜌𝑙𝑏ℎ 𝜋𝜌𝑙

6(𝑟𝑜

4 − 𝑟𝑖4)

𝜔𝑛 = √𝑘

𝑚

ℎ

2𝑙2 √𝑌

0.23𝜌

1

𝑙√

3𝐺

𝜌

|𝑃𝑜𝑢𝑡|𝑜𝑝𝑡 = 𝑚𝑤

2

16𝜁𝑚𝜔𝑛

√0.233𝜌3𝑙3𝑏𝑤𝐵,𝑐 2

8𝜁𝑚√𝑌

𝜋√𝜌3𝑙2(𝑟𝑜4−𝑟𝑖

4)𝑤𝐵,𝑡 2

96𝜁𝑚√3𝐺

|𝑃𝑜𝑢𝑡|𝑜𝑝𝑡,𝑡𝑜𝑟𝑠𝑖𝑜𝑛

|𝑃𝑜𝑢𝑡|𝑜𝑝𝑡,𝑐𝑎𝑛𝑡𝑖𝑙𝑒𝑣𝑒𝑟

2.37(𝑟𝑜4−𝑟𝑖

4)

𝑙𝑏

𝑤𝐵,𝑡 2

𝑤𝐵,𝑐 2 √𝑌

3𝐺 (2-3)

≈2.21(𝑟𝑜

4−𝑟𝑖4)

𝑙𝑏

𝑤𝐵,𝑡 2

𝑤𝐵,𝑐 2 (2-4)

In the above expressions, Y is the elastic modulus (Young’s modulus) of the cantilever harvester,

G is the shear modulus of the torsion harvester, ρ is the material density for both cantilever and

torsion harvesters, and 𝑤𝐵,𝑐 and 𝑤𝐵,𝑡 are the base accelerations exciting the cantilever and

torsion harvesters, respectively. The simplification from (2-3) to (2-4) is made using the

assumption that most materials have a Poisson’s ratio of ν ≈ 0.3, making Y ≈ 2.6G.

Equation (2-4) summarizes the relative theoretical power output ratios for the torsion

harvester and the common cantilever harvester. If the expression (2-4) is evaluated to be greater

than unity, then this indicates that the torsion harvester produces higher outputs than the

32

equivalent cantilever harvester. By inspection, for this to occur, the product of geometric

properties lb and base excitation 𝑤𝐵,𝑐 2 for the cantilever harvester must be less than 2.21 times

the geometric properties (𝑟𝑜4 − 𝑟𝑖

4) and base excitation 𝑤𝐵,𝑡 2 for the torsion harvester. For

scenarios where this is the case, such as with short cylinders and large radii, the torsion harvester

provides higher power outputs compared to the equivalent cantilever beam harvester.

Thus, beyond having a more uniform strain distribution on its accessible surfaces, the torsion

harvester has potentially higher outputs when utilized in the low aspect ratio cylinder scenarios.

2.4 Summary

A concept of a torsion based shear mode energy harvester to overcome some limitations of

conventional cantilever beam harvesters was presented in this chapter. This concept makes use of

an eccentric mass attached to a circular tube to induce torsional stresses on the tube from linear

vibrations. A finite element model of the harvester was created to predict the performance of the

harvester concept at resonance and a prototype was constructed with PZT piezoceramics to

validate this model. It was found that the prototype was able to produce 0.403 mW of average

dissipative power through an optimal load resistance of 4.35 MΩ with an output voltage of 41.9

VRMS. When compared to conventional cantilever beam harvester configurations, it was found

that the torsion harvester experienced a more uniform strain distribution, thus minimizing losses

in the harvester as a result of charge cancellation. Furthermore, a simple mathematical model

indicated that in scenarios involving short harvesters with large cross-sections, the torsion

harvester would result in higher power outputs than the equivalent cantilever harvester. Thus, the

novel torsion harvester has the potential to produce much higher power outputs than possible

with conventional designs and may help overcome one of the primary limitations of energy

harvesting technology.

33

Comparison of Harvester Geometries 3.

With the conceptual design of the torsion harvester validated with a proof-of-concept device,

further insight into the behaviour of this harvester is necessary to determine the factors that affect

its performance. This insight can then be used to find methods of improving upon the power

outputs of the harvesters as well as overcoming any limitations that may arise. This chapter

explores the behaviour of the torsion harvester concept by examining the effects of tube cross-

section and eccentric mass geometries on the harvester’s performance along with the influence of

mechanical damping. These effects are explored using three comparison metrics that are then

utilized through finite element simulations and experimental characterization of prototypes to

compare different harvesters.

3.1 Comparison Metrics

In order to compare the impact of various cross-section designs, metrics for characterizing the

performance of an energy harvesting configuration are first necessary. Since the ultimate goal for

energy harvesters is to power electronic systems, the simplest way to compare cross-section

designs is to compare the peak power output of each configuration through a resistive load. With

this, higher peak power outputs naturally indicate more effective designs. For this study, since

the designs explored are resonant structures, the peak power outputs were compared at resonant

frequencies of the harvester through an optimal load resistance.

The performance of various harvester configurations can also be compared through their open

circuit voltage outputs. This metric correlates to the mechanical strain produced at the locations

of PZT elements in the harvester and also dictates the necessary electrical components required

to interface the harvester with sensor electronics. Commonly used rectifying circuits can suffer

from voltage losses through diode-bias voltages and so a harvester with high power outputs at

low voltages when measured with a purely resistive load can suffer high losses when interfaced

with other electronics [40]. Hence, the open circuit voltage of the harvester serves as a metric to

34

highlight electrical considerations that may be required to interface the harvester with real-world

applications.

The above two metrics do not allow for differentiation between input conditions such as

excitation accelerations or differing resonant frequencies between harvesters based on different

tube cross-section. For instance, the energy input provided to an energy harvester by a 1 gpk

vibration environment at 10 Hz is different from the energy input to the harvester provided by a

1 gpk vibration environment at 500 Hz and metrics such as power output alone do not account for

the different input conditions. Hence, to take such factors into account, a third energy-based

metric becomes necessary.

A piezoelectric energy harvester works on the principle of converting mechanical energy in the

form of vibrations into electrical energy. Thus, the effectiveness of the energy harvester can be

quantified by comparing the electrical energy output to the total mechanical energy input into the

harvester as a result of ambient vibrations at the desired frequency. With such a metric, it is then

possible to account for excitation conditions such as base acceleration and frequency since these

are reflected in the energy input into the harvester. To develop this metric, the energy harvesting

structure is represented by a simplified single degree-of-freedom (SDOF) lumped parameter

model with a sinusoidal base excitation as shown in Figure 3.1.

Figure 3.1: SDOF model of harvester with sinusoidal base excitation

The differential equation to describe the motion of this system is then expressed as:

𝑚 + 𝑐 + 𝑘𝑧 = −𝑚 (3-1)

35

where m is the equivalent mass of the system, k is the equivalent stiffness, and c is the damping

coefficient in the system. It should be noted that c represents all damping in the system as a

result of mechanical losses and energy harvesting effects. The relative motion of the mass m with

respect to the base is represented by z(t). In this work, a dot over a variable (e.g. ) represents a

first-order derivative of that variable with respect to time. Similarly, two dots represent a second-

order time derivative of the variable. If transient behavior is neglected, the steady-state response

of this system to base excitation can then be expressed as:

𝑧(𝑡) = 𝑍 sin (𝜔𝑡 − 𝜙) (3-2)

𝑍 =Ω2

√(1−Ω2)2+(2𝜁Ω)2 (3-3)

𝜙 = arctan (2𝜁𝛺

1−𝛺2) (3-4)

Here, Ω is the excitation frequency ratio 𝜔

𝜔𝑛 and ζ is the damping ratio

𝑐

2√𝑚𝑘 . The total energy of

the system is then the sum of the potential and kinetic energies of the system.

𝑈 =1

2𝑘𝑧2 +

1

2𝑚2 (3-5)

𝑑𝑈

𝑑𝑡= 𝑘𝑧 + 𝑚 (5)

𝑑𝐸

𝑑𝑡= −𝑐2 + 𝑚 (5)

𝑑𝐸

𝑑𝑡= 𝑚(−2𝜁𝜔𝑛2 + ) (3-6)

Equation (3-6) indicates that the total energy in the system changes with time as a result of

damping in the system and vibrational motion of the mass. Plotting the results of (3-6) over time

gives a graph as in Figure 3.2 that is sinusoidal and time-varying but with a negative off-set,

indicating that energy is steadily lost from the system. In the absence of external excitation, this

energy loss would result in a decaying response from the system. Thus, to maintain steady state

behavior as is assumed for the harvester, the average power input from mechanical vibrations

must equal the average power dissipated from the system due to all system losses.

36

Figure 3.2: Energy dissipation from SDOF harvester model over time

Consequently, the average power input into the harvester as a result of mechanical vibrations

can be expressed as:

𝑃𝑖𝑛,𝑎𝑣𝑔 =1

𝑇𝑓∫ −

𝑑𝑈

𝑑𝑡𝑑𝑡

𝑇𝑓

0=

1

𝑇𝑓∫ 𝑚(2𝜁𝜔𝑛2 − )𝑑𝑡

𝑇𝑓

0 (3-7)

Here, Tf is the period of vibration corresponding to the excitation frequency. The effectiveness of

the harvester (𝜅, kappa) can then be calculated as the ratio of the average electrical power output

to the average mechanical power input calculated in (3-7).

𝜅 =𝑃𝑒𝑙𝑒𝑐,𝑎𝑣𝑔

𝑃𝑖𝑛,𝑎𝑣𝑔 (3-8)

The average electrical power produced for the harvester can be estimated easily by measuring

the RMS voltage produced by the harvester through a resistive load, R, and the expression:

𝑃𝑒𝑙𝑒𝑐,𝑎𝑣𝑔 =𝑉𝑅𝑀𝑆

2

𝑅 (3-9)

Estimating the SDOF equivalent stiffness, k, and mass, m, for the torsion harvester concept can

be achieved using finite element models or experimental results for torsional stiffness under

static or quasi-static loading conditions:

𝑘 =𝑀𝑇

𝜃 (3-10)

In this, MT is the applied torque on the torsion tube geometry and θ is the resulting angle of twist

in the tube.

37

For the torsion harvesters under consideration, the equivalent mass of the torsion harvester can

be estimated by assuming that the contribution of the eccentric mass to the SDOF model is much

greater than the mass contributions from the torsion tube in the harvester. Under this assumption,

the equivalent mass of the torsion harvester is then approximately equal to mass moment of

inertia of the eccentric mass about the centre of torsion of the torsion tube. It is shown in

subsequent sections that a SDOF model of the harvester developed in this manner closely models

finite element and experimental results.

3.2 Harvester Configurations

Off-the-shelf wireless sensor modules are typically centimeter scaled devices. If energy

harvesters are expected to power such devices, the size of the harvester must be of the same scale

or smaller than the sensor module. As a result, the harvester configurations pursued in this study

were constrained to a volume of 1 cm3 regardless of their geometry.

Ambient vibrations can typically be found at frequencies below 350 Hz [41]. For a centimeter-

scaled torsion harvester to work in this frequency range, the natural frequency of its torsional

mode must be minimized. This can be achieved by maximizing the rotational inertia of the

eccentric mass while minimizing the torsional rigidity of the torsion tube. The mass inertia of the

eccentric mass can be maximized by increasing its eccentricity and the density of its

compositional material. For the tube, the torsional rigidity is a product of the shear modulus of

the composite PZT and substrate cross section and the torsion constant of the tube cross-section.

Thus, the torsional rigidity of the tube can be reduced by selecting materials with low shear

moduli or by designing cross-sections with low torsion constants.

PZT shear mode elements purchased off-the-shelf for the torsion harvesters are available in the

form of rectangular plates. In order to bond these to the torsion structure, a variety of rectangular

cross-sections were explored for the harvester design. Three cross-sections were selected for

their ease of bonding and machinability and are shown below in Figure 3.3. Each of these cross-

sections in turn was used to form the tube in the torsion harvester depicted in Figure 3.4.

38

Figure 3.3: Torsion tube cross-sections examined in order of decreasing torsion constants: (a) Solid (b) C-Channel

(c) Slotted-C

Figure 3.4: Illustration of centimetre-scaled torsion harvester

The dimensions of the harvester designs were chosen such that six off-the-shelf PZT-5A

elements measuring 2 mm x 2 mm x 0.5 mm could be bonded to the tube. The remaining

dimensions of the harvester were chosen to yield a total device volume of 1 cm3. The major

dimensions of each harvester configuration are summarized in Table 3.A.

39

Table 3.A: Characteristics and geometry of various torsion harvester configurations explored

Solid C-Channel Slotted-C

Torsion Tube

Material HDPE Aluminum Aluminum

Length [mm] 10 9 9

Width [mm] 3.25 3.25 2.50

Height [mm] 3.25 3.25 -

Wall Thickness [mm] - 0.25 0.25

Eccentric Mass

Material Tungsten

Carbide

Tungsten

Carbide

Tungsten

Carbide

Total Length [mm] 18 22 22

Width [mm] 3 3 3

Height [mm] 5 4 4

Mass [g] 4.72 4.20 4.20

Eccentricity [mm] 6.72 9.70 9.70

For each torsion tube, the constituent material was selected from one of three easily accessible

and machinable materials: high density polyethylene (HDPE), aluminum and steel. From these,

aluminum was chosen as the material for the C-channel and slotted-C harvesters since its

stiffness was adequate to maintain the structural integrity of the thin-walled cross sections while

keeping the resonant frequencies of the harvester comparable to those found in ambient

vibrations. For the solid cross-section, the geometry of the torsion tube meant that its structural

integrity was not as severely impacted by the material choice, and so the material for this

configuration was chosen to be (HDPE) polymer to minimize its resonant frequency. Due to

dimension constraints placed by available raw materials, this resulted in some variation in the

length of each torsion tube and the lengths of the eccentric masses, but all harvesters were

designed to comply with the 1cm3 size limit.

Unlike the proof-of-concept prototype presented in the previous chapter, the miniaturized

harvesters are not supported against bending at the eccentric mass. This was to reduce the

machining complexity required to create a support that will allow rotation of the eccentric mass

but prevent vertical motion to prevent bending. Furthermore, with the dimensions of the tube

shown in Table 3.A, it was observed that the slenderness ratio of the torsion tube was sufficiently

small that it underwent predominantly torsional modes of vibrations over bending modes.

40

3.3 Simulation Results

The three torsion tube configurations proposed were modeled in finite element software

(ANSYS, ANSYS Inc., Canonsburg, Pennsylvania, USA) to observe their behavior and compare

their performance with the metrics identified previously. Each configuration was subject to two

analyses: a modal analysis to identify resonant modes and a harmonic analysis with vertical

sinusoidal vibrations applied to the configuration base to observe a steady-state energy

harvesting response. To determine peak open circuit voltage and power outputs, the harmonic

analysis was conducted at the first resonant frequency with an applied modal damping of 3% and

a peak sinusoidal base excitation acceleration of 9.81 m/s2 (1 g). This base acceleration was

achieved in ANSYS by applying a sinusoidal non-zero nodal displacement excitation at the

relevant support nodes in the structure and defining this displacement parametrically as a

function of the excitation frequency. The structural materials in the harvester were modeled

using SOLID186 second order 3D elements and the piezoelectric elements were modeled with

SOLID226 coupled field elements using the dimensions outlined in previous sections. The PZT

sections modeled with these elements were assumed to be purely capacitive in nature with

infinite leakage resistance between the positive and negative electrodes. An epoxy adhesive layer

measuring 0.010” was added between the PZT elements and the torsion tube to account for the

effects of the adhesive used for bonding the PZT elements and all components of the harvester

were assumed to be uniformly and perfectly bonded. To estimate power outputs, a resistive load

was modeled with CIRCU94 elements to provide power dissipation of harvested electrical

energy. For each harvester configuration, the resistive load was varied with an ANSYS

subroutine to find the optimal load resistance that would yield the highest power outputs from

each configuration. The results of the finite element analyses are summarized in Table 3.B.

Table 3.B: Summary of simulation results

Solid C-Channel Slotted-C

Torsional Stiffness [N-m/rad] 1.95 3.53 1.88

Eccentric Mass Inertia [mg – m2] 0.42 0.67 0.67

Resonant Frequency [Hz] 333.5 360.0 246.6

Open Circuit Voltage [V] 25.92 45.84 61.44

Optimal Load Resistance [MΩ] 10.37 9.98 14.35

Peak Power Output [µW] 32.04 101.5 126.4

Harvester Effectiveness (𝜅) [%] 0.25 0.98 1.09

41

These results indicate that the C-channel and slotted-C configurations utilizing the more

eccentric masses performed better in all metrics when compared to the solid tube harvester. This

was a result of the greater eccentricity of the two respective masses serving to increase the

torsional stresses induced in these configurations, thus providing better performance when

compared to the solid configuration. Furthermore, when comparing the slotted-C and C-channel

configurations, the slotted-C tube configuration was found to have the highest open circuit

voltage, power outputs and effectiveness. Upon closer inspection of torsion tube’s structure for

this configuration, it can be observed that the harvester’s effectiveness is the result of the

proximity of the substructure to the PZT elements. As a result, a larger fraction of the stress

developed in the torsion tube is converted by the PZT elements into electrical energy. On the

other hand, in the solid tube configuration and C-channel configurations, stresses produced in the

centre of the tube and corners of the tube do not contribute towards energy production resulting

in lower harvester effectiveness. Thus, of the three configurations under consideration, the

slotted-C torsion harvester configuration was deemed to be the superior design with an ability to

convert 1.1% of input mechanical energy into output energy. This effectiveness metric for the

non-optimized slotted-C harvester agrees well with the general range of 0.2 % – 5 % reported in

[42] and [43] for piezoelectric energy harvesters.

It is important to note that the simulation results presented above are based on a set of

assumptions and their limitations in predicting the response of any real world prototypes are

addressed here.

The predicted response of the torsion harvesters was based on the piezoelectric material

properties presented in [38]. However, manufacturers of PZT elements often claim a variation of

5 – 10 % from these quoted properties. Furthermore, piezoelectric and dielectric constants also

show strong frequency and dynamic stress dependence and can change at different operating

frequencies [44]. Using the previous simulations, this variation in material properties is expected

to produce up to 20 % variation in the predicted electrical response of each harvester outlined in

Table 3.B.

The above simulation results also assume a mechanical modal damping of 3% in each torsion

harvester along with an adhesive thickness of 0.010” used to bond the PZT elements to the

torsion tube. Both of these parameters can influence the harvesters’ performance and are difficult

42

to predict before prototyping. Mechanical damping in particular is influenced by factors such as

geometry, material selection, and boundary conditions and may differ for each harvester

prototype. However, since the exact value of mechanical damping cannot be predicted, the

sensitivity of the harvester configurations’ power output to modal damping was explored through

more simulations and is shown below in Figure 3.5. These results indicate that the power output

for each configuration decreases exponentially with increasing modal damping but at each

damping value, the slotted-C and C-channel cross-sections still perform better than the solid

cross-section. However, if the C-channel or slotted-C cross-section have higher damping ratios

than the solid cross-section, their power output may drop below the power outputs of the solid

cross-section harvester.

It is of interest to note that while the harvester effectiveness factor of each harvester

configuration also decreases exponentially with increasing damping, this does not change the

relative comparison between the harvester configurations. (see Figure 3.6). Across all damping

level considered, the C-channel and slotted-C harvesters have a higher effectiveness than the

solid harvester indicating a superior design. Hence, using the harvester effectiveness metric, the

C-channel and slotted-C harvesters remain the superior, or at least equally effective, designs for

all damping ratios.

Figure 3.5: Influence of modal damping ratio on peak power output at resonance through optimal load resistance

of each cross-section configuration

0.00E+00

2.00E-05

4.00E-05

6.00E-05

8.00E-05

1.00E-04

1.20E-04

1.40E-04

0 0.02 0.04 0.06 0.08 0.1 0.12

Pea

k P

ow

er O

utp

ut

[W]

Damping Ratio

Solid C-Channel Slotted-C

43

Figure 3.6: Influence of modal damping ratio on harvester effectiveness at resonance through optimal load

resistance of each cross-section configuration

3.4 Experimental Results

After simulating these designs, the solid tube and the slotted-C tube configurations were

selected for prototype fabrication as the worst and best configuration, respectively. As per the

design, the solid tube was machined from HDPE with a cross section of 3.25 mm x 3.25 mm

while the slotted-C tube was machined from aluminum with the dimensions as listed in Table

3.A. For each prototype, the torsion tube and its base were machined as one piece out of a raw

material piece to overcome any issues associated with bonding the tube to the harvester base. Six

off-the-shelf PZT-5A shear mode plates were bonded to the torsion tube using epoxy adhesives

(LePage Speed Set) with the underside of each tube left free of PZT elements. Adhesive

thickness of 0.010” was ensured by sandwiching the desired parts between two parallel plates

controlled by leadscrew stages and a set of calipers to ensure the desired plate separation. All

PZT elements were connected electrically in parallel by wires bonded to the elements by silver

conductive epoxy (MG Chemicals 8331). The eccentric mass of each prototype was then bonded

to the tube and the prototype base was secured to the shaker mounting plate using epoxy

adhesives. The prototypes are shown in Figure 3.7 and are depicted as bonded to the shaker

mounting plate.

0.00E+00

2.00E-03

4.00E-03

6.00E-03

8.00E-03

1.00E-02

1.20E-02

0 0.02 0.04 0.06 0.08 0.1 0.12

Har

ves

ter

Eff

ecti

ven

ess

(κ)

Damping Ratio

Solid C Channel Slotted-C

44

Figure 3.7: Prototypes of two centimetre-scaled torsion harvesters with a Canadian 10¢ coin for scale: (a) solid

cross-section and (b) slotted-C cross-section

The performance of these prototypes was tested by mounting them on an electrodynamic

shaker (Modal Shop 2110E) and exciting them with a constant acceleration amplitude swept-sine

signal from 10 Hz to 1000 Hz. The mechanical response was first characterized to identify the

torsional modes of vibration and this was done using a laser vibrometer (Polytec Inc.) to provide

non-contact velocity measurements at three locations on the harvester shown in Figure 3.8.

Figure 3.8: Locations of interest in characterizing mechanical response of each torsion harvester prototype

Mechanical response characterization of the solid cross-section harvester was done with a

swept-sine signal with a constant acceleration of 1 gpk while the slotted-C harvester was excited

with a constant acceleration of 0.25 gpk. The slotted-C harvester was characterized at a lower

acceleration magnitude because higher values were found to cause large angular rotations of the

eccentric mass making it difficult to measure velocity at the eccentric mass tip with a uniaxial

laser vibrometer. As a result, the two different acceleration amplitudes were used for the

mechanical characterizations. Further electrical characterization of the prototype was done using

the same acceleration amplitude of 1 gpk for both prototypes. The resulting mechanical response

45

of each harvester prototype at the three measured locations is shown in Figure 3.9 and Figure

3.10.

Figure 3.9: Mechanical response of solid tube harvester

Figure 3.10: Mechanical response of slotted-C tube harvester

The mechanical response plots indicate that both the solid tube and slotted-C harvester

prototypes perform as expected from the simulation results (see Table 3.B) and undergo their

first torsional resonant mode at a frequency of 351 Hz and 253 Hz, respectively. This mode is

characterized in both harvesters’ response plot by high velocities at the eccentric mass tip along

with low velocities at the torsion tube tip. The close agreement between the simulation and

measured resonant frequencies of the prototype also indicates that the mechanical assembly of

both harvester prototypes was achieved with minimal flaws and defects.

46

The mechanical response plot of each harvester prototype was also utilized to measure the

mechanical modal damping at resonance by identifying the resonant frequency and the half-

power frequencies (frequencies at which the velocity magnitude is 70.7% of the resonant

velocity magnitude) in the eccentric mass tip response plots. This method for calculating modal

damping is sensitive to accurate frequency resolution for determining the mechanical damping

ratio, but since both harvester prototypes exhibited very dominant resonant peaks, identifying the

appropriate frequencies was easily achieved using data analysis software. Using this approach

for the solid tube prototype, the resonant peak was found to occur at a frequency of 350.54 Hz

with a peak velocity of 153.54 mm/s. The corresponding half-power points of 108.55 mm/s

occurred at frequencies of 346.01 Hz and 355.92 Hz, yielding a damping ratio of 2.8% that was

in close agreement to the assumed damping ratio used for simulation predictions. For the slotted-

C prototype, the resonant peak occurred at a frequency of 253.05 Hz with a velocity of 28.78

mm/s. The corresponding half-power points of 20.35 mm/s occur at frequencies of 242.15 Hz

and 265.52 Hz, giving this prototype a damping ratio of 9.2%. Unlike the solid harvester

prototype, the damping ratio for the slotted-C harvester was much higher than assumed in the

simulations, indicating a large dissipation of input power that could otherwise be utilized for

energy harvesting purposes.

The SDOF parameters for the harvester prototypes were calculated next to estimate the

harvester effectiveness metric for each prototype. The stiffness k for each prototype was

estimated by statically applying a known force at a predetermined location along the eccentric

mass to create a torque on the torsion tube. By measuring the resulting difference in vertical

deformation at the eccentric mass tip and torsion tube tip locations shown in Figure 3.8, and

knowing the distance between these points, the angle of twist of the tube could be estimated

yielding the torsional stiffness using (3-10). The SDOF mass m was estimated using moment of

inertia expressions for a thick plate with effective dimensions listed below in Table 3.C. These

effective dimensions are different from the total dimensions outlined in Table 3.A as they

account for the rectangular slot cut into the eccentric mass for attaching the torsion tube along

with any machining discrepancies. The eccentricity of each mass was then used with the parallel

axis theorem to calculate the moment of inertia of the mass about the centre of the torsion tube.

The results of this analysis are summarized below in TABLE 3.C.

47

Table 3.C: Stiffness and mass estimation for SDOF harvester model

Solid Slotted-C

Tors

ion T

ube

Force [mN] 11.7 10.5

Torque Arm [mm] 6.8 9.7

Torque [µN-m] 79.5 101.9

Torsion Tip Deflection [µm] 0.9 0.3

Eccentric Mass Tip Deflection [µm] 1.6 1.4

Measurement point separation [mm]* 17 20

Angle of Twist [µrad] 41.1 55.0

Torsional Stiffness, k [N-m/rad] 1.91 1.85

Ecc

entr

ic

Mas

s

Mass [g] 4.8 4.3

Effective Length [mm] 13.6 19.4

Width [mm] 5.0 4.0

Eccentricity [mm] 6.7 9.7

Moment of inertia, m [mg-m2] 0.42 0.67

* This parameter refers to the distance between the torsion tip and eccentric mass tip points used in measuring the

vertical deflection.

The resulting k and m parameters from this analysis are then used in conjunction with the

previously measured damping ratios and (3-6) to estimate the harvester effectiveness factor for

each prototype.

After observing the mechanical response, the electrical performance of each prototype was

characterized by first measuring the source impedance of all PZT elements on each harvester.

This was done using an Agilent 4288A capacitance meter at 1 kHz that modeled the PZT source

impedance as an equivalent capacitance (Cp) in parallel with a leakage resistance (Rp). The

measurement results from this are listed in Table 3.D. It should be noted that although these

measurements are not taken at the operating frequencies of the two harvester prototypes, the

values are not expected to change significantly since all frequencies under consideration are well

below the resonant frequency of the piezoceramic elements.

Table 3.D: Source impedance measurements of each harvester prototype with associated

measurement accuracy

Cp [pF] Rp [MΩ]

Solid 36.4417 ± 0.9100 164.136 ±17.189

Slotted-C 33.5629 ± 0.2634 128.596 ± 13.336

The impedance measurements of the PZT elements indicate that the solid and slotted-C

harvester both have non-infinite leakage resistances that contribute to voltage and power losses

48

in the harvesters as was observed with the prototype in the previous chapter. As a result of this

leakage current, the two prototypes are expected to produce lower outputs than anticipated in the

simulations.

Following the impedance measurements, the open circuit voltages of the harvester were

characterized by performing a mechanical excitation sinusoidal sweep with 1 gpk constant

acceleration for each harvester with open circuit conditions. The results of this are shown in

Figure 3.11. After the open circuit voltage measurements, peak power outputs of each harvester

prototype were measured by exciting the prototypes with 1 gpk sinusoidal acceleration at their

resonant frequencies and attaching a variable resistor to the electrical leads of the harvester. The

load resistance was varied to find the optimal value for power dissipation. The results of the

resistance sweep thus obtained are summarized below in Figure 3.12. A summary of all

simulation and experimental results is shown in Table 3.E.

Figure 3.11: Open circuit voltage frequency response of torsion harvester prototypes

Figure 3.12: Output power of harvester prototypes over varying load resistance values at resonance

1.00

10.00

100.00

0 10 20 30 40 50

Ou

tpu

t P

ow

er [

µW

]

Load Resistance [MΩ]

Solid

Slotted-C

49

Table 3.E: Summary of simulation and prototype performance

Solid Slotted-C

ANSYS Prototype ANSYS Prototype ANSYS*

Torsional Stiffness [N-m] 1.95 1.91 1.88 1.85 1.88

Damping Ratio 0.03 0.028 0.03 0.092 0.09

Resonant Frequency [Hz] 333.5 351 246.6 253 246.6

Maximum Open Circuit Voltage [V] 25.92 22.26 61.44 17.10 18.72

Optimal Load Resistance [MΩ] 10.37 9.8 14.35 14.9 14.35

Peak Power Output [µW] 32.04 40.1 126.4 9.79 13.21

Harvester Effectiveness (𝜅) [%] 0.25 0.28 1.09 0.25 0.33

* Simulation repeated after experiments with new damping ratio

The electrical performance of the solid harvester prototype was found to be in close overall

agreement to the simulation predictions. The prototype produced a slightly lower open circuit

voltage than the simulation predictions, but as a result of its lower source impedance and

damping ratio, was able to produce higher than expected peak power outputs through an optimal

load of 9 MΩ. On the other hand, the slotted-C harvester’s electrical performance was much

poorer than the expected simulated values. The majority of these power reductions were

attributed to its significantly higher damping ratio of 9% than the simulated 3% damping.

Repeating the previous ANSYS simulations for the slotted-C harvester at the measured damping

ratio of 9% shows that with this damping, the prototype was capable of producing an open circuit

voltage of 18.7 V, a peak power output of 13 µW and a harvester effectiveness of 0.33%, all

figures that correspond well with the measured data. Since the resonant frequency of the slotted-

C prototype is in close agreement with the simulation predictions, this high damping ratio is

unlikely to be a result of assembly or manufacturing flaws and may be inherent to the harvester

configuration due to its size, material selection or geometry. Additional minor losses in the

slotted-C harvester can be attributed to its low leakage resistance that results in current losses

between the positive and negative electrodes, further reducing the voltage and power outputs of

the prototype to the values observed in Table 3.E. However, despite the large losses in the

slotted-C harvester, its effectiveness (𝜅) was only slightly lower to the solid harvester, indicating

its overall better design.

50

Thus, while the compliant slotted-C tube harvester prototype was predicted to produce higher

power and voltage outputs, as a result of its higher damping ratio, the performance of this

harvester prototype was found to be inferior to the prototype of the solid tube harvester.

3.5 Summary

In conclusion, three torsion tube configurations were examined in this work to explore the

design space of the torsion harvester concept and observe its influence on the device’s

performance. Upon analyzing the configurations through finite element models, it was observed

that configurations with increasing torsional compliance and eccentric masses with larger

rotational inertias resulted in more efficient harvester designs that produced higher power outputs

and open circuit voltages. However, the centimeter-scaled harvester configurations were found to

be very susceptible to mechanical damping and after fabricating prototypes, the slotted-C

harvester was found to have a much higher damping ratio than was initially anticipated, resulting

in significant performance losses. Thus, while the most compliant slotted-C harvester was

expected to have the highest power outputs, its prototype was unable to match the predicted

performance. Thus, mechanical damping in the torsion harvester prototypes was identified as a

key performance factor and its influence must be minimized to fully realize the potential of the

slotted-C torsion harvester.

51

Analytical Modelling and Design Guidelines 4.

The previous chapter presented the effects of factors such as torsion tube cross-section, mass

eccentricity and damping on the output of the torsion harvester through the development and

comparison of multiple prototypes. These influences are now examined in greater detail with the

development of a mathematical model and analytical expressions. These expressions are then

used to develop a set of design guidelines that can be used to optimize and customize a torsion

harvester for a desired application environment.

4.1 Analytical Expressions

The torsion harvester concept described in chapter two operates by inducing torsion on a tube

under the action of non-rotational vibrations. Analyzing this structure as presented before is a

difficult task since the presence of the eccentric mass induces complex bending and torsional

vibrations that may interact with each other. Furthermore, practical implementation of this

harvester introduces other non-uniformities such as the use of discrete piezoelectric patches that

are difficult to capture with a comprehensive analytical model. Hence, a simplified model is

considered in this section that preserves the operating principle but with simplified geometry.

This allows for the accurate determination of trends affecting the torsion harvester without the

computational investment required for high fidelity analytical models. It is shown later with the

finite element models that the conclusions drawn from this analytical model are valid for the

harvester concept presented in earlier chapters.

Consider the harvester shown in Figure 4.1 consisting of a tube with an arbitrary but uniform

cross-section with a piezoelectric element of uniform thickness attached to its outer surface. The

piezoelectric element is poled tangentially and harvesting electrodes placed at the base of the

tube and at its tip to provide d15 shear mode energy harvesting. Unlike in the original torsion

harvester, to simplify the analysis, this geometry is assumed to be excited by rotational vibrations

thus directly inducing torsional stresses in the tube. As a result, the presence of the eccentric

mass is neglected.

52

Figure 4.1: Simplified model of torsion harvester

Figure 4.2: Cross-section of the simplified harvester. Domain I represents the substrate material and domain II

corresponds to the piezoelectric element

The torsion of a non-circular cross-section results in both twisting and warping and the process

of analyzing such sections is discussed in depth in [45]. Using this approach for the model

depicted in Figure 4.1, the governing equation of undamped motion for a differential torsional

element in the absence of external forces can be expressed as:

𝜌𝐼𝑝𝜕2𝜃(𝑧,𝑡)

𝜕𝑡2 =𝜕

𝜕𝑧(𝑀(𝑧, 𝑡)) (4-1)

53

where ρ is the effective density of the cross-section, Ip is the effective polar moment of inertia of

cross-section, θ(z,t) is the angle of twist as a function of position z and time t and M(z,t) is the

internal resisting torque as a result of angular twist θ(z,t).

The internal torque M(z,t) can be expressed as the sum of the torques generated in the substrate

Ms(z,t) and the torque generated in the piezoelectric element Mp(z,t) and each of these can be

evaluated by integrating the resulting shear stresses and their respective moment arms over the

cross-section.

𝑀(𝑧, 𝑡) = 𝑀𝑠(𝑧, 𝑡) + 𝑀𝑝(𝑧, 𝑡) (4-2)

𝑀(𝑧, 𝑡) = ∫ τxz𝑦𝑑𝐴 −𝐴

∫ τyz𝑥𝑑𝐴𝐴

(4-3)

If a single warping function ψ(x,y) is assumed to correspond to the entire non-circular cross-

section of the tube, then the internal torques Ms(z,t) and Mp(z,t) can evaluated by expressing the

shearing stresses in terms of the warping function and integrating over the cross-section:

𝑀𝑠(𝑧, 𝑡) = 𝐺𝑠𝐽𝑠𝜕𝜃(𝑧,𝑡)

𝜕𝑧 (4-4)

Here, Gs refers to the shear modulus of the substrate material and Js is a constant corresponding

to the integral of the warping function over domain I as shown in Figure 4.2. Physically, Js can

be considered as the substructure’s contribution to the overall torsion constant of the entire cross

section.

𝐽𝑠 = ∬ [𝑦 (𝜕𝜓

𝜕𝑥− 𝑦) − 𝑥 (

𝜕𝜓

𝜕𝑦+ 𝑥)] 𝑑𝑥𝑑𝑦

𝐼 (4-5)

Similarly evaluating Mp(z,t) and accounting for piezoelectric effects yields

𝑀𝑝(𝑧, 𝑡) = 𝑐55𝐸 𝐽𝑝

𝜕𝜃(𝑧,𝑡)

𝜕𝑧+

𝑐55𝐸 𝑑15𝑄𝑝

𝑠𝑒𝑣(𝑡) (4-6)

In (4-6), 𝑐55𝐸 is a stiffness term relating to the stiffness matrix of the piezoelectric material, Jp is

the integral of the warping function as in (4-5) but evaluated over domain II, d15 is the shear-

mode piezoelectric strain constant, se is the separation between the electrodes in the harvester, Qp

is the first polar moment of area (𝑄𝑝 = ∫ 𝑟𝑑𝐴𝐼𝐼

) of the piezoelectric cross-section (domain II)

54

about the centre of rotation of the cross-section, and v(t) is the potential difference between the

electrodes of the harvester as a function of time. Using these expressions, (4-1) can be expressed

as:

𝜌𝐼𝑝𝜕2𝜃(𝑧,𝑡)

𝜕𝑡2 = (𝐺𝑠𝐽𝑠 + 𝑐55𝐸 𝐽𝑝)

𝜕2𝜃

𝜕𝑧2 + 𝜕

𝜕𝑧(

𝑐55𝐸 𝑑15𝑄𝑝

𝑠𝑒𝑣(𝑡)𝐻(𝑧) − 𝐻(𝑧 − 𝐿)) (4-7)

The addition of the Heaviside function H(z) at z = 0 and z = L in (4-7) is to ensure the survival of

the third term in the differential equation. The quantity represented by (𝐺𝑠𝐽𝑠 + 𝑐55𝐸 𝐽𝑝) in the

above equation can be thought of as the total torsional rigidity of the composite cross-section of

the harvester. These terms will be replaced by κT in subsequent expressions. If the angular

twisting θ(z,t) is expressed as the superposition of the base excitation θb(z,t) and the relative

twisting θrel(z,t), (4-7) can be expressed as:

𝜃(𝑧, 𝑡) = 𝜃𝑏(𝑧, 𝑡) + 𝜃𝑟𝑒𝑙(𝑧, 𝑡) (4-8)

𝜌𝐼𝑝𝜕2𝜃𝑟𝑒𝑙(𝑧,𝑡)

𝜕𝑡2 − 𝜅𝑇𝜕2𝜃𝑟𝑒𝑙

𝜕𝑧2 −𝑐55

𝐸 𝑑15𝑄𝑝

𝑠𝑒𝛿(𝑧) − 𝛿(𝑧 − 𝐿)𝑣(𝑡) = −𝜌𝐼𝑝

𝜕2𝜃𝑏(𝑧,𝑡)

𝜕𝑡2 (4-9)

In (4-9), δ(z) is the Dirac-delta function and occurs as a result of evaluating the derivative of the

Heaviside function. This equation represents the mechanical half of the coupled-set of equations

that describe the behavior of the harvester. In the absence of piezoelectric phenomena, the term

containing v(t) in the equation vanishes, yielding a standalone differential equation describing

the torsional behavior of the tube under base excitations. To obtain the second of the coupled

equations, the electrical domain of the harvester must be examined.

The charge collected by the positive electrode at the base of the tube (z = 0) can be expressed

by integrating the piezoelectric equation 𝐷1 = 𝑑15𝑇5 + 𝜖11𝑇 𝐸1 over domain II from Figure 4.2.

The charge collected can then be expressed as

𝑄 =𝑐55

𝐸 𝐽𝑝𝑑15

𝑟𝑝,𝑐

𝜕𝜃𝑟𝑒𝑙(𝑧=0,𝑡)

𝜕𝑧−

𝜖11𝑆 𝐴𝑝

𝑠𝑒𝑣(𝑡) (4-10)

In (4-10), Q represents the charge collected on the positive electrode at the base of the tube (z =

0), rp,c represents the effective radial distance at which the entire piezoelectric element’s torque

Mp(z=0,t) can be represented with a single force, 𝜖11𝑆 is the constant strain permittivity of the

55

piezoelectric material and Ap is the area of the electrode (equal to area of domain II in the

considered geometry). The charge collected by the positive electrode can then be related to the

current and voltage through the applied load resistor, RL as:

𝑖 =𝑑𝑄

𝑑𝑡=

𝑣(𝑡)

𝑅𝐿 (4-11)

𝜖11𝑆 𝐴𝑝

𝑠𝑒

𝑑𝑣(𝑡)

𝑑𝑡+

𝑣(𝑡)

𝑅𝐿=

𝑐55𝐸 𝐽𝑝𝑑15

𝑟𝑝,𝑐

𝜕2𝜃𝑟𝑒𝑙(𝑧=0,𝑡)

𝜕𝑧𝜕𝑡 (4-12)

Equation (4-12) is the second coupled field equation that relates the two degrees of freedom

θ(z,t) and v(t) of the harvester model.

Using modal analysis, the relative twisting of the tube can be expressed as a superposition of

all the modes of the harvester:

𝜃𝑟𝑒𝑙(𝑧, 𝑡) = ∑ 𝜙𝑟(𝑧)𝜂𝑟(𝑡)∞𝑟=0 (4-13)

where Φr(z) represents the normalized mode shape, 𝜂𝑟(𝑡) is the modal coordinate and r is the

mode number. Substituting (4-13) into (4-9) and (4-12) yields:

𝜌𝐼𝑝𝜙𝑟𝑑2𝜂𝑟

𝑑𝑡2 − 𝜅𝑇𝜂𝑟𝑑2𝜙𝑟

𝑑𝑧2 −𝑐55

𝐸 𝑑15𝑄𝑝

𝑠𝑒𝛿(𝑧) − 𝛿(𝑧 − 𝐿)𝑣(𝑡) = −𝜌𝐼𝑝

𝜕2𝜃𝑏(𝑧,𝑡)

𝜕𝑡2 (4-14)

𝜖11𝑆 𝐴𝑝

𝑠𝑒

𝑑𝑣(𝑡)

𝑑𝑡+

𝑣(𝑡)

𝑅𝐿=

𝑐55𝐸 𝐽𝑝𝑑15

𝑟𝑝,𝑐

𝑑𝜂𝑟

𝑑𝑡 𝑑𝜙𝑟

𝑑𝑧|

𝑧=0 (4-15)

Knowing the mode shape for the fixed-free boundary conditions of the torsion tube, the mode

shape can be expressed as:

𝜙𝑟(𝑧) = √2

𝐿sin

(2𝑟+1)𝜋𝑧

2𝐿 (4-16)

Equations (4-14) and (4-15) can then be rewritten as:

𝜙𝑟𝑑2𝜂𝑟

𝑑𝑡2 + 𝜔𝑟

2𝜙𝑟𝜂𝑟 −𝑐55

𝐸 𝑑15𝑄𝑝

𝜌𝐼𝑝𝑠𝑒𝛿(𝑧) − 𝛿(𝑧 − 𝐿)𝑣(𝑡) =

𝜕2𝜃𝑏(𝑧,𝑡)

𝜕𝑡2 (4-17)

𝐶𝑝𝑑𝑣(𝑡)

𝑑𝑡+

𝑣(𝑡)

𝑅𝐿=

𝑐55𝐸 𝐽𝑝𝑑15

𝑟𝑝,𝑐

𝐶𝑟(2𝑟+1)𝜋

2𝐿

𝑑𝜂𝑟

𝑑𝑡 (4-18)

56

with the following substitutions for the rth resonant frequency, ωr, and the piezoelectric

capacitance Cp:

𝜔𝑟2 =

𝜅𝑇(2𝑟+1)2𝜋2

4𝜌𝐼𝑝𝐿2 (4-19)

𝐶𝑝 = 𝜖11

𝑆 𝐴𝑝

𝑠𝑒 (4-20)

Multiplying (4-17) by Φr(z) and integrating z from 0 to L yields the following undamped modal

equation:

𝑑2𝜂𝑟

𝑑𝑡2 + 𝜔𝑟2𝜂𝑟 + √

2

𝐿

𝑐55𝐸 𝑑15𝑄𝑝(−1)𝑟

𝜌𝐼𝑝𝑠𝑒𝑣(𝑡) = −

2√2𝐿

2𝜋𝑟+𝜋

𝜕2𝜃𝑏(𝑧,𝑡)

𝜕𝑡2 (4-21)

Adding proportional modal damping ζr to this equation gives:

𝑑2𝜂𝑟

𝑑𝑡2 + 2𝜁𝑟𝜔𝑟𝑑𝜂𝑟

𝑑𝑡+ 𝜔𝑟

2𝜂𝑟 + √2

𝐿

𝑐55𝐸 𝑑15𝑄𝑝(−1)𝑟

𝜌𝐼𝑝𝑠𝑒𝑣(𝑡) = −

2√2𝐿

2𝜋𝑟+𝜋

𝜕2𝜃𝑏(𝑧,𝑡)

𝜕𝑡2 (4-22)

The coupling factors in the above equations can be replaced as:

𝜒𝑚 = √2

𝐿

𝑐55𝐸 𝑑15𝑄𝑝(−1)𝑟

𝜌𝐼𝑝𝑠𝑒 (4-23)

𝜒𝑒 =𝑐55

𝐸 𝐽𝑝𝑑15

𝑟𝑝,𝑐

(2𝑟+1)𝜋

2𝐿√

2

𝐿 (4-24)

𝜇 = −2√2𝐿

2𝜋𝑟+𝜋 (4-25)

Here, (4-23) and (4-24) characterize the electromechanical coupling in the system and χm

represents the contribution of the voltage in the mechanical equation (4-22) and χe represents the

contribution of the mechanical degree of freedom in the electrical domain (4-18).

Equations (4-22) and (4-18) give two coupled time domain equations. If the excitation motion

is assumed to be sinusoidal and continuous with time, the steady-state solutions to these

equations can be obtained for v(t) and the power output P(t).

57

|𝑣𝑟(𝑡)| =−Ω𝑟𝜇𝜒𝑒𝑅𝐿𝜃

√[𝜔𝑟2(1−Ω𝑟

21+2𝜁𝑟𝜔𝑟𝐶𝑝𝑅𝐿)2

+Ω𝑟2(2𝜁𝑟𝜔𝑟+𝐶𝑝𝑅𝐿𝜔𝑟

21−Ω𝑟2+𝜒𝑒𝜒𝑚𝑅𝐿)

2]

(4-26)

|𝑃𝑟(𝑡)| =Ω𝑟

2𝜇2𝜒𝑒2𝜃

2𝑅𝐿

[𝜔𝑟2(1−Ω𝑟

21+2𝜁𝑟𝜔𝑟𝐶𝑝𝑅𝐿)2

+Ω𝑟2(2𝜁𝑟𝜔𝑟+𝐶𝑝𝑅𝐿𝜔𝑟

21−Ω𝑟2+𝜒𝑒𝜒𝑚𝑅𝐿)

2] (4-27)

In the above expressions, Ωr represents the frequency ratio 𝜔

𝜔𝑟 between the excitation

frequency ω and the resonant frequency of each mode and 𝜃 represents the angular acceleration

of the base excitations. Finally, if the device is excited close to its natural resonant frequency,

(Ω0 ~ 1), only the contributions from the fundamental mode (r = 0) need to be considered and the

expressions voltage and power can be simplified as:

|𝑣(𝑡)| =−𝜇𝜒𝑒𝑅𝐿𝜃

√(2𝐶𝑝𝑅𝐿𝜁0𝜔02)

2+(2𝜁0𝜔0+𝜒𝑒𝜒𝑚𝑅𝐿)2

(4-28)

|𝑃(𝑡)| =𝜇2𝜒𝑒

2𝜃2

𝑅𝐿

(2𝐶𝑝𝑅𝐿𝜁0𝜔02)

2+(2𝜁0𝜔0+𝜒𝑒𝜒𝑚𝑅𝐿)2

(4-29)

The above two expressions reflect the electrical performance of the torsion harvester under a

given base excitation and with a load resistance RL.

4.2 Influence of Various Parameters

The expressions from the analytical model indicate the output response of the simplified

torsion harvester model to a variety of input parameters. Since the analytical expressions are

based on a simplified geometry and excitation conditions, (4-28) and (4-29) are not expected to

accurately predict the values of the voltage and power outputs of the harvesters presented in

previous chapters. Instead, these expressions are used to highlight the trends caused by

manipulating any one parameter and can be used to guide the design of the torsion harvester.

The influence of environmental factors such as mechanical damping ζ0 and the excitation

acceleration 𝜃 can be readily observed since these expressions are explicitly present in (4-29). It

can be seen that the power output of the harvester is proportional to the square of base

acceleration amplitude and so, for optimal performance, the harvester must be excited with the

58

highest base accelerations present in its target environment. Furthermore, the power output of the

harvester is inversely proportional to the mechanical damping ζ0 in the harvester and so

mechanical damping should be minimized in the harvester to maximize power output.

Equation (4-29) also indicates that the load resistance attached to the harvester plays a role in

the power output of the harvester. Since the RL terms appear in both the numerator and

denominator, it is expected that there is an optimal load resistance, where power output is

expected to be maximized. Differentiating |P(t)| with respect to RL and setting it equal to zero

yields the optimal resistance as:

𝑅𝐿,𝑜𝑝𝑡 = √4𝜁0

2𝜔02

4𝜁02𝜔0

4𝐶𝑝2+𝜒𝑒

2𝜒𝑚2 (4-30)

Equation (4-30) indicates that the optimal load resistance for the harvester is not only

dependent on its piezoelectric capacitance (Cp), but also on the coupling factors from the

electromechanical equations. In scenarios where the coupling is weak, such as large

substructures with small piezoelectric elements, (4-30) can be simplified to yield the optimal

load resistance as simply the source impedance of the capacitive piezoelectric elements.

However, for efficient harvester designs that convert a large amount of input strain energy into

electrical energy, the optimal load resistance cannot be simply assumed as being equal to the

electrical source impedance of the harvester. The optimal load resistance for such harvesters

must either be predicted by the accurate determination of the coupling factors or through

experimental means.

The influence of piezoelectric material properties on the harvester’s power output can be

determined by tracing the piezoelectric constants d15 and 𝜖11𝑆 in (4-29). Using this approach and

the expression for optimal resistance in (4-30), the influence of piezoelectric properties was

determined and is shown in Figure 4.3. It can be observed that to maximize power output of the

torsion harvester, the d15 constant must be maximized while minimizing the dielectric

permittivity of the piezoelectric material. This result confirms the established d-g figure of merit

commonly used to evaluate piezoelectric materials for energy harvesting [2]. It should also be

noted from Figure 4.3 that although high d15 values are desired for high power outputs, beyond a

certain value, increasing d15 produces diminishing returns as the power output of the harvester

59

becomes limited by other factors. Thus, materials with significantly higher d15 values may not

necessarily produce proportionally higher power outputs.

Figure 4.3: Influence of piezoelectric coefficients d15 and 𝜖11𝑆 on the power output of the simplified torsion

harvester

Equation (4-29) also indicates that more compliant harvesters may produce higher power

outputs since decreasing torsional rigidity (𝜅𝑇) also reduces the resonant frequency of the

harvester (ω0). However, achieving this requires manipulation of various factors such as Ap, Jp,

Js, Gs and 𝑐55𝐸 , that also influence other terms in (4-29) and the interaction between all these is

difficult to predict from the analytical model alone. Hence, the influence of these torsional

rigidity parameters is explored in more detail in the next section with finite element models.

Finally, (4-29) can also be used to identify the influence of scaling in the harvester. If a torsion

tube harvester is scaled in size by a single factor for all geometric dimensions while maintaining

shape, material composition, base excitation conditions and mechanical damping, the power

output of the scaled harvester is found to vary with the sixth power of the scaling factor. Thus, in

this model, a harvester twice as large as the reference design produces 64 times the amount of

output power. This relationship occurs because under identical excitation conditions, the energy

input into the scaled harvester also varies with the sixth power of the scaling factor allowing the

scaled harvester to produce energy outputs proportional to the energy input. From a harvester

design perspective, this model dictates that for highest power outputs in a given vibration

environment, it is ideal to use the largest harvester that can be accommodated by the

environment without severely affecting the sources of vibrations.

log(|

P(t

)|)

d15

|P(t

)|

𝜖11

60

4.3 Finite Element Model Results

After observing the influence of various parameters in the analytical model of the simplified

torsion harvester geometry, the full harvester concept with the eccentric mass and excited by

non-rotational vibrations as presented in chapter two (referred to as “original torsion harvester”)

was analyzed to determine if the predicted trends using the simplified geometry were still valid

for the full harvester concept. In order to observe the influence of various parameters, a finite

element model of this harvester was created and analyzed in a commercial finite element

software (ANSYS, ANSYS Inc., Canonsburg, Pennsylvania, USA). As before, the geometry of

the harvester was subject to two analyses: a modal analysis to identify resonant modes and a

harmonic analysis with vertical sinusoidal vibrations applied to the base to observe a steady-state

energy harvesting response. Furthermore, as in the analytical model, the harmonic analysis was

conducted at the first resonant frequency with the addition of a constant modal damping of 3%

and a constant peak sinusoidal base excitation acceleration of 9.81 m/s2 (1 gpk). The torsion tube

substructure, eccentric mass and base in the harvester were modeled using SOLID186 second

order 3D elements while the piezoelectric elements were modeled with SOLID226 coupled field

elements with displacement and volt degrees of freedom. To measure power outputs, a resistive

load was modeled between the assigned electrode surfaces with CIRCU94 elements and its

resistance was varied to identify optimal load resistance values. The baseline model used for this

analysis was identical to the solid harvester configuration presented in chapter three. This

baseline model was then analyzed over varying sets of influencing parameters to observe their

effect on the power output. The results of the simulations and prototype testing are summarized

in the figures below.

Among these, it can be observed that the peak power output decreases inversely with

increasing damping ratio (Figure 4.4), increases with the square of the base acceleration (Figure

4.5), and furthermore piezoelectric properties d15 (Figure 4.6) and 𝜖11𝑆 (Figure 4.7) all behave as

predicted by the analytical expressions developed previously, validating their design guidelines.

The influence of piezoelectric properties such as d15 and 𝜖11𝑆 is particularly accurate when

compared to the trends shown in Figure 4.3. However, the effect of the scaling factor is found to

differ from the predicted response from the simplified model (Figure 4.8). In the finite element

analysis of the original harvester geometry, the power output is observed to vary with the fourth

power of the scaling factor as opposed to the predicted sixth power. This discrepancy arises due

61

to the difference in operation of the analytical model from the original torsion harvester. In the

analytical model, the torsion tube is assumed to be excited directly by torsional vibrations and

hence, their magnitude is assumed to remain constant when examining the influence of scaling.

However, in the original torsion harvester, torsion is induced in the torsion tube due to the action

of the eccentric mass under non-rotational vibrations. Thus, the angle of twist at the free end of

the tube is not an independent environmental parameter but is actually inversely proportional to

the eccentricity of the mass under a constant vertical excitation environment (𝜃𝑡𝑖𝑝 ≈𝑢

𝑒, assuming

a rigid eccentric mass; where u is the non-rotational vibration amplitude and e is the eccentricity

of the tip mass). Thus, when examining the influence of scaling, the magnitude of 𝜃 at the free

tip and thus , is inversely proportional to the scaling factor, thus yielding a total impact of the

fourth power of the scaling factor on the power output of the harvester. However, despite this

discrepancy, the original design guideline on using the largest harvester possible in a given

environment still remains valid.

Figure 4.4: Influence of mechanical damping on peak power output of original torsion harvester

0.00E+00

5.00E-06

1.00E-05

1.50E-05

2.00E-05

2.50E-05

3.00E-05

3.50E-05

0 0.02 0.04 0.06 0.08 0.1 0.12

Pea

k P

ow

er O

utp

ut

[W]

Damping Ratio

62

Figure 4.5: Influence of base acceleration magnitude on peak power output of original torsion harvester

Figure 4.6: Influence of d15 piezoelectric constant on peak power output of original torsion harvester

Figure 4.7: Influence of dielectric permittivity (𝜖11𝑆 ) on peak power output of original torsion harvester

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

0 1 2 3 4 5

Pea

k P

ow

er O

utp

ut

[W]

Base Acceleration [g]

0.E+00

1.E-05

2.E-05

3.E-05

4.E-05

5.E-05

0 500 1000 1500 2000

Pea

k P

ow

er O

utp

ut

[W]

d15 [pC/N]

0.E+00

1.E-05

2.E-05

3.E-05

4.E-05

0 500 1000 1500 2000 2500 3000

Pea

k P

ow

er O

utp

ut

[W]

Relative Permittivity (ϵ)

63

Figure 4.8: Influence of scaling factor on peak power output of original torsion harvester

After verifying these trends, the influence of interacting geometric and material parameters

that contribute to the overall torsional rigidity of the harvester was also observed. For this, the

three different torsion tube cross-sections considered in the previous chapter were analyzed

through finite element models. For each cross-section, the substrate material modulus was varied

while PZT material properties and geometry was kept constant. The results of this analysis are

shown in Figure 4.9.

Figure 4.9: Influence of cross-section geometry and substrate Young’s modulus on peak power output of original

torsion harvester

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

0 2 4 6 8 10 12

Pea

k P

ow

er O

utp

ut

[W]

Scaling Factor

0.0E+00

5.0E-05

1.0E-04

1.5E-04

2.0E-04

2.5E-04

3.0E-04

1.E+09 1.E+10 1.E+11 1.E+12

Pea

k P

ow

er O

uptu

t [W

]

Substrate Young's Modulus [Pa]

ANSYS Square ANSYS C-Channel ANSYS Slotted-C

64

The results of this analysis indicate that for each cross-section geometry, there is a substrate

modulus that produces the highest peak power output in the harvester. The modulus at which this

peak occurs varies with cross-section geometry but appears to increase as the partial torsion

constant evaluated over the substrate (Js) decreases. This indicates that the optimal material for

the substrate (Gs from the analytical model) is related to the torsion constants of the substrate and

piezoelectric elements (Js and Jp) and the material properties of PZT (𝑐55𝐸 ). As Js decreases for a

certain cross section geometry, the optimal value of substrate modulus increases. The exact

proportionality between these constants is not fully known at this point, but in order to maximize

the peak power output from the torsion harvester, it is evident that a degree of matching is

necessary between the torsional rigidity of the PZT elements (𝑐55𝐸 𝐽𝑝) and that of the torsion tube

(𝐺𝑠𝐽𝑠).

4.4 Summary

In conclusion, the influence of a wide variety of parameters on the power output of a torsion

based shear mode harvester was presented in this chapter through analytical and finite element

models. Using these, it was observed that to maximize power output from such harvesters, it is

ideal to use the largest harvester that can be supported by a vibration environment with the

highest base accelerations. Furthermore, the ideal piezoelectric materials for these harvesters

maximized the d15 piezoelectric constant while minimizing the dielectric permittivity. Finally, the

ideal torsion tube geometry for these harvesters contained some form of matching between the

torsional rigidity of the PZT elements (𝑐55𝐸 𝐽𝑝) and that of the torsion tube (𝐺𝑠𝐽𝑠). These

guidelines in conjunction with finite element simulations presented can be used to design the

optimal torsion based shear mode harvester for a desired application.

65

Improved Torsion Harvester for a Wireless Sensor 5.

Module

The design guidelines presented previously outlined strategies for designing a torsion harvester

for a specific application. These are now utilized to develop a torsion harvester for powering a

wireless sensor module. Electronics necessary for integrating the torsion harvester are also

explored and the capability of the torsion harvester to recharge a capacitor is demonstrated.

5.1 Harvester Design and Performance

Based on the design guidelines from the previous chapter, the size of the harvester was first

determined. With the scale of off-the-shelf wireless sensor components under consideration, the

largest harvester that could feasibly be used to power them was assumed to have a size of 10 cm3

to allow integration with the sensor module. With this size constraint, a variety of torsion tube

designs were explored and evaluated for their torsional compliance, ease of manufacturing and

ease of assembly with commercially available piezoelectric elements. Since most commercially

available shear mode piezoelectric elements are thin rectangular plates, the cross-section shown

in Figure 5.1 was selected for the design as a result of its high compliance that would minimize

torsional vibration frequencies and ease of manufacturing. To minimize mechanical damping in

the system, the torsion tube and support were designed as a single piece to minimize assembly

defects. This harvester design is shown in Figure 5.2 and its major dimensions are outlined in

Table 5.A.

66

Figure 5.1: (a) Torsion tube geometry and (b) cross section of improved torsion harvester

Figure 5.2: Schematic of improved torsion tube harvester for the wireless sensor module showing all components

Table 5.A: Characteristics of torsion harvester for wireless sensor module

Torsion

Tube

Material Aluminum

Length [mm] 22

Width [mm] 5.60

Wall Thickness [mm] 0.50

Eccentric

Mass

Material Tungsten Carbide

Length [mm] 44.5

Width [mm] 7

Height [mm] 10

The piezoelectric elements used for this harvester were off-the-shelf PZT-5A piezoceramics

with dimensions of 5 mm x 5 mm x 0.5 mm. In this scenario, since the material properties (𝑐55𝐸 ).

and dimensions of the piezoelectric elements (Jp) were fixed due to available off-the-shelf

manufacturer specifications and the cross-section geometry of the torsion tube had been selected

(Js), the material properties of the torsion tube (G) could be optimized using the mechanical

impedance matching phenomenon demonstrated in the previous chapter. This was achieved

67

through a set of finite element simulations to yield the maximum power output for the harvester

at each torsion tube shear modulus. In these, the harvester geometry from Figure 5.2 was

modeled in ANSYS and was subject to a coupled piezoelectric analysis at its resonant frequency

with a base acceleration of 0.25 gpk. Each design was subject to an applied modal damping of 3%

and structural materials in the harvester were modeled using SOLID186 second order 3D

elements and the piezoelectric elements were modeled with SOLID226 coupled field elements.

For each torsion tube material examined, the power output of the harvester was measured

through an optimized resistive load. The results of these simulations are summarized in Figure

5.3.

¸

Figure 5.3: Effect of torsion tube material’s elastic modulus on power output of torsion harvester

From these results, it was observed that the optimal material for the torsion tube had a shear

modulus of 17 GPa. However, this did not correspond to any feasibly available material and the

closest practically accessible material was instead found to be aluminum with a shear modulus of

27 GPa. With this material selected for the torsion tube, additional finite element simulations

were performed on the harvester to examine its behavior. Characteristics such as open circuit

voltage and peak power output were studied by performing a harmonic analysis of the harvester

at its resonant frequency. The results of these simulations at a base acceleration of 0.25 gpk are

outlined in Table 5.B and a schematic of the voltage distribution in the harvester with an optimal

load resistance is shown in Figure 5.4.

0.00E+00

5.00E-05

1.00E-04

1.50E-04

2.00E-04

5.00E+08 5.00E+09 5.00E+10

Max

imum

Pow

er O

utp

ut

[W]

Torsion Tube Shear Modulus, G [Pa]

68

Table 5.B: Summary of simulation results

Resonance Frequency [Hz] 244

Open Circuit Voltage [VRMS] 61.8

Maximum Average Power Output [µW] 220

Optimal Load Resistance [MΩ] 9.40

Figure 5.4: Voltage distribution in torsion harvester at resonance with optimal load resistance. Units are in Volts

RMS. Voltage at the output electrodes is 45.7 VRMS.

From these results, it was observed that the torsion harvester under consideration was capable

of producing an open circuit voltage of 61.8 VRMS and dissipating a maximum power output of

220 μW (average continuous power) by producing 45.7 VRMS across a 9.40 MΩ resistive load at

its resonant frequency of 244 Hz.

5.2 Experimental Results

After simulating the performance of the harvester, a prototype was constructed with the

dimensions listed in Table 5.A. A schematic of this prototype is shown below in Figure 5.5.

Eight off-the-shelf PZT-5A shear mode plates were bonded to the torsion tube (two on each face)

using epoxy adhesives (LePage Speed Set). An adhesive bond line of 0.010” was ensured in this

by sandwiching the desired parts between two parallel plates controlled by leadscrew stages and

a set of calipers to ensure the desired plate separation. All PZT elements were connected

1

MN

MX

X

Y

Z

4 Beam Optimized Harvester - PZT

05.71341

11.426817.1402

22.853628.5671

34.280539.9939

45.707351.4207

SEP 16 2014

21:03:59

NODAL SOLUTION

STEP=1

SUB =1

FREQ=243.728

AMPLITUDE

VOLT (AVG)

RSYS=0

DMX =.309E-04

SMX =51.4207

69

electrically in parallel by wires bonded to the elements by silver conductive epoxy (MG

Chemicals 8331).

Figure 5.5: Torsion harvester prototype

The performance of the prototype was tested by mounting it on an electrodynamic shaker

(Modal Shop 2110E) and exciting it with a 0.25 gpk constant acceleration amplitude swept-sine

signal from 10 Hz to 1000 Hz. The mechanical response was first characterized to identify the

torsional modes of vibration and this was done using a laser vibrometer (Polytec Inc.) to provide

non-contact velocity measurements at three locations on the harvester shown in Figure 5.6. The

results of the mechanical characterization are shown in Figure 5.7.

Figure 5.6: Locations of interest in characterizing mechanical response of each torsion harvester prototype

70

Figure 5.7: Mechanical frequency response of torsion harvester

Figure 5.7 indicates that this torsion harvester undergoes its first resonant torsional mode at a

frequency of 237 Hz. As with previous prototypes, this mode is characterized by high vertical

velocities of the eccentric mass tip with corresponding lower velocities at the torsion tip tub and

base. This observed response matches the predicted performance from the ANSYS modal

analysis indicating that the assembly of the harvester was performed without flaws.

After observing the mechanical response, the electrical response of the torsion harvester was

characterized. The electrical source impedance of the harvester was measured using an Agilent

4288A capacitance meter at 1 kHz. This meter models the PZT source impedance as an

equivalent capacitance (Cp) in parallel with a leakage resistance (Rp) and using this, the total

source impedance of the eight PZT elements was measured as 196.97 ± 0.21 pF in parallel with a

resistance of 5.78 ± 0.05 MΩ. At the resonant frequency of 237 Hz of the harvester, this yielded

a combined impedance of the harvester as 2.15 MΩ. This value is lower than the ANSYS

predicted value of 9.40 MΩ at resonance but, as in previous prototypes, can be attributed to a

variety of factors such as imperfect and non-ideal bonding between the PZT plates and the tube

structure and non-infinite leakage resistance between the PZT electrodes. The non-uniform

bonding creates imperfect clamped boundary conditions on the PZT plates resulting in a higher

measured capacitance (and therefore, lower source impedance) than the ANSYS model. The

ANSYS model also does not account for non-ideal leakage resistance between the two electrodes

of the PZT element, further decreasing source impedance. As a result of this lower source

impedance, it was expected that the harvester prototype would produce lower open circuit

voltages compared to the simulation predictions. However, as was observed later, the

71

discrepancy in source impedance did not affect the predicted and measured power output from

the torsion harvester.

The open circuit voltage and short circuit current output of the harvester were then tested to

further characterize the electrical performance of the harvester. This was undertaken by exciting

the harvester with a swept sine signal from 10 Hz to 1000 Hz with a constant base excitation

amplitude of 0.25 gpk acceleration. The open circuit voltages were measured using a National

Instruments compactRio data acquisition system and the short circuit currents were measured

using a Stanford Research Systems SR570 current pre-amplifier. The frequency response results

from these measurements are shown below in Figure 5.8 and Figure 5.9.

Figure 5.8: Open circuit voltage frequency response of torsion harvester under 0.25 gpk base excitation

Figure 5.9: Short circuit current frequency response of torsion harvester under 0.25 gpk base excitation

From these tests, the harvester prototype was found to have a maximum open circuit voltage

of 37 VRMS at an open circuit resonant frequency of 237.1 Hz and a maximum short circuit

current of 23 μARMS at a short-circuit resonant frequency of 235.9 Hz. As expected due to the

72

prototype’s lower source impedance, the measured open circuit voltage was lower than the

simulation predictions for reasons discussed previously.

The power output of this harvester was measured next by attaching a variable load resistor to

the harvester and measuring the voltage across this resistor at various resistance values. For this

test, the harvester was once again excited with a base acceleration of 0.25 gpk at its resonant

frequency and the results are summarized in Figure 5.10.

Figure 5.10: Output power of harvester prototypes over varying load resistance values at resonance

As expected, it was found that peak power transfer from the harvester to the load resistor

occurred when the load resistance was close to the source impedance of the harvester. In this

case, at a load resistance of 2.50 MΩ, the harvester prototype was found to dissipate a maximum

output power of approximately 210 μW as predicted from the ANSYS simulations.

To further gauge this harvester’s capability, the load resistance of 2.50 MΩ was also used to

excite the harvester at higher excitation amplitudes. The power outputs measured using this

approach are shown in Figure 5.11. Through this, it was observed that under 0.60 gpk base

excitation at its resonant frequency, this torsion harvester prototype was able to generate 1.16

mW of output electrical power.

0

50

100

150

200

250

0 1 2 3 4 5 6

Aver

age

Outp

ut

Po

wer

[μ

W]

Load Resistance [MΩ]

73

Figure 5.11: Output power from improved torsion harvester prototype under various excitation accelerations at

resonance

Finally, as in chapter three, the harvester effectiveness of this torsion harvester was calculated

using the procedure outlined in Section 3.4 and Table 3.C. Using the mechanical response plot

from Figure 5.7, the modal damping of this harvester was calculated to be 3%, matching the

assumed damping from the simulations. The remaining calculations for the SDOF parameters

required for the harvester effectiveness are summarized in Table 5.C using the approach outlined

in Chapter 3.

Table 5.C: Stiffness and mass estimation for SDOF model

Solid

Tors

ion T

ube

Force [mN] 114

Torque Arm [mm] 18.6

Torque [mN-m] 2.12

Torsion Tip Deflection [µm] 0.2

Eccentric Mass Tip Deflection [µm] 1.5

Measurement point separation [mm] 35.2

Angle of Twist [µrad] 36.9

Torsional Stiffness, k [N-m/rad] 57.5

Ecc

entr

ic

Mas

s

Mass [g] 46.6

Effective Length [mm] 37.2

Width [mm] 10.0

Eccentricity [mm] 18.6

Moment of inertia, m [mg-m2] 29.1

Using these parameters in (3-6) along with the acceleration input and measured output power

in Figure 5.10, the harvester effectiveness (κ) for this prototype was calculated to be 3.5 %. This

value approaches the 5% value reported for the best cantilever harvester [43] (including those

0

500

1000

1500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Aver

age

Outp

ut

Po

wer

[µ

W]

Excitation Acceleration [gpk]

74

utilizing single crystal materials), and is much higher than those observed with previous torsion

harvester prototypes.

5.3 Interface Circuits

With the performance of this harvester thus characterized, electronic circuits necessary to

integrate this harvester with a wireless sensor module were examined. The power generated from

the torsion harvester is sinusoidal in nature and interfacing this with electronics requiring DC

power such as wireless sensor modules requires intermediate conditioning. Hence, two different

interface circuits were fabricated and their performance was evaluated for potentially powering a

wireless sensor module. The first of these was a standard interface diode bridge rectifier that was

chosen for its simplicity in converting the AC voltages developed by the harvester into a DC

voltage. A schematic of this circuit is shown in Figure 5.12.

Figure 5.12: Circuit layout of prototyped diode bridge rectifier

The second circuit examined in this investigation was a non-linear voltage circuit described as

a synchronous electrical charge extraction (SECE) circuit. A detailed overview of this circuit is

outlined in [46] so its operation is briefly summarized here. The SECE circuit builds on the diode

bridge rectifier with the addition of a transistor switch and a transformer that acts as a buffer

between the output and the energy harvester. Unlike the diode rectifier, charge does not flow

constantly from the harvester to the circuit output, but instead is extracted at periodic intervals

defined by a trigger wave fed into the transistor switch. When the switch is triggered, charge is

allowed to flow through the transformer and then onto the output terminal of the circuit. For

optimal charge extraction, the trigger wave is synchronized with peak torsional displacements in

75

the harvester. As a result of the trigger source, this circuit requires a power input for operation. In

this study, the trigger wave was generated by a separate function generator and the power input

into the circuit was not characterized. The SECE circuit itself was chosen for this application due

to its simplistic implementation and potential to significantly improve power outputs compared

to the diode rectifier circuit. A schematic of the prototyped SECE circuit is shown in Figure 5.13

and the resulting voltage waveforms in the circuit are shown in Figure 5.14.

Figure 5.13: Circuit layout of prototyped synchronous electrical charge extraction (SECE) circuit

Figure 5.14: Voltage waveforms in the SECE circuit

After the circuits were constructed, two characterizations of power outputs were done for each

at their respective output terminals. First, a variable load resistance was attached at the output to

measure the dissipative power output performance of each circuit. Following this, the capability

of each circuit to deliver power to a DC load such as a battery was characterized by attaching a

variable DC power supply at the output terminals and measuring the current flow from the

-50

-30

-10

10

30

50

0 0.002 0.004 0.006 0.008 0.01

Volt

age

[V]

Time [s]

Harvester Output [V] SECE Output [V] Switch Trigger [V]

76

interface circuit into the DC load. Both of these characterizations were carried out at a base

acceleration of 0.25 gpk at resonance and the results are shown below in Figure 5.15 and Figure

5.16.

Figure 5.15: Output power of the two circuits considered with a variable resistive load

Figure 5.16: Output power of the two circuits considered with a variable DC load

In all scenarios considered, the power outputs from the interface circuits were lower than those

observed in the previous test with a simple resistive load as a result of electrical losses in the

interfacing components. In the case of the resistive load, the diode rectifier exhibited the same

load matching behavior as was previously seen with a pure resistive load with maximum power

output occurring at approximately 2 MΩ. At this resistance, the rectifier circuit produced a

0.0

20.0

40.0

60.0

80.0

100.0

120.0

0.00 1.00 2.00 3.00 4.00

Ou

tpu

t P

ow

er [

μW

]

Load Resistance [MΩ]

Diode Rectifier SECE

0.0

20.0

40.0

60.0

80.0

100.0

0.00 2.00 4.00 6.00 8.00 10.00 12.00

Outp

ut

Pow

er [

μW

]

DC Load [V]

Diode Rectifier SECE

77

maximum output of 165 μW (compared to 210 μW with just a load resistor). This indicates that

the presence of the diodes in the rectifier circuit results in power losses when conditioning the

voltage waveforms of the torsion harvester. Re-building the diode rectifier circuit with different

diodes resulted in similar losses suggesting that these losses are inherent to the circuit.

With the SECE circuit with a resistive load, it was observed that the power output was

significantly lower than the dissipated power measured in the previous section. At 0.25 gpk,

excitation, this circuit was found to produce a maximum power output of approximately 36 μW.

However, unlike the diode rectifier circuit, the power output from the SECE circuit was

relatively uniform across varying load resistances, indicating a decoupling between the output

terminal of the circuit from the input impedance of the energy harvester. Thus, with a resistive

load attached to interface circuits, the diode rectifier circuit was found to provide much higher

power outputs, but its optimal performance was influenced by the input impedance of the

harvester. The SECE circuit served to remove the impedance influence, but based on the tested

circuit, this was achieved with significant reductions in power output.

In the presence of a DC load, the diode rectifier once again demonstrated a large influence

between the load characteristics and the output power. In this case, in the voltage range

considered, the power delivered through the diode rectifier circuit increased as the voltage of the

DC load increased. With the SECE circuit, however, the output power was once again relatively

constant despite the range of voltages considered. Furthermore, with DC loads operating at

voltages below 3.5 V, the SECE circuit was found to provide higher power outputs than the

diode rectifier circuit. This result indicates that for battery recharging applications, particularly

with batteries operating at voltages below 3.5 V, the SECE circuit may provide higher power

outputs and would therefore be better choice amongst the two circuits.

After comparing the two circuits with resistive and DC loads, the ability of each to deliver

charge was examined by placing a fully discharged 470 μF capacitor across the output terminals

of each circuit. As the capacitor charged, the voltage across the capacitor was recorded as a

function of time for each circuit with a 0.25 gpk, base acceleration at the resonant frequency of

the energy harvester. The results are shown in Figure 5.17. In this, the switch to begin charging

the capacitor is closed at the time instant of 10 seconds.

78

Figure 5.17: Charging a 470 μF capacitor with both circuits under varying excitation conditions

In the recorded time window, the capacitor with the diode rectifier was observed to exhibit a

roughly linear increase in voltage and reached a voltage of approximately 2.3 V over 90 s of

charging time. On the other hand, the SECE circuit exhibited a much faster and non-linear

charging response in the capacitor and reached a voltage of almost 4 V over 90 s charging time.

Hence, since it demonstrated faster capacitor charging behavior and had higher power outputs at

microcontroller operating voltages of 3.5 V, the SECE circuit was chosen to interface the torsion

harvester with sensor electronics.

5.4 Sensor Electronics

After characterizing the behavior of the interface circuits, a sensor circuit was prototyped to

demonstrate the feasibility of the torsion harvester to power a wireless sensor module. This

sensor circuit consisted of a thermoresistive temperature sensor attached to a microcontroller and

a RFID memory chip. The sensor module was not fitted with any transmission capabilities and

the temperature history recorded by the sensor was instead accessed from the RFID memory chip

by a designated RFID reader. Together, these components simulated the operation of a wireless

sensor that would record the ambient temperature history between cycles of maintenance. The

overall schematic of this sensor module is shown in Figure 5.18.

0

1

2

3

4

5

0 20 40 60 80 100

Vo

ltag

e [V

]

Time [s]

Diode Rectifier SECE

79

Figure 5.18: Layout of wireless temperature sensor module with torsion harvester and interface circuitry

To determine the possible sensing duty cycles for this sensor module, the energy consumption

of the module was characterized. For this, the sensor module was powered at various voltages

and the current flowing into the sensor module was recorded on an oscilloscope. The results of

this are summarized in Table 5.D. At voltages below 1.7 V, it was observed that the

microcontroller entered into a sleep mode and exhibited negligible energy consumption. When

the input voltage reached 1.7 V, the microcontroller performed a series of wake-up tasks that

required an energy consumption of 34 µJ. For continuous operation, it was observed that the

microcontroller required a minimum input voltage of 2.0 V.

Table 5.D: Energy consumption of sensor and microcontroller circuit per active cycle of sensing

Operating Voltage [V] Energy Consumption [µJ]

1.7 34

2.0 21

2.5 28

3.0 36

3.5 44

Since, the energy consumption of the microcontroller circuit increased with operating voltage,

it was desired to operate the circuit at the lowest possible voltage of 2.0 V. At this voltage, the

total energy consumption including wake up and operation of the microcontroller was

approximately 55 µJ. From the previous results, it was observed that with a DC load of 2.0 V –

2.5 V, the SECE circuit was capable of producing 37 µW of power and therefore, the

microcontroller was capable of being powered every 1.5 s. To account for any other losses in the

system and for sleep mode power consumption, the microcontroller was then programmed to

80

wake up every 2 seconds and draw power from a charging capacitor connected to the SECE

circuit and the torsion harvester. Upon waking up, the microcontroller was programmed to read

the current temperature from the sensor and record this value into the RFID memory chip before

entering a commanded sleep-mode cycle. The wireless sensor module and electronics are shown

in Figure 5.19.

Figure 5.19: Wireless sensor module and interface circuitry

To test the operation of the sensor circuit, a heat gun was applied and removed from the

temperature sensor in cycles of approximately 30 seconds. The raw data from the temperature

sensor generated during this test was stored in the RFID memory using microcontroller’s analog

to digital converter (ADC). Using the calibration data for the sensor, this raw data was then

converted into temperature readings that are plotted in Figure 5.20 to show the heating and

cooling cycles.

81

Figure 5.20: Temperature measurements as recorded on the RF memory by temperature sensor and microcontroller

system

5.5 Summary

In conclusion, an improved torsion harvester prototype with a demonstrated ability to produce

1.16 mW of electrical power and a harvester effectiveness of 3.5 % was outlined in this chapter.

The ability of this torsion based shear mode harvester to power a wireless sensor module was

also successfully demonstrated in this chapter by powering a temperature sensor consisting of the

sensor, a microcontroller and RFID memory chip. The temperature sensor module was interfaced

to the torsion energy harvester by means of a SECE non-linear conditioning circuit that was

found to have superior performance than a diode bridge rectifier when recharging a capacitor and

delivering power to a DC load at low voltages. However, when dissipating power through a

resistive load or delivering power to high voltage DC loads, the diode rectifier bridge was found

to result in higher power outputs. With both circuits examined however, the circuits appeared to

experience significant losses and so further work is necessary to improve their efficiency.

Improved efficiency, particularly with the SECE circuit, may allow for integration of harvesters

with high power sensors such as three axis accelerometers.

0

10

20

30

40

50

60

70

80

0 100 200 300 400 500 600

Tem

per

atu

re [

C]

Time [s]

82

Evaluation of Single Crystal Materials 6.

With the ability of the torsion harvester to power a wireless sensor module and recharge a

capacitor thus demonstrated, novel single crystal materials are evaluated for their potential to

provide additional improvements in energy harvesting performance of the torsion harvester. This

evaluation is summarized here.

6.1 Overview of Single Crystal Materials

If the piezoceramic properties summarized Table 2.A are expanded to include single crystal

piezoceramic compositions as in Table 6.A, it is quickly evident that the shear mode d-g product

of the Pb(Mg1/3Nb2/3)O3-PbTiO3 (PMN-PT) single crystal is found to be much higher than any

PZT composition. This indicates that if these PMN-PT single crystals are utilized in the torsion

mode harvester, they have the potential to significantly improve the output performance of these

harvesters.

Table 6.A: Piezoelectric properties of various PZT compositions and PMN-PT single crystal

Property

Material

PZT-8 PZT-4 PZT-5A PZT-5H PMN-PT

Single Crystal

d33 [pC/N] 225 289 374 593 2000 a

g33 [mV-m/N] 25.4 26.1 24.8 19.7 34.8

d33-g33 5,715 7,543 9,275 11,682 69,504

d31 [pC/N] -97 -123 -171 -274 -1750 b

g31 [mV-m/N] -10.9 -11.1 -11.4 -9.11 -43.0

d31-g31 1,057 1,365 1,949 2,496 75,193

d15 [pC/N] 330 496 584 741 5190 c

g15 [mV-m/N] 28.9 39.4 38.2 26.8 90.2

d15-g15 9,537 19,542 22,309 19,859 468,039

d36 [pC/N]

N/A

2600 b

g36 [mV-m/N] 63.8

d36-g36 165,978

a <001> poled single crystal

b <011> poled single crystal

c <111> poled single crystal

83

PMN-PT single crystals are solid state solutions of lead magnesium niobate (PMN) and lead

titanate (PT). In these, the crystal structure of the solvent (PMN) remains unchanged with the

addition of solute (PT) and the solute is incorporated substitutionally or interstitially into the

solvent crystal structure. The addition of lead titanate also makes the previously electrostrictive

PMN into a ferroelectric material. The crystal structure of the resulting solid solution is

rhombohedral up to a PT content of 35% at which point, the crystal structure undergoes

morphotropic change into tetragonal crystal structure. The highest piezoelectric response occurs

in the rhombohedral crystals near the morphotropic phase boundary and so, most commercially

available PMN-PT single crystals have PT compositions ranging from 30-33% [47].

PMN-PT single crystals have a spontaneous polarization along the <111> crystal direction (i.e.

the long diagonal of a unit crystal). Under the application of an external poling field, the single

crystal sample may divide itself into various electrical micro and nano-domains each with a

specific <111> poling direction but with a net polarization vector in parallel with the applied

field. These electrical domains in a single crystal lattice structure produce an overall macroscopic

effect that is different than the original crystal with <111> polarization and so this process is

referred to as domain engineering and can be used to produce a desired performance from the

single crystal [48]. This concept is shown below in Figure 6.1.

Figure 6.1: Example of domain engineering with (a) unpoled single crystal with possible spontaneous polarization

directions and (b) <001> poled crystal with possible polarization directions [48]

Among the poling fields possible, applying poling along <011> crystal direction produces a

crystal with mm2 crystal class symmetry that is particularly applicable for shear mode energy

harvesting. Furthermore, it is observed by Han et al., that if a sample of <011> single crystal is

rotated by 45o degrees about the poling axis, the resulting single crystal has a unique d36 shear

mode that is present only in single crystal ceramics [49]. Unlike the d15 mode, this mode is re-

polable and hence requires the same electrodes for the poling process as harvesting applications,

84

significantly reducing manufacturing time and cost. Using the process outlined in their work, a

plate of such of piezoelectric sample is expected to have its thickness oriented along the <011>

crystal direction and length and widths along <111> and <111> respectively. A sample of the

piezoceramic operating in this mode is shown in Figure 6.2.

Figure 6.2: Crystal orientations for PMN-PT with d36 response [50]. P is the poling direction and T is the applied

stress.

6.2 Manufacturing Methods

There are two common methods of manufacturing single crystals of PMN-PT: the high

temperature flux method and the modified Bridgman technique [48]. Among these, the modified

Bridgman technique is more commonly used by piezoceramic manufacturers due to its ability to

produce larger single crystal boules.

The high temperature flux method involves first measuring and dry mixing the appropriate

amounts of high purity Pb3O4, ZnO, MgCO3, Nb2O5 and TiO2. The powders are loaded into a

Platinum crucible which is then placed in an Alumina crucible with a lid and then sealed to

minimize the release of volatile PbO. The crucibles are placed in a tube furnace at a temperature

of 1100oC – 1200

oC and slowly cooled at a rate of 1-5

oC/h to room temperature. Hot nitric acid

is then used to separate the grown crystal from the remaining melt products.

The modified Bridgman method uses a similar melt as the hot flux method along with a seed

single crystal in the desired crystallographic orientation. Unlike the hot flux method, the crucible

containing the melt is instead placed in a special tubular furnace and translated across a

temperature gradient in the furnace. As the melt passes through the temperature gradient, the

85

melt solidifies at the seed-melt boundary resulting in crystal growth. This method requires

precise temperature control and position control of the crucible within the furnace and is

typically achieved in a custom built furnace. However, the results from this method are highly

repeatable and single crystal manufacturers use this method for commercially available single

crystal products [48].

Figure 6.3: Schematic of modified Bridgman technique for crystal growth [48]

6.3 Characterization Considerations

After manufacturing the single crystals using one of the two methods listed above,

characterizing the resulting single crystal requires two additional steps. First, the composition of

the crystal is checked using X-ray diffraction methods to identify the compounds and proportions

of materials present in the crystal. Second, the crystal is oriented along the desired

crystallographic orientations using a live Laue back-reflection camera. With the appropriate

crystal directions determined, the necessary cuts and poling fields can be applied to the crystal to

obtain the desired performance. Poling is achieved using standard industry practices with an

electrical field of approximately 5 kV/cm [48].

6.4 Single Crystals for Torsion Harvester Concept

To investigate how the potential benefits of single crystal piezoceramics would translate into

increased energy harvesting outputs, the improved torsion harvester developed in chapter five for

wireless sensor module was modeled in ANSYS and the PZT-5A piezoceramic elements were

86

replaced with PMN-PT singles of equivalent size. Since two distinct shear modes, d15 and d36,

can be observed in the PMN-PT single crystals, each of these was analyzed in turn for the torsion

harvester.

6.4.1 d15 Shear Mode

The d15 shear mode of the PMN-PT single crystal is analogous to the d15 mode of the PZT

piezoceramics used for previous simulations and prototypes. As a result, the same process as the

previous simulations was used to observe the effects of the PMN-PT single crystal in the torsion

harvester. The necessary material properties for the single crystal were obtained from [51] for an

Orthorhombic – 1O PMN-PT crystal with large shear mode constants. The results of this

simulation are summarized below and compared with PZT-5A results from chapter five in Table

6.D. The voltage distribution in the d15 single crystal harvester with an optimal resistive load is

shown in Figure 6.4 (compare with Figure 5.4).

Table 6.B: Summary of simulation results with d15 PMN-PT single crystal compared to PZT-5A

results from chapter five

d15 PMN-PT PZT-5A

Resonance Frequency [Hz] 249 244

Open Circuit Voltage [VRMS] 92.3 61.8

Maximum Average Power Output [µW] 603 220

Optimal Load Resistance [kΩ] 7670 9400

Equivalent SDOF Stiffness, k [N-m/rad] 57.6 53.7

Harvester Effectiveness [%] 10.7 3.77

Figure 6.4: Voltage distribution in torsion harvester with d15 PMN-PT single crystal at resonance with optimal load

resistance. Units are in Volts RMS. Voltage at the output electrodes is 67.9 VRMS.

87

From these results, the harvester with the d15 single crystal has similar mechanical

performance as the PZT-5A harvester and hence has a similar resonant frequency and torsional

stiffness. However, due to the superior piezoelectric performance of the single crystal, the d15

single crystal harvester is able to produce nearly three times the power output of the PZT-5A

harvester. As a result, the single crystal torsion harvester is also much more effective than the

PZT harvester with an effectiveness metric of 11%, nearly twice that observed from the best

cantilever beam harvester [43] reported in literature. Thus, the d15 mode of PMN-PT single

crystals was found to have significant benefits to the torsion harvester concept.

6.4.2 d36 Shear Mode

With the appropriate domain-engineering and crystal cut orientations, PMN-PT single crystals

have a unique d36 shear mode that has a high d-g product and is re-polable. To use this mode in

the torsion harvester concept, the orientation of the poling direction of the piezoceramic element

must be changed from the original concept described in Figure 2.2 to the one shown Figure 6.7

with the poling direction aligned along the thickness of the piezoceramic elements. As a result,

the energy harvesting electrodes for this harvester are oriented perpendicular to the thickness of

the piezoceramic elements as opposed to the sides in the case of the d15 mode.

Figure 6.5: Piezoceramic poling directions to utilize d36 shear mode in the torsion harvester

With this poling orientation, the simulations for the improved torsion harvester from chapter

five were repeated with the d36 mode PMN-PT single crystal. The required material properties

for this crystal orientation were obtained from [52]. The results of the simulation are summarized

below and compared with PZT-5A results from chapter five in Table 6.D. The voltage

distribution in the d15 single crystal harvester with an optimal resistive load is shown in Figure

6.6 (compare with Figure 5.4 and Figure 6.4).

88

Table 6.C: Summary of simulation results with d36 PMN-PT single crystal compared to PZT-5A

results from chapter five

d36 PMN-PT PZT-5A

Resonance Frequency [Hz] 243 244

Open Circuit Voltage [VRMS] 3.51 61.8

Maximum Average Power Output [µW] 345 220

Optimal Load Resistance [kΩ] 28.3 9400

Equivalent SDOF Stiffness, k [N-m/rad] 55.2 53.7

Harvester Effectiveness [%] 5.99 3.77

Figure 6.6: Voltage distribution in torsion harvester with d36 PMN-PT single crystal at resonance with optimal load

resistance. Units are in Volts RMS. Voltage at the output electrodes is 3.12 VRMS.

From these results, it was observed that the mechanical performance of this harvester was very

similar to the PZT harvester and the d15 single crystal harvester. However, due to the poling

orientation of the d36, the electrical performance of this harvester had some significant

differences from the other two. The across-thickness poling and harvesting in the d36 harvester

resulted in significantly lower output voltages because of the much smaller electrode separation

in the thin piezoceramic elements. This smaller separation also resulted in significantly lower

impedance of the piezoceramic elements in this configuration and can be observed in the much

lower optimal load resistance required for the d36 harvester. However, despite the low voltages,

the significantly lower source impedance resulted in this harvester configuration producing

higher power outputs and effectiveness than was observed with the PZT torsion harvester. The

harvester effectiveness with the d36 harvester was also higher than commonly reported values of

89

the cantilever harvester. Hence, the d36 PMN-PT mode also has the potential to significantly

improve the performance of the torsion harvester concept.

6.5 Characterization of Commercial Single Crystals Elements

After determining the potential for single crystals in the torsion harvester, samples of PMN-PT

single crystals were tested to determine if their performance met the values described in the

literature. Due to the re-polability of the d36 shear mode in PMN-PT single crystals, this mode

was chosen for experimental testing. Plates of PMN-33%PT were obtained from a commercial

piezoceramic manufacturer with crystal directions as shown in Figure 6.2. These plates were

fabricated to be 5 mm x 5 mm in length and width and 0.5 mm across their <011> thickness. The

PMN-PT single crystals were also polarized across their thickness with an electric field of 5

kV/cm.

The characterization of the piezoelectric properties of the single crystals was achieved using

the procedure outlined in [53] using an impedance sweep of the sample. In this approach, using

the resonant and anti-resonant points of the sample’s impedance frequency response (points

corresponding to lowest and highest impedance, respectively), the coupling coefficient k36 can be

determined. The coupling coefficient relates to how efficiently the material can convert energy

from the mechanical to the electrical domains and can be used to calculate all other piezoelectric

constants. Determining k36 from the impedance (Z) spectrum is achieved using:

𝑘362 =

1

1+0.6540𝑟 (6-1)

1

𝑟=

𝑓𝑎2−𝑓𝑟

2

𝑓𝑟2 (6-2)

In (6-2), fa is the anti-resonant frequency of the sample and fr is the resonant frequency of the

sample. Using this approach, four distinct shear plates of PMN-PT were measured under

unclamped boundary conditions (all surfaces free to deform) to obtain impedance plots as shown

in Figure 6.7. The measured k36 for each sample are then summarized in Table 6.D.

90

Figure 6.7: Example impedance spectrum plot from one PMN-PT sample

Table 6.D: Results from impedance analyses of PMN-PT samples

Element fr [kHz] fa [kHz] k36

1 2935 3070 0.35

2 2840 3020 0.41

3 2907 3065 0.38

4 2903 3065 0.39

The results of these analyses indicate that single crystal samples obtained do not meet the

reported piezoelectric properties. In literature, the d36 samples have a reported k36 of 0.80 – 0.85

[52] compared to the measured values of 0.35 – 0.41, indicating that the obtained samples are not

as efficient as reported in converting energy between the electrical and mechanical domains. The

measured PMN-PT k36 was found to be even lower than the reported k15 for PZT materials of

0.60 – 0.65 [2], thus indicating a severe flaw in the obtained samples. To ensure that the obtained

samples had not depolarized during transport, the PMN-PT plates were re-poled with an electric

field of 5 kV/cm and tested again with the above process, however, this did not yield any new

results. Thus, the obtained samples were not suitable for providing improvements from the

torsion harvesters and were not tested with any torsion harvester prototypes.

There are a few possible reasons for the inconsistencies between the measured and reported

piezoelectric values of the PMN-PT samples. It is possible that there were manufacturing

10

100

1000

1 2 3 4 5 6

Imped

ance

[Ω

]

Frequency [MHz]

PMN-PT Sample: Z vs f

91

difficulties in producing the samples or that the samples were not manufactured to the provided

specifications. The d36 PMN-PT samples are not offered as off-the-shelf samples by the supplier

and were custom manufactured to the specifications provided from the literature. Most of the

results provided in the literature are obtained by in-house manufactured samples by laboratory

groups and hence it is possible that the literature results were interpreted differently by the

manufacturer resulting in deviations from the reported values. If any of the compositions or

crystal directions were altered in the process, this could result in reduction in performance. This

theory can be tested using processes such as X-ray diffraction to check the chemical composition

of the samples and Laue back scatter camera to check crystallographic orientations. However,

due to limited availability of this equipment, this verification was not conducted during this

research project.

6.6 Summary

PMN-PT single crystals have enormous potential to improve upon the power output from the

torsion harvester concepts if they are utilized in place of traditional PZT piezoceramics. Using

finite element analysis in ANSYS, the improved torsion harvester presented in the previous

chapter was found to have significantly improved performance with an output power

improvement of nearly 300% and improvement in harvester effectiveness from 3.5% to 11%.

However, when samples of PMN-PT were obtained, experimental characterization of their

material properties indicated that the sample properties were significantly lower than expected.

This was likely caused due to manufacturing defects in making these single crystal samples but

could not be investigated in detail. Thus, without experimental validation with prototype results,

the potential for single crystal ceramics for torsion harvester cannot be confirmed with certainty.

92

Concluding Remarks 7.

This report summarizes the development of a novel harvester that attempts to improve upon

the power output limitations of current piezoelectric harvesting technology. Utilizing the shear

mode of piezoceramic materials and its improved electromechanical conversion properties

compared to other modes, this harvester operates by inducing torsion on a tube under the action

of an eccentric mass when exposed to ambient vibrations. The validity of this concept was

established with experimental characterization of the first prototype that matched the

performance predicted by finite element simulations. Furthermore, when compared with

traditional cantilever beam harvesters, the novel torsion harvester was found to have more

uniform strain distribution that resulted in lower losses among attached piezoelectric elements. In

addition, a mathematical comparison of the novel torsion harvester with the cantilever beam

harvester showed that in applications requiring short beams with large cross-sections, the torsion

harvester provides higher outputs than the equivalent cantilever beam.

After this, the behaviour of this novel harvester was explored through comparison of three

cross-section geometries of the torsion tube and varying geometries of the eccentric mass. The

effects of damping on the performance output were also examined. For this comparison, a new

metric, harvester effectiveness, involving input energy and output energy from the harvester was

also developed to allow for comparison of different geometries and different operating

frequencies of the harvesters under consideration. Through this, it was observed that cross-

sections with smaller torsion constants and high eccentric mass inertias had the potential for the

highest power output and highest harvester. Using finite element analyses, the best performing

configuration from the harvesters configured were found to have a potential output of 126 µW at

resonance and a harvester effectiveness metric of 1.1 %. However, mechanical damping in the

harvester was found difficult to predict and when prototypes of the best and worst harvester

configuration were tested experimentally, the mechanical damping in the harvester was found to

vary from the simulation assumptions. As a result, the predicted best configuration, the slotted-C

harvester prototype, was able to produce only approximately 10 µW at a resonance with a

harvester effectiveness of only 0.25 %.

93

As a result of this discrepancy between prototypes and simulation performance, the factors

affecting the performance of the harvester were analyzed in greater detail through the

development of a mathematical model representing the harvester. This examination and the

subsequent analysis of the influencing parameters resulted in a set of design guidelines that could

be used to direct the design of a torsion harvester for a desired application. These guidelines

were then used to develop an improved torsion harvester that was capable of producing up to 1.2

mW of power with a harvester effectiveness of 3.5 % to power a wireless sensor module.

The wireless sensor module selected for this application consisted of a temperature sensor

attached to a low-power microcontroller and a memory module. Together, these components

simulated the operation of a sensor that would track the temperature history of a location without

transmitting the information to a receiving module. Means of interfacing this sensor module with

the harvester were then examined by comparing two circuits found in literature: a diode-rectifier

circuit and a synchronous extraction of electric charge (SECE) circuit. Through this comparison,

the SECE was found to be better suited at recharging a capacitor and at powering DC loads with

voltages around 2.0 V. As a result, this circuit was used for interfacing between the energy

harvester and the sensor electronics. With this arrangement, the sensor electronics were

successfully powered by the energy harvester under base excitations of 0.25 gpk with a sensing

duty period of 2 seconds, thus demonstrating the feasibility of the torsion harvester to power a

sensor module.

Finally, the use of alternative materials such as single crystals of PMN-PT in the torsion

harvester was examined. Through finite element simulations and with material properties

reported in the literature, the improved torsion harvester used with the sensor module was found

to benefit from significant improvements in its performance with the single crystal materials. The

power output of this harvester was found to increase from 220 µW observed with the PZT

materials to 600 µW with the PMN-PT single crystals. The harvester also saw an improvement

in harvester effectiveness from 3.5 % to 11 %. However, when samples of PMN-PT were

obtained from the manufacturer to test the reported claims, they were unable to match

piezoelectric properties from the literature. Due to limited resources, the exact cause of this

discrepancy was not identified but potential causes to investigate include improper crystal cuts or

manufacturing flaws in the piezoelectric material. The investigation of PMN-PT single crystals is

left as future work to fully validate their potential in improving power outputs.

94

7.1 Contributions

This research has resulted in three major contributions to the field of energy harvesting.

7.1.1 Torsion Harvester

The torsion harvester developed in this work is a novel development that is able to induce

torsional stresses in a tube despite operating under non-rotational vibrations. No such harvester

has been reported in the literature. Furthermore, much like a cantilever beam harvester, this

harvester has the advantage of geometric variability in the design of the torsion tube and

eccentric mass to allow its resonant frequency, and hence, its operating point to be tuned to the

desired ambient frequency point. Unlike a cantilever beam harvester, however, this torsion

harvester has uniform strain energy distribution along its outer surface which results in minimal

losses in the piezoelectric elements when the entirety of the available surfaces are utilized.

7.1.2 Design Guidelines for Torsion Harvester

After demonstrating the validity of the torsion concept, a number of design guidelines were

developed to identify a process of designing a harvester for a target application. This led to the

identification of criteria such as the size of the harvester, identification of the highest base

acceleration in the ambient vibration spectrum, and mechanical impedance matching

phenomenon that are counter-intuitive at a cursory glance but have significant influence on the

overall output of the harvester. The development of these guidelines is a valuable tool in the

implementation of this harvester for various applications as was demonstrated in the

development of a prototype used to power a wireless sensor module.

7.1.3 Investigation of Single Crystals for Torsion Harvester

Finally, a significant contribution of this work is in the examination of single crystal materials

in the context of the torsion harvester. Much literature has been devoted in recent years into the

potential for single crystals to significantly improve the power output of conventional energy

harvester. When the torsion harvester was simulated with reported properties of these materials,

its performance likewise was found to have significantly improved. However, testing samples of

the single crystal from an established piezoceramic manufacturer indicates that despite the

research interest, single crystals are not established enough in the industry to ensure guaranteed

success.

95

7.2 Future Work

Despite the successes of this project, three key areas were also identified as the focus of future

work for the torsion harvester

7.2.1 Mechanical Damping

Mechanical damping was found to significantly and inversely affect the output performance of

each harvester prototype. However, mechanical damping was also difficult to predict before the

assembly of the prototype. Thus, further investigation into the causes of mechanical damping is

necessary to its effects can be accurately predicted before prototypes can be fabricated.

Investigation into the causes of mechanical damping can also lead to methods of reducing

damping in torsion harvesters resulting in these devices fully achieving their potential.

7.2.2 Interface Electronics

Despite their previously mentioned advantages, the interface electronics considered in this

work were both found to suffer from high losses that reduced the amount of useable power

generated by the harvester prototype. The SECE circuit in particular suffered from losses of

almost 80%, significantly reducing the advantages provided by the torsion harvester. Developing

interface circuitry with greater efficiency, including the use of higher efficiency components can

then allow the use of wireless sensor modules with greater power consumption and greater

functionality.

7.2.3 Single Crystals

The samples of PMN-PT single crystals tested in this work did not match the reported

properties of these materials in the literature. However, the as the simulations indicate, the

potential for single crystals for energy harvesting applications is immense and cannot be

dismissed easily. Hence, further research is necessary into the sample obtained to identify the

causes of their poor performance and identify ways in which PMN-PT samples can match

reported performance. Using the processes outlined in chapter six, methods of manufacturing

single crystals in-house can also be pursued to ensure quality. This can help fully realize the

potential of the torsion harvester presented in this work.

96

7.3 Conclusion

In conclusion, this research work has achieved the following achievements to meet the goals

set out at the start of the project:

Developed a novel energy harvesting structure using shear mode of piezoceramic

materials under the application of torsion with the goal of improving output power

Investigated and demonstrated the potential for novel single crystal materials such as

PMN-PT to significantly improve the torsion harvester’s outputs compared to bulk

piezoceramics

Demonstrated this novel energy harvester’s capability with a prototype capable of

producing 1.2 mW under base accelerations 0.60 gpk at a frequency of 237 Hz

Demonstrated this harvester’s ability to power a wireless sensor module by integrating

harvester prototype with appropriate conditioning electronics to create a compact

standalone self-powered system

97

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103

Appendix A: Buffer Circuit for Voltage Measurements

from Piezoelectric Harvesters

The energy harvesters under consideration in this thesis have source impedances in the range

of 2 MΩ – 20 MΩ. As a result, voltages generated by these devices cannot be directly measured

by most digital multimeters or oscilloscopes which have an input impedance of 1 MΩ. If these

measurement tools are directly connected to the harvester, their low input impedance causes

loading effects, where current flows into the measurement tool thus lowering the measured

output voltage. Hence, it is necessary to add an impedance buffer circuit between the harvester

and measurement device to prevent this impedance mismatch. This circuit is described below.

This buffer impedance circuit shown in Figure A.1 is based around a TL-071 op-amp with an

input impedance of 1000 MΩ that is significantly higher than the source impedance of the energy

harvesters. As a result, there is minimal current flow into this op-amp resulting in accurate

measurements of the harvester voltage. This op-amp is then configured in a voltage follower so

that it produces this harvester voltage at its own low-impedance output terminal that can be read

by an oscilloscope or a data acquisition device. Since the data acquisition device available for

this research work had a maximum voltage rating of +/- 10V, this circuit is additionally provided

with a capacitance based voltage divider to further lower the voltages produced by the harvester

to tolerable limits of the data acquisition device.

Figure A.1: Impedance buffer circuit layout

104

To ensure this circuit behaves in a predictable manner across all voltages and frequencies

generated by the energy harvester. This was achieved using a function generator to provide a

known sinusoidal voltage input at the input terminal of the buffer circuit and measuring the

output voltage to determine the gain characteristics of this circuit. These results are plotted below

in Figure A.2.

Figure A.2: Input/output voltage attenuation in impedance buffer circuit

The results from this test indicate that the output voltage from the buffer circuit is consistently

measured to be 11.36 times lower than the input voltage into the circuit at input amplitudes up to

100 V and frequencies up to 1000 Hz. Thus, this buffer circuit is suitable for use for all the

energy harvesters considered in this research.

0

5

10

15

20

10 100 1000

Vin

/Vo

ut

Frequency [Hz] 25 V Amplitude 50 V Amplitude 100 V Amplitude