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Piezoelectric Vibration Energy Harvesters for Low Frequency Applications

A thesis submitted to the University of Manchester for the degree of Master of Philosophy in

the Faulty of Science and Engineering

2020

Jie Wang

Department of Mechanical, Aerospace & Civil Engineering

School of Engineering

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Contents

List of Figures .................................................................................................................................. 6

List of Tables ................................................................................................................................... 11

Abstract.............................................................................................................................................. 12

Acknowledgement .......................................................................................................................... 15

Scientific Outputs ........................................................................................................................... 16

1. Introduction .............................................................................................................................. 17

1.1. Background ...................................................................................................................... 17

1.2. Aim ....................................................................................................................................... 20

1.3. Objectives and Thesis Outline...................................................................................... 20

2. Literature Review .................................................................................................................... 21

2.1. Basics of Piezoelectricity ............................................................................................... 21

2.2. Vibration Sources ............................................................................................................ 22

2.3. Piezoelectric Materials .................................................................................................... 23

2.4. Hybrid Harvesting ............................................................................................................ 25

2.5. Harvester Layout .............................................................................................................. 27

2.5.1. Harvester configuration .......................................................................................... 27

2.5.2. Harvester area distribution .................................................................................... 28

2.5.3. Tip mass ..................................................................................................................... 32

2.6. Degradation ....................................................................................................................... 34

2.7. Motivation from Previous Studies ............................................................................... 36

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2.7.1. The Potential for Using Solar Panels as Suitable Tip Masses for Low

Frequency PVDF-Based Energy Harvesters ..................................................................... 36

2.7.2 The Effect of Planform Geometry and Excitation Level on the Energy

Harvesting Performance for Low Frequency PVDF-Based Energy Harvesters ...... 37

2.7.3 The Effect of Prolonged Operation on the Power Generation Degradation for

Cantilevered Vibration Energy Harvesters ........................................................................ 38

3. Solar Panels as Tip Masses in Low Frequency Vibration Harvesters ....................... 40

3.1. Materials and Methods.................................................................................................... 40

3.1.1. Harvester design and realisation ......................................................................... 40

3.1.2. Harvester identification .......................................................................................... 44

3.1.3. Experimental apparatus ......................................................................................... 47

3.2. Considerations for Harvesters with Flexible Solar Panels ................................... 49

3.2.1. Model for an equivalent concentrated tip mass ............................................... 49

3.2.2 Concentrated tip mass model vs. experimental measurements ................. 53

3.3 Considerations for Harvesters with Rigid Solar Panels ........................................ 56

3.3.1 Model for a distributed tip mass .......................................................................... 56

3.3.2 Flexible vs. rigid solar panels ............................................................................... 61

3.4 Optimum Tip Mass Configuration ................................................................................ 62

3.5 Solar Power ....................................................................................................................... 63

4. Systematic Experimental Study on Planform Geometry and Excitation Effects .... 66

4.1 Materials and Methods.................................................................................................... 66

4.1.1 Harvesters design and realization ....................................................................... 66

4.1.2 Experimental set-up ................................................................................................ 69

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4.1.3 Harvesters dynamic properties ............................................................................ 71

4.1.4 Linear electro-mechanical model ........................................................................ 73

4.2 Results and Discussion.................................................................................................. 76

4.2.1 Dynamics and power generation measurements ............................................ 76

4.2.2 Planform effects ....................................................................................................... 78

4.2.3 Aspect ratio vs second moment of area ............................................................ 80

5. Long-term Power Degradation Testing of Piezoelectric Vibration Energy Harvesters

for Low Frequency Applications .................................................................................................. 84

5.1. Materials and Methods.................................................................................................... 84

5.1.1. Harvesting materials and configurations .......................................................... 84

5.1.2. Experimental setup .................................................................................................. 88

5.1.3. Testing conditions ................................................................................................... 89

5.1.4. Data processing........................................................................................................ 92

5.2. Results and Discussions ............................................................................................... 94

5.2.1. Power degradation ................................................................................................... 94

5.2.2. Power density ........................................................................................................... 99

5.2.3. Active tuning of natural frequency .................................................................... 101

6. Conclusions and Future Work ............................................................................................ 103

6.1. Solar Panels as Tip Masses (Chapter 3)................................................................... 103

6.2. Planform Geometry and Excitation Effects (Chapter 4) ....................................... 105

6.3. Long-term Power Degradation Effects (Chapter 5) ............................................... 107

6.4. Summary ........................................................................................................................... 108

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7. References ............................................................................................................................... 110

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List of Figures

Figure 1-1: An example of a vibration energy harvesting solution - A completely self-powered

train brake force measurement device that uses a wireless network to communicate the brake

force data to the locomotive (developed by Chris Ludlow), [piezo.com]. ............................ 17

Figure 2-1: (a) Schematic of the magnetoelectric laminate configuration with polarization units

[27]. (b) Schematic of thermoelectric and vibration energy harvester on the cantilever [28].

........................................................................................................................................... 26

Figure 2-2:Schematic configurations of unimorph (left) and bimorph(right). Figure taken from

[4]. ....................................................................................................................................... 28

Figure 2-3: Strain profiles for alternative planforms. The red circles indicate the point at which

load is applied. Image taken from [40] ................................................................................ 29

Figure 2-4: (a) The shape of the root segment candidate cantilever beam from [38]. The

dotted lines represent the extent of the steel beam. (b) Schematics of inverted taper in thick

and width cantilever beam [39]. .......................................................................................... 30

Figure 2-5: Schematic configuration of the PVDF-based piezoelectric energy harvesting

device from [18]. ............................................................................................................... 31

Figure 2-6: Schematic of the piezoelectric MEMS micro power generator from [44] ........... 32

Figure 2-7: Design of piezoelectric harvester with a polymer spring attached to piezoelectric

bimorph – Image taken from [46]......................................................................................... 33

Figure 3-1:The configuration of the harvesters considered in this study: (a) and (b) schematics

of the harvester construction; (c) schematic representation of the custom-made mounting

system to connect the harvester to the shaker; (d) four harvesters realised without

incorporating solar panels; (e) four harvesters with solar panels. ........................................ 42

Figure 3-2: Experimental set-up used in the current study. ................................................. 48

Figure 3-3: First modelling approach used in the current study where solar panels are

represented as an equivalent concentrated tip mass. Equivalent tip mass is highlighted in blue.

........................................................................................................................................... 50

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Figure 3-4: Measured and predicted tip displacement output FRFs for the eight harvesters. (a)

Harvesters without solar panels (empty circle markers). (b) Harvesters with solar panels (filled

circle markers). Markers represent the experimental measurements whereas solid lines

represent the equivalent concentrated tip mass model predictions...................................... 54

Figure 3-5: Measured and predicted tip displacement output FRFs for the eight harvesters. (a)

Harvesters without solar panels (empty circle markers). (b) Harvesters with solar panels (filled

circle markers). Markers represent the experimental measurements whereas .................... 55

Figure 3-6: Measured and predicted power output FRFs for the PVDF elements of the eight

harvesters. (a) Harvesters without solar panels (empty circle markers). (b) Harvesters with

solar panels (filled circle markers). Markers represent the experimental measurements

whereas solid lines represent concentrated tip mass model predictions. ............................. 56

Figure 3-7: Second modelling approach used in the current study where solar panels are

represented as a distributed tip mass. Tip mass is highlighted in blue. ............................... 57

Figure 3-8: Flexible vs. rigid tip model predictions for harvesters with solar panels. (a) Tip

displacement frequency response function. (b) PVDF power frequency response function. 61

Figure 3-9: Flexible vs. rigid tip model predictions for harvesters with solar panels. (a) Tip

displacement frequency response function. (b) PVDF power frequency response function. 62

Figure 3-10: Flexible vs. rigid tip model predictions for harvesters with solar panels. (a) Tip

displacement frequency response function. (b) PVDF power frequency response function. 62

Figure 3-11: Experimentally measured power from a single solar panel in milli-Watt versus

light lux. Markers represent the experimentally measured values whereas lines represent

curve fitting for the measured data. ..................................................................................... 64

Figure 4-1: The harvesters considered in this study: (a) design schematics (the example

shown includes four PVDF elements on each side); (b) one PVDF element; (c) schematic

representation of the custom-made mounting system to connect the harvester to the shaker

(the upper plate is removed to show internal details); (d) harvesters H1-H4 with length of 78

mm; (e) harvesters H5-H8 with length of 155 mm. .............................................................. 68

Figure 4-2: Experimental set-up used to test the harvesters. ............................................... 70

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Figure 4-3: Measured elastic restoring force (top) and damping ratio (bottom) of the harvesters

(the continuous fitting lines are included to help visualize the trend in the data). ................. 72

Figure 4-4: Tip displacement and RMS power measurements versus excitation frequency.

Markers represent measured responses whereas continuous lines are the linear model

predictions. For the long harvesters (length = 155 mm), the theoretical model predictions are

greyed out for better visualisation of the experimental results as theoretical results are just

included to assess the presence of non-linear effects. ........................................................ 77

Figure 4-5: Tip displacement and RMS power measurements versus excitation frequency.

Markers represent measured responses whereas continuous lines are the linear model

predictions. For the long harvesters (length = 155 mm), the theoretical model predictions are

greyed out for better visualisation of the experimental results as theoretical results are just

included to assess the presence of non-linear effects. ........................................................ 78

Figure 4-6: Peak power as a function of harvester width for different base excitation levels; (a)

set of short harvesters (H1-H4); (b) set of long harvesters (H5-H8). Dashed lines are added to

help visualize the trends but do not imply a linear variation between measurement points. . 79

Figure 4-7: Relative performance of the two sets of harvesters; (a) quotient of peak power of

long harvesters divided by that of short harvesters; (b) quotient of peak power density of short

harvesters divided by that of long harvesters. Dashed lines are added to help visualize the

trends. ................................................................................................................................. 80

Figure 4-8: Variation of the peak power (left) and the peak power density (right) with harvester

aspect ratio. Open circles represent the set of short harvesters whereas filled circles represent

the set of long harvesters. ................................................................................................... 81

Figure 4-9: Variation of the peak power (left) and the peak power density (right) with harvester

second moment of area. Open circles represent the set of short harvesters whereas filled

circles represent the set of long harvesters ......................................................................... 82

Figure 5-1: Schematic diagram of the configuration of the passive and active layers

construction for the three harvesting devices employed in the current study. (a) PVDF; (b)

MFC; and (c) QP. Thicknesses are not drawn to scale and are exaggerated for better

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visualization of the components. In MFC, polyimide film contains interdigitated patterned

electrodes but are not shown in the side view for simplicity. Also, layers of structural epoxy

sandwich the PZT-5A1 fibers as shown but structural epoxy additionally exists between fibers.

........................................................................................................................................... 86

Figure 5-2: Harvesting devices attached to a custom-made acrylic clamp. (a) Devices with no

tip mass; (b) and (c) example devices with tip mass set at different angles for better

visualisation. ....................................................................................................................... 88

Figure 5-3: Harvesting devices attached to a custom-made acrylic clamp. (a) Devices with no

tip mass; (b) and (c) example devices with tip mass set at different angles for better

visualisation. ....................................................................................................................... 88

Figure 5-4: Experimental Setup used in the current study. .................................................. 89

Figure 5-5: An example of power time-series smoothing (case of MFC-10): the red line is the

original time-series, whilst the blue line is the smoothed time-series. .................................. 93

Figure 5-6: Change in (a) natural frequency, and (b) optimum load resistance for the test cases

considered in this study. ...................................................................................................... 95

Figure 5-7: Power output (top) and normalized power output (bottom) recorded during testing

for the present harvesters. .................................................................................................. 95

Figure 5-8: Power output (top) and normalized power output (bottom) recorded during testing

for the present harvesters. .................................................................................................. 96

Figure 5-9: Comparison of initial and final power output from the harvesters. (a) Normal scale,

and (b) log-scale to better compare orders of magnitude. ................................................... 98

Figure 5-10: Comparison of the (a) power density, and (b) power density to cost ratio for tests

conducted. (c) and (d) are the same plots as (a) and (b) but on a log-scale for better

visualisation of the orders of magnitude. ........................................................................... 100

Figure 5-11: Results for active natural frequency tuning case of the QP-30 device showing

change in natural frequency, and power output with reference to corresponding initial

measured value. Red markers denote instants where tuning of the input signal was conducted

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to match the shift in natural frequency. Due to the variation in input signal frequency, x-axis is

shown in time rather than vibration cycle. .......................................................................... 102

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List of Tables

Table 1: Acceleration (m/s2) magnitude and frequency of fundamental vibration mode from

various sources [36]. .......................................................................................................... 22

Table 3-1: Geometric characteristics of the harvesters considered in this study. ................. 43

Table 3-2: Mechanical and electric properties of the harvesters considered in this study. ... 46

Table 3-3: Optimum load resistance values for solar panels. .............................................. 47

Table 4-1: Characteristics of the harvesters considered in this study. ................................. 69

Table 5-1: Geometric characteristics of the developed harvesting devices to achieve desired

natural frequency. ............................................................................................................... 90

Table 5-2: Testing conditions for the current experiment. .................................................... 91

Table 5-3: Temperature variation (in °C) during testing. ...................................................... 92

Table 5-4: Variations of the harvested power. ..................................................................... 96

Table 5-5: Comparison of power density and power density to cost ratio. ........................... 99

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Abstract

Piezoelectric Vibration Energy Harvesters for Low Frequency Applications

Jie Wang

The University of Manchester Master of Philosophy

Piezoelectric energy harvesters are a promising power generation solution for widespread adoption of wireless sensors in remote locations. A well-recognized class of these devices employs highly flexible PVDF (Poly-vinylidene fluoride) polymers known for low resonant frequencies and are thus adequate for harvesting mechanical energy within low frequency applications. This thesis contributes to the state-of-art in the design of PVDF-based vibration energy harvesters through the provision of three novel studies into the effects of tip mass, planform geometry, and prolonged operation on the dynamics and power generation performance for this class of harvesters.

The first study proposes the use of solar panels as active tip masses to understand the effect of their inclusion on the dynamics and power generation performance of cantilevered PVDF-based energy harvesters. Four different harvester planforms with and without solar panels are tested using off-the-shelf components. The experimental results show that the flexible solar panels considered are capable of reducing resonance frequency by up to 14% and increasing the PVDF power generated by up to 54%. Two analytical models are developed to further investigate this concept; employing both an equivalent concentrated tip mass model to represent the case of flexible solar panels and a distributed tip mass model to represent rigid panels. For the flexible solar panel model, it is found that the highest PVDF power output is produced when the length of solar panels is two thirds of the total length. On the other hand, results from the rigid solar panel model show that the optimum length of solar panels increases with the relative tip mass ratio, approaching an asymptotic value of half of the total length as the relative tip mass ratio increases significantly.

Meanwhile, there is currently very limited understanding on how to size this class of harvesting devices to operate efficiently in response to a given excitation level. As such, the second study provides a thorough experimental investigation into the effect of planform geometries and excitation levels on the dynamics and power generation performance of PVDF-based cantilevered vibration energy harvesters. Three main findings are obtained: First, there is an optimal width for the harvester where the output power is maximized. This optimal width value depends on the excitation amplitude in such a way that narrower harvesters are more suited for small excitations whereas wider harvesters perform better upon experiencing large excitations. Second, it is shown that the selection of length is critical in that it should be decided to ensure a linear device response to the operation excitation as if non-linear effects are triggered, they will drastically deteriorate the power density performance. Finally, the second moment of area for the harvester cantilever is particularly effective at capturing the geometric effect on the power density.

Finally, the third study in this thesis considers investigating how piezoelectric vibration energy harvesters typically employed within low-frequency applications degrade during long-term operation in realistic harvesting conditions. Here, not only PVDF-based harvesters were investigated but other piezoelectric material options were considered for cross-comparison. The harvesters tested are unimorph cantilevers based on three of the most commonly used piezoelectric options: polyvinylidene fluoride (PVDF), macro fiber composite (MFC), and Quick

Pack (QP). Testing was carried out under single-frequency excitation (10−40 Hz) of 1g

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amplitude for three million vibration cycles. The results show that larger cumulative variation in natural frequency and optimum load resistance yields a larger variation in power output, thereby linking the power performance degradation to the degradation of the mechanical and/or electrical properties of the harvesters. The study also indicates that increasing the tip mass does not necessarily improve the average power output, suggesting that a larger tip mass may exacerbate the degradation of the mechanical and/or electrical properties of the harvester. It was also shown that stiffer harvesters show the highest signs of power degradation. Nevertheless, QP harvesters managed to demonstrate the highest power density values. When cost is considered within the assessment, PVDF harvesters managed to demonstrate the highest power density to cost ratio.

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Copyright

i. The author of this thesis (including any appendices and/or schedules to this thesis)

owns certain copyright or related rights in it (the “Copyright”) and s/he has given The

University of Manchester certain rights to use such Copyright, including for

administrative purposes.

ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic copy,

may be made only in accordance with the Copyright, De- signs and Patents Act 1988

(as amended) and regulations issued under it or, where appropriate, in accordance

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property (the “Intellectual Property”) and any reproductions of copyright works in the

thesis, for example graphs and tables (“Reproductions”), which may be described in

this thesis, may not be owned by the author and may be owned by third parties. Such

Intellectual Property and Reproductions cannot and must not be made available for

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Property and/or Reproductions.

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Reproductions described in it may take place is available in the University IP Policy

(see http://documents.manchester.ac.uk/DocuInfo.aspx?DocID=487), in any relevant

Thesis restriction declarations deposited in the University Library, The University

Library’s regulations (see http://www.manchester.ac.uk/library/aboutus/regulations)

and in The University’s policy on presentation of Theses

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Acknowledgement

To begin with, I want to express my sincere gratitude to my supervisors Dr Mostafa Nabawy,

Dr Alistair Revell and Dr Andrea Cioncolini for offering me guidance and valuable

instructions at all times. They are the most supportive people and have encouraged me from

day one till the final level. With all of you by my side, it made me fearless in the last two years,

and I was able to fight hard for this extremely difficult journey. You made me a better person

in the end. Especially, I must thank Dr Andrew Kennaugh who was also one of the strongest

supports during this period. His kindness and caring brought hope while I went through

hardship in the laboratory.

At same time, I was blessed by meeting so many kind people in the University, such as Marta

Camps, Andrew Mole, and other colleagues in George Begg who provided me plenty of help

with huge patience. I want to offer my regards and appreciation to all my friends that supported

and believed in me without question: Lina Mehyar, Fatma Abdullah, Nonye Azikiwe, Tian

Tian, Andrei Velcescu and Seongin Na, etc. Without you, it would be impossible for me to

manage the darkness and loneliness. I also owe so much to Robin Aziz. With your patience

and encouragement, I have been able to complete this impossible mission. I cannot imagine

the time without you by my side.

Lastly, I must give all the great thanks to my parents and sister. You are the reason I am here

today. There are no words that can express how much I love you and how deeply I miss you.

Loving me unconditionally and supporting me no matter what. One day, I will grow and flourish

for you and make you proud.

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Scientific Outputs

The work presented in this thesis has contributed to the following research outputs:

Wang, J., Nabawy, M. R. A., Cioncolini, A. & Revell, A (2019). Solar Panels as Tip Masses in Low Frequency Vibration Harvesters. Energies, 12(20):3815. (doi: 10.3390/en12203815) Wang, J., Nabawy, M. R. A., Cioncolini, A., Weigert, S. & Revell, A. Low-Frequency PVDF-Based Vibration Energy Harvesters: A Systematic Experimental Study on Planform Geometry & Excitation Effects. Journal of Intelligent Materials Systems and Structures, In review. Jacob H., Wang, J., Nabawy, M., Cioncolini, A. & Revell, A. Long-Term Power Degradation Testing of Piezoelectric Vibration Energy Harvesters for Low-Frequency Applications. Engineering Research Express, In review.

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1. Introduction

This chapter provides the context of the carried research and defines the aim and objectives

of this dissertation.

1.1. Background

In the modern omni-connected world of Big-Data and the Internet of Things, there is an

intense need for the provision of power to small wireless electronic devices. Applications

include environmental sensing, equipment and process monitoring, and smart city

applications - most of which require robust long-duration operation in remote and

sometimes harsh environmental conditions. This demand has led to an increasing interest

in developing energy harvesting solutions that would act as reliable, affordable, and

environmentally friendly power sources for these devices and sensors [1][2]. Recent

advances in developing technologies, such as portable electronics, wireless sensors,

transmitter, and receivers, led to a dramatic decrease in the amount of power required to

operate these devices compared to conventional devices. In fact, for small electronic

Figure 1-1: An example of a vibration energy harvesting solution - A completely self-powered train brake force measurement device that uses a wireless network to communicate the brake force data to the locomotive (developed by Chris Ludlow), [piezo.com].

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devices, the power required for successful operation is usually in the range of micro-Watt

and milli-Watt [3]. Motivated by these developments, finding viable solutions for

harvesting energy from ambient wasted energy has had increasing attractiveness over

the past few decades.

Small-scale energy harvesters provide a viable replacement for conventional batteries which

typically suffer from issues such as restricted lifetime, power efficiency, and maintenance and

replacement. Vibration-based energy harvesters have received most attention in recent years,

mainly because ambient vibration (motion) sources are ubiquitous and versatile. Examples of

vibration sources involve direct human/animal motion from walking, finger tapping, breathing

and heartbeat, structural vibration from industry equipment, public architecture, bridges and

vehicles, etc. [4]. In general, there are three main ways to convert mechanical energy to

electricity: these are electromagnetic, electrostatic and piezoelectric mechanisms [1].

Compared to the other two mechanisms, piezoelectric transduction is the most effective in

terms of offering the highest power density values [5][6][7]. Another main advantage of

piezoelectric harvesters is their readiness for use in practical applications, due to the well-

established technology and their simple configurations [8]. Furthermore, piezoelectric energy

harvesting offers longer lifetime operations, and are a perfect solution for remote application,

such as in sensory nodes in unreachable locations, biomedical devices, and larger-scale

sensor networks [9]. The potential of future demand for piezoelectric devices can be seen by

the worldwide annual revenue of piezoelectric devices which is reported to have increased

from $22 billion in 2012 to $37 billion in 2017 [9].

There exist about 200 different piezoelectric materials that can be used for energy harvesting

[2]. Piezoelectric material can be classified into different types based on the material structure.

Among those, piezoelectric ceramics, such as PZT (lead zirconate titanate), are the most

investigated since they have favourable characteristics in generating higher power values [2].

However, ceramics also possess some limitations such as rigidity, brittleness, and toxicity.

This motivated several research groups over the recent years to consider piezoelectric

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polymers, such as polyvinylidene fluoride (PVDF) and its copolymers, as a productive

alternative to piezoelectric ceramics, due to their flexibility, tailor-ability, low cost, and

availability [10]. As such, energy harvesting based on piezoelectric polymers has gained

attention as a promising means that can overcome many of the practical issues associated

with harvesters based on piezoceramics.

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1.2. Aim

The aim of this thesis is:

To contribute to the state of the art in design of low frequency PVDF-based vibration energy

harvesting devices through the provision of novel experimental investigations into the effects

of tip mass, planform geometry, and prolonged operation on the dynamics and power

generation performance, and supporting these investigations with closed form theoretical

models.

1.3. Objectives and Thesis Outline

• Review current concepts for low frequency piezoelectric vibration energy harvesting

solutions corresponding to the topics of this thesis, particularly focusing on aspects such

as potential for hybrid harvesting, planform variation effect, and performance degradation

due to prolonged operations. (Chapter 2)

• Study the effect of inclusion of solar panels as active tip masses on the dynamics and

power generation performance of PVDF-based cantilever energy harvesters. (Chapter 3)

• Systematically explore the effect of planform geometry and excitation amplitude on the

dynamics and power generation of PVDF-based cantilever energy harvesters. (Chapter

4)

• Assess power generation degradation of vibration piezoelectric harvesters under long

duration operations including polymer and ceramic based piezoelectric devices. (Chapter

5)

• Summarise the key research findings, discuss their implications to the field of

vibration energy harvesting, and identify routes for future research. (Chapter 6)

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2. Literature Review

This chapter provides an overview of the current state of the art in relation to piezoelectric

energy harvesting solutions proposed for low frequency applications. Whilst there is a vast

amount of research conducted on piezoelectric energy harvesting solutions over the past few

decades, this review will only focus on the most relevant aspects to the current thesis.

2.1. Basics of Piezoelectricity

The principle of the piezoelectric effect is based on the structure of crystal lattice. When

the crystalline structure experiences external stress, the mechanical energy is converted

into electricity [5]. The behaviour of a piezoelectric material can be classified into two

types, direct and converse [11]. The direct effect is the mechanism where the mechanical

strain energy is converted into electrical energy, hence is suitable for devices such as

sensors or energy harvesters. On the other hand, the converse effect refers to the

opposite mechanism where that applied electrical potential is transformed into mechanical

strain energy, hence is suitable for actuators. The governing constitutive equations for

piezoelectricity are generally written in the form:

[Converse

Direct ] = [

SD

] = [sE dt

d εT] [TE

] ; (2.1)

Where T is the stress, S is the strain, E is the electric field, D is the charge density, sEis

the compliance under the constant electrical field E, εT is the dielectric permittivity under

the constant stress T, d and dt are the matrices for the direct and converse piezoelectric

effect, and dt represents the transpose of d [12].

Typically, piezoelectric materials are employed under two coupling modes, namely, 31

and 33, which have the corresponding piezoelectric strain coefficients, d31 and d33 ,

referring to the direction of applied mechanical force to the poling direction of the single

crystal [13] [14]. As such, the 31-mode is such that the external force is perpendicular to

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the poling direction, whereas for the 33-mode the force has the same direction as the

crystal align direction [14]. d33 is larger than d31 because external stress is applied

alongside the polar direction of material [3]. Consequently, the 33-mode has a higher

coupling coefficient, k , than the 31-mode, where k is defined in the harvesting context

as:

k = √Electric energy output

Mechanical energy input (2.2)

2.2. Vibration Sources

Among the many ambient energy sources, each solution of scavenging energy has its features.

One of the most utilised energy sources is solar power as it provides excellent power output,

but only under direct sunlight [15]. This is why the efficiency of a solar system is dramatically

affected by the environment including factors such as presence of clouds and rain. Vibration

energy, on the other hand, is an easily accessible alternative source of energy that is abundant

in our environment but is usually wasted [10].

Table 1: Acceleration (m/s2) magnitude and frequency of fundamental vibration mode from various sources [36].

Vibration source A (m/s2) 𝒇𝒑𝒆𝒂𝒌(Hz)

Car engine compartment 12 200

Base of 3-axis machine tool 10 70

Blender casing 6.4 121

Clothes dyer 3.5 121

Person nervously tapping their heel 3 1

Car instrumental panel 3 13

Door Frame just after door closes 3 125

Small microwave oven 2.5 121

HVAC vents in office building 0.2-1.5 60

Window next to a bust road 0.7 100

CD on notebook computer 0.6 75

Second story floor of busy office 0.2 1000

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Ahmed et al. classified vibration sources into two categories, continuous and intermitter

sources [11]. A continuous source refers to the host system vibrating with a constant or a

specific range of frequencies irrespective to the amplitude, e.g. engine machine vibration and

monitoring system. On the contrary, an intermitter source depends on the host structure

deformation instead of frequency, e.g. motion of human body, building, or bridge [11]. Most

research studies focus on simulating continuous vibration sources as these sources represent

a unique input against which performance can be judged in a controlled fashion. When

considering continuous vibration sources, the interest is typically directed towards the

resonance performance, since it results in the maximum power output for energy harvesting.

Acceleration magnitude and frequency of the continuous vibration source are the main

quantities required to characterise the source. Table 2-1 provides some examples of

acceleration magnitude and frequency for some vibration sources whilst operating at

resonance based on the data collected by Roundy et al. [15].

2.3. Piezoelectric Materials

Piezoelectric energy harvesters can be realized from a range of materials; however, most

efforts have considered ceramic (PZT [lead zirconate titanate]) or polymer (PVDF [Poly-

vinylidene fluoride]) materials with the former being the most studied configuration for base

excitations whereas the latter was the most studied for wind excitations [14]. Other studies

focused on comparing the performance of PZT and PVDF based harvesters when subjected

to different sources of excitations. Considering wind and rain drop excitations, Vatansever et

al. assessed the effect of material dimensions, wind speed, drop mass, and releasing height

of the drops on the amount of harvested power for polymer (using PVDF, part numbers: LDT1-

028K and LDT4-028K) and ceramic (single layer and bimorph PZTs) based devices [16]. For

excitations from rain drop impact, they showed that the PVDF LDT1-028K was by far

generating the highest voltages. For wind excitation, they measured a power density value for

the PVDF LDT4-028K of 157.9 μW/cm3 at 10 m/s wind speed whereas ceramic-based

harvesters showed less power density values of 9.67 μW/cm3 for the single layer configuration

24

and 2.28 μW/cm3 for the bimorph configuration when tested at the same wind speed. Using

an array of inline back-to-back cantilevered piezoelectric harvesters, Hobeck and Inman

compared the performance of these arrays when realized with PVDFs (LDT2-028K/L) versus

when realized with PZTs (QuickPack) [17]. They demonstrated the superiority of the PZT

harvester array showing its capability to achieve 1 mW per cantilever at a mean wind speed

of 11.5 m/s whereas the PVDF harvester array achieved 1.2 μW per cantilever at wind speed

of 7 m/s.

The existing literature clearly indicates that each of the PZT and PVDF options has its

strengths and weaknesses. PZT-based harvesters have relatively higher electromechanical

coupling coefficient, higher mechanical quality factors and larger stiffness [2]; however, they

also possess some serious disadvantages such as having a heavy content of lead which is a

serious environmental hazard. PZT-based harvesters are also fragile, only limited to small

deformations, and have high economic cost [18][19][20]. On the other hand, and despite of

the relatively low electromechanical coupling coefficients, PVDF-based harvesters are

environmentally friendly, allow large deformations, have greater resistance to mechanical

shocks, and are much lower in their economic cost [18][19].

Piezoelectric vibration energy harvesters incorporate thin layers of piezoelectric material and

typically have a thin and flat geometric shape to promptly react to the motion of the host

structure. To date, the cantilever geometry is by far the most frequently used structural design

in piezoelectric vibration energy harvesters [3]. The choice of the piezoelectric material,

ceramics such as PZT or polymers such as PVDF, essentially depends on the frequency of

the ambient vibration of interest. Piezoelectric ceramics are rigid and have high resonance

frequencies, making them suitable for high frequency applications (50-100 Hz or higher) [3].

Piezoelectric polymers, on the other hand, are highly flexible and have low resonance

frequencies, and are therefore suitable for applications with low vibration frequency (up to 50-

100 Hz) [3].

25

The present thesis will focus on PVDF-based vibration energy harvesters of cantilever design,

a configuration that has not been extensively investigated to date for low frequency

applications, and only a relatively low number of studies is available [18][21]. In particular,

Song et al. tested a bimorph harvester comprising two PVDF films bonded together with an

adhesive layer and a load mass at the free end of the beam, measuring a power output of

112.8 μW and a power density of 8.61 mW/cm3 for an excitation with frequency of 35 Hz and

amplitude of 0.5 g [22]. The harvester tested by Jiang et al. was fabricated by laminating one

PVDF layer with a polyester layer, and provided a maximum power output of 16 μW for an

excitation with frequency of 17 Hz and amplitude of 1.2 g [23]. With the aim of assessing the

effect of air damping on the performance of low frequency (about 100 Hz) cantilever PVDF

harvesters, Cao et al. carried out tests in air and in vacuum. They showed superior (almost

twice) output power generation in vacuum and recorded a peak power of about 101 μW at

4.31 g acceleration in vacuum [15]. Rammohan et al. tested an array of three bimorph

harvesters, each comprising a copper foil between two PVDF layers, measuring a power

output of 2.8 μW at 33 Hz with input acceleration of 0.8 g [24]. Tsukamoto et al. tested a

bimorph harvester comprising a flexible 3D meshed-core elastic layer sandwiched between

two PVDF layers, recording a maximum power of 24.6 μW under resonance conditions at 18.7

Hz and 0.2g acceleration amplitude [25]. Chandwani et al. investigated multi-band harvesters,

measuring an average power of 6 μW for the frequency band 21-35 Hz and an average power

of 7.7 μW for the frequency band 45-60 Hz [26].

2.4. Hybrid Harvesting

The concept of hybrid energy harvesting is to simultaneously harvest energy from several

ambient energy sources. This strategy, therefore, allows for more robustness and resilience

to deal with unpredictable environmental conditions. In such cases, multiple technologies can

be combined and play their respective advantage and compensate for each other’s shortage.

This approach of hybrid energy harvesting has already shown success in simultaneously

26

harvesting wind and wave energies, wind and structural vibration energies, and wind and solar

energies [2][27][28][29][30].

Other examples of hybrid energy harvesting include the simultaneous harvesting of vibration

and magnetic energies using a magneto-strictive layer attached to the piezoelectric cantilever

beam with tip mass as proposed by Dong et al. and shown in Figure 2-11(a) [15][31]. It is

reported that this configuration was able to output a peak-to-peak voltage of 8 V at a vibration

frequency of 20 Hz. Similarly, Tӧreyin et al. introduced a hybrid cantilever beam to harvest

energy from ambient vibration and heat. The schematic structure of this concept is shown in

Figure 2-1(b) and is capable of employing both electromagnetic and thermoelectric energy

conversion [32]. The reported generated power from vibration was 1.12 nW and 0.79 nW from

the thermoelectric.

Tadesse et al. also used a PZT cantilever beam combined with an electromagnetic

mechanism that uses a magnet at the tip together with a stationary coil. 0.25 W of power were

generated from the electromagnetic mechanism and 0.025 mW from the piezoelectric part

under 35 g acceleration and a frequency of 20 Hz [33]. Magoteaux et al. also implemented the

idea of hybrid energy harvesting into unmanned air vehicles by using solar cells together with

piezoelectric cantilever beams in an independent fashion [34]. They demonstrated that this

energy harvesting system is able to provide usable energy for powering a sensor or a small

capacity battery.

Figure 2-1: (a) Schematic of the magnetoelectric laminate configuration with polarization units [27]. (b) Schematic of thermoelectric and vibration energy harvester on the cantilever [28].

27

Gambier et al. considered a cantilevered structure capable of harvesting energies from base

acceleration, solar and thermal excitations [35]. They developed a multilayer cantilever with

piezoceramic, thin-film battery and metallic substructure layers. This cantilever was then

bonded and fully covered with flexible solar panels. Jiang et al. proposed using a magnet as

a tip mass for their PVDF based cantilever harvester [36]. A coil was located below this

magnetic tip allowing simultaneous electromagnetic energy harvesting.

Despite of the previous attempts for hybrid harvesting, the combination of PVDF-based

vibration harvesters with solar panels acting as tip masses and additionally harvesting energy

from light sources have never been attempted before and therefore will be studied in this thesis.

2.5. Harvester Layout

2.5.1. Harvester configuration

Piezoelectric materials can be employed within a particular harvester configuration in a

number of ways. The configuration can be modified through a number of approaches including

changing the piezoelectric material, changing the electrode pattern, altering the poling and

stress direction and arrangement of layers, etc. As such, there is room for conducting a range

of studies to attempt to understand the optimal design of a harvesting device configuration.

Thin and flat structures, like beams and plates, allow piezoelectric materials to respond to the

excitation from host structure efficiently and result in less space and weight occupation for the

whole system. As such, most studies on energy harvesting are established based on thin-

layer geometric shapes. In fact, the most popular studied configuration within energy

harvesting applications is the cantilever beam configuration [3].

Conventionally, a cantilever harvester structure involves an active piezoelectric layer bonded

to a metal substrate layer. This configuration is usually achieved through two categories:

unimorph and bimorph [3]. The difference is in that whether one or two active piezoelectric

layers are bonded to the metal substrate as shown in Figure 2-2. A bimorph configuration is

sometimes preferred since the two active layers attached on both sides of the substrate will

28

increase the power output without significant change in volume as the thickness dimension is

typically much lower compared to width and length [10].

There are more other modulations that can be introduced to energy harvesting device

configurations and these have been proposed by many research groups in recent years. For

example, there has been successful demonstrations for changing the mechanical boundary

conditions with varying support point using dynamic magnifier, multiple inertia masses, and

two degrees of freedom beam [37][38]. However, these configurations are out of the scope for

the current thesis, hence will not be considered.

2.5.2. Harvester area distribution

The other important factor for energy conversion is how the strain is distributed over the

piezoelectric material layer, as output energy counts on the volume subjected to the

mechanical stress from external load [39]. This distribution is mainly a function of the

harvester’s layout arrangement. In general, the aims from modifying the geometric layout of a

harvester are to: (1) improve the scavenged power; (2) enhance the robustness of the device;

(3) meet the resonate frequency spectrum from practical environment; (4) minimize the

damping losses from mechanical structure; and (5) achieve good industrial realizability [12].

Figure 2-2:Schematic configurations of unimorph (left) and bimorph(right). Figure taken from[4].

29

For a classical cantilever beam structure, the highest bending strain happens at the region

near the fixed end of cantilever, where the strain varies linearly along the length direction, see

Figure 2-3 [40][41]. As a result, the piezoelectric material at the free end has very little

contribution to energy generation. Therefore, many different designs of beam

shape/geometries have proposed to achieve near constant/uniform strain level over the entire

cantilever, so that most of the cantilever material contributes to power generation. Analytical

and experimental studies are both presented by many research groups, including for

trapezoidal and triangular shapes as shown in Figure 2-3 (b) and (c) [40].

Roundy et al. showed that a tapered cantilever beam can produce 30% more power output

compared to a rectangular beam [40]. Friswell et al. analytically considered optimal shape

design based on uniform, triangular, and root segment shapes, where the root segment has

half-length of the uniform model [42], Figure 2-4(a). They showed that the maximum power

output is achieved through the root segment structure. Later, a study by Chen et al. showed,

both analytically and experimentally, that a triangular cantilever beam can generated more

voltage than rectangular and trapezoidal beams under the same condition due to its superior

strain distribution [37].

Figure 2-3: Strain profiles for alternative planforms. The red circles indicate the point at which load is applied. Image taken from [40].

30

A comprehensive study on the effect of the beam geometry was proposed by Pradeesh et al.

[43]. They investigated various structures including rectangle, triangle, taper in width, taper in

thickness, taper in thickness and width, inverted taper in width, and inverted taper in thickness

and width. They analytically and numerically showed that the beam geometry with inverted

taper in thick and width, as shown in Figure 2-4(b), has the largest stress and strain. Moreover,

it also leads to less natural frequency compared with a rectangular beam, where the natural

frequency was reduced from 94.25 Hz to 44 Hz. In fact, in terms of efficiency, the shape of

beam has minimal influence, but it increases the tolerable excitation amplitude which leads to

significant power output.

Thickness influence was also studied. Paquin and Amant developed a semi-analytical

mechanical model, and by varying the beam slope angles, they showed that a variable

thickness configuration can increase power output by 3.6 times compared to a constant

thickness beam [44].

Figure 2-4: (a) The shape of the root segment candidate cantilever beam from [38]. The dotted lines represent the extent of the steel beam. (b) Schematics of inverted taper in thick and width cantilever beam [39].

31

A relatively comprehensive analysis was presented by Cho et al. who included material

parameters, areal and volumetric dimensions, and configuration of the passive and active

layers of cantilever beam into consideration [45]. They reported that a maximum power density

was produced when the cantilever beam is close to an aspect ratio of 1 with a constant area

and bending level showing a maximum power density of 13.47 mW/cm3 for a beam aspect

ratio of 0.86. Song et al. developed an optimisation model based on the uncoupled point-

mass-model to analyse PVDF-based energy harvesters’ performance [18]. They studied a

bimorph cantilever beam with no metal core shim but with a steel load mass attached at the

tip, Figure 2-5. They demonstrated that the maximum power output depends on the frequency

and stress in the PVDF layer, controlled by the yield strength (𝜎𝑦𝑖𝑒𝑙𝑑 ) of the material. In

particular, they showed that a harvester would reach an optimum performance with proper

design of thickness and length of the beam as 𝜎𝑦𝑖𝑒𝑙𝑑 is affected by ratio of thickness and length

squared.

A more advanced model was presented by Patel et al., combining a finite element model to

simulate the mechanical structure and a distributed parameter model for the

electromechanical aspects [46]. The model illustrated that there exists an optimal beam length,

Figure 2-5: Schematic configuration of the PVDF-based piezoelectric energy harvesting device from [18].

32

since the mechanical damping increases with the longer length. They also found the substrate

thickness and tip mass inertia are additional potential parameters to enhance performance.

In summary, from a planform design perspective, the achievable maximum power output can

be modulated and hence optimised by adjusting the geometry particularly through the length

and width of the beam. One of the aims for this study is, therefore, to investigate the connection

between the harvester performance (dynamics and power) and its planform geometry (length

and width) for PVDF-based harvesters, a topic that has not been considered in previous

research works.

2.5.3. Tip mass

Tip masses are typically included within a harvester to tune the resonance frequency to that

of the operational frequency of the host source of vibration. The general understanding is that

the more massive the tip mass is, the lower the resonance frequency and the better power

generation performance are. Jiang et al. modelled the effect of using tip mass at the free end

of beam showing that the output power was increased by either reducing the elastic layer

thickness or attaching heavier tip mass, with both ways reducing the system resonate

frequency [22]. Roundy et al. also pointed out the larger the tip mass is, a better performance

from the harvester will be obtained, as mass and power have a linear relationship [47].

On the other hand, Ammar et al. reported a piezoelectric generator with 1 μm aluminium nitride

thickness for a compact sensor node and used a seismic tip mass as shown in Figure 2-6 [48].

However, the usage of tip mass raised the problem of overstrain in the piezoelectric layer and

Figure 2-6: Schematic of the piezoelectric MEMS micro power generator from [44].

33

caused damage through brittle fractures. Consistently, Cho et al. pointed out that a dense tip

mass will result in too much stress on a beam as the tensile strength of piezoelectric ceramics

are typically in the range of 100 MPa approximately, and hence from a durability perspective,

it is undesirable to use large tip mass to maximise output power [45]. Furthermore, Li et al.

pointed that maximising the tip mass is not always a sufficient approach to adjust frequency

in many applications, due to the high elastic modulus of piezoceramic, but not in piezopolymer

[3]. As such, to reduce the stiffness of cantilever beam, it may be more favourable to extend

the length of cantilever and/or use an alternative material.

Cornwell et al. found out that the usage of an auxiliary beam can increase the power output

by 25 times under a significant displacement [49]. Later, Dhakar et al. designed a polymer

cantilever beam with PZT-5A bimorph attached to it and an extra tip mass as shown in Figure

2-7 [50]. An improvement of power was achieved demonstrating a 32% increase at 0.1 g, while

the natural frequency decreased from 125 Hz to 36 Hz.

In summary, employing a tip mass will modulate the resonance frequency and can enhance

the power generation performance, as long as it is done within strain constraints. However,

the issue of the effect of tip mass on the durability and lifetime performance of a harvester has

not been fully addressed in this field. In fact, only a few previous studies mentioned/considered

this potential problem. As such, one of the objectives of the current thesis is to address this

Figure 2-7: Design of piezoelectric harvester with a polymer spring attached to piezoelectric bimorph – Image taken from [46].

34

limitation and provide a better understanding into the effect of tip mass inclusion on the long

duration performance of piezoelectric energy harvesters.

2.6. Degradation

The technology of piezoelectric vibration energy harvesters has gradually matured through

the extensive research carried out in the last decade, and the successful self-powering of

sensors and small electronic devices using piezoelectric vibration energy harvester has been

repeatedly demonstrated [51]. Whilst current research is looking into design optimization and

performance enhancement, one key aspect that has seemingly been overlooked is the long-

term stability of Piezoelectric vibration energy harvesters [27][52]. Because of the laws of

physics, and notably of the second law of thermodynamics, wear and tear naturally and

inevitably occur in any product as a result of normal aging, thus limiting the product service

life. As a necessary step in product development before commercialization, endurance testing

is routinely performed to gauge the capacity of a product to last or withstand wear and tear

and provide evidence of how its performance varies with time, thereby assessing the durability

and performance stability of the product over prolonged operation in representative operating

conditions. Piezoelectric vibration energy harvesters are electromechanical systems that

generate electric power by inducing a cyclic stress in the piezoelectric material using the

ambient vibration. Any degradation in piezoelectric vibration energy harvester’s performance

can therefore be attributed to a degradation of the mechanical properties (stiffness and/or

damping), a degradation of the electrical properties (impedance), or a combination of both.

With piezoelectric vibration energy harvesters of cantilever beam geometry, in particular, the

induced stress is concentrated near the fixed root of the beam. This location is where

mechanical degradation would be expected to occur from stress-induced fatigue due to the

cyclical nature of the loading. As mentioned in the previous section, more often than not, the

resonant frequency of the cantilever beam structure is higher than the frequency of the

external exciting vibration, and a proof mass is usually added on the tip end of the beam so

that the resonant frequency can be reduced to match the input frequency of the host structure

35

[53]. Clearly, the inclusion of a proof mass can play a role in the durability and performance

stability of the device.

Degradation tests on piezoelectric materials have been carried out in a number of studies for

different PZT materials experiencing different loading conditions. For example, Tai and Kim

studied the degradation of piezoelectric properties under compressive cyclic loading as well

as mechanical fatigue behaviours for Pb(Zr,Ti)03 ceramics of tetragonal, morphotropic phase

boundary and rhombohedral compositions [53]. They showed that fatigue resistance was

highest for the rhombohedral composition, and that fatigue resistance could be lowered for all

three compositions using poling treatment. In another comprehensive study, Cain et al.

investigated candidates having the shapes of flat discs and long cylinders. The study dealt

with simple uniaxial mechanical cycling of monolithic materials parallel to the polar direction

under short circuit conditions [54]. Electrical stressing has also been investigated. It has been

shown that for repeated mechanical loading, soft materials are very sensitive with the softer

composition PC 5H having a greater rate of degradation compared to PZT-5A when stress is

increased. Hard materials showed less degradation due to mechanical cycling but were

sensitive to extended operation under constant load. Cyclic electrical stressing results showed

that soft materials are again more sensitive compared to hard materials; however, degradation

due to electrical stressing was of less significance compared to that due to moderate

mechanical cyclic loading. Elahi et al. also examined degradation of rectangular PZT-5A4E

patches over a range of conditions for temperature (20 oC to 180 oC), load resistance (0 Ohm

to 90 Ohm), and frequency (50 Hz to 250 Hz). They investigated up to 2500 stress cycles

demonstrating that after 1500 cycles, electromechanical properties showed signs of

degradation [55].

Some documented endurance tests carried out for vibration piezoelectric beams are also

present. Pillatsch et al. studied degradation effects of a bimorph cantilever bending beam with

tip mass whilst excited at the tip using magnetic actuation under symmetrical and

asymmetrical sinusoidal loading [56]. Besides loss of power, they demonstrated that extended

36

use causes significant changes in the beam’s material properties including both mechanical

(stiffness) and electrical (impedance) properties. The change in stiffness would in turn affect

the natural frequency of the device and a change in impedance would change the ideal load

resistance of a circuit for maximum power. This result is consistent with a study by Selten et

al. which found that, in the case of mechanical compressive tests, the Young’s modulus and

piezoelectric coefficients are strongly affected by loading history. Note that micro-cracks in the

piezoelectric ceramic layer have been found in a number of studies, e.g. [55][57][58]. and

formation of such micro-cracks was typically considered a potential leading cause of

degradation. However, possible fabrication solutions such as pre-stressing the material and

laser lift-off could decrease degradation effects and enhance the strain capability of the

piezoelectric material. In a recent study, Wang et al. considered testing of beams with bonded

MFC patches. They showed that stress depolarization can cause actuation function

degradation of the piezoelectric devices ultimately causing the failure of the structural system.

2.7. Motivation from Previous Studies

The previous sections provided an overview of the current state of the art and have identified

a number of gaps in our understanding for a number of aspects related to piezoelectric

vibration energy harvesting. This thesis aims to contribute to improving our current

understanding on three main topics.

2.7.1. The Potential for Using Solar Panels as Suitable Tip Masses for Low

Frequency PVDF-Based Energy Harvesters

An energy harvesting device is likely to be small physically in comparison to traditional power

generation technologies, and as such achieving relevant power output is important. An

interesting strategy, therefore, is to consider the hybrid harvesting of multiple ambient energies

within the same device at the same time. A number of current approaches to tackle this aspect

has been presented in Section 2.4. These studies (discussed in Section 2.4), provided

motivation for the first study of the current thesis: to develop PVDF energy harvesting

37

cantilevers for low frequency vibration applications that use functional tip masses rather than

the traditionally used passive tip masses. To do so, the decision was made to understand the

feasibility/benefit of using solar panels to act as these functional tip masses. The objective of

the first study of this thesis is, therefore, to investigate the extent to which the additional mass

of the panels can be exploited to favourably modulate the dynamics of the harvester and

therefore improve its piezoelectric power generation. The effect of varying the geometry

(length and width) of these harvesters was also, partially, investigated in this study, and this

led to the recognition of the importance to conduct a more thorough investigation of the

planform geometry effects.

2.7.2 The Effect of Planform Geometry and Excitation Level on the Energy

Harvesting Performance for Low Frequency PVDF-Based Energy

Harvesters

Regarding the effect of planform geometry on the energy harvesting performance of a

cantilever beam, there are key questions that need to be answered including: are there optimal

length and width values for this class of harvesters that would maximize the power generation?

Can the length and width effects be combined into a single geometrical parameter capable of

predicting the output performance, or do they influence the performance in an independent

fashion? The second study of this thesis will show that there is an optimal width per excitation

amplitude where the output power is maximized, and that the second area moment is

particularly effective at capturing the geometric effect on the power generation. Despite the

investigations reviewed in section 2.5, a systematic study on how the planform geometry and

the excitation input affect the dynamics and power generation of PVDF-based cantilever

energy harvesters operating at low frequency is still missing, hence creating the motivation for

the second study of the present thesis. Another point of interest is the fact that PVDF elements

are particularly flexible and can undergo large deflections without significant loss of structural

integrity. Since the power generated is proportional to the strain, there is in principle an

38

incentive to develop cantilever harvesters that can undergo larger deflections in order to

maximize the power output from the PVDF elements. As with most elastic structures,

cantilever bending remains linear for only small deflection to length ratios, whilst for large

deflection to length ratios non-linearity becomes inevitable. Increasing response amplitude to

increase power output is clearly possible when the harvester responds linearly, but this is no

longer necessarily guaranteed when non-linear effects are present. As will be shown hereafter,

the data collected in the second study confirm this, showing that non-linearity in the harvester

response is detrimental for power generation.

2.7.3 The Effect of Prolonged Operation on the Power Generation Degradation

for Cantilevered Vibration Energy Harvesters

Overall, experimental studies have shown that piezoelectric devices do show degradation

under different loading conditions. Most of the available investigations have considered cyclic

compressive loading, and comparatively fewer studies have considered cyclic bending loading.

There are also some studies comparing energy harvesting performance of a number of

piezoelectric material options. However, the performance degradation over prolonged

operations of different piezoelectric harvesters has not been compared. Piezoelectric

polymers are known for higher flexibility, environmental compatibility, and resistance to

mechanical shocks than piezoelectric ceramics. These qualities show promise in energy

harvesting applications at low frequencies; however, their performance degradation has never

been assessed. Finally, piezoelectric vibration energy harvesters typically employ a tip mass

to lower the device resonant frequency to the required harvesting frequency as well as to

increase the harvested power. However, there is currently no systematic study that considers

tip mass effect on the harvesting performance over prolonged operations. Because of the

aforementioned limitations, in the third study of this thesis the variation in performance during

prolonged operation of three of the most frequently used piezoelectric options (PVDF, MFC,

and QP) will be thoroughly investigated within the context of piezoelectric vibration energy

39

harvesters for low frequency applications. To the best of the author’s knowledge, no previous

study has compared the performance degradation of these alternatives when employed as

piezoelectric vibration energy harvester in bending mode, creating the motivation for this study.

As such, the third study of this thesis will analyse how the power output from these

piezoelectric energy harvesting devices changes over time whilst shedding more light on the

effect that the inclusion of a tip mass has on performance degradation.

40

3. Solar Panels as Tip Masses in Low Frequency Vibration

Harvesters1

The main objective of this chapter is to investigate the potential for using solar panel as

effective tip masses to favourably modulate the dynamics of the harvester and therefore

improve its piezoelectric power generation. Moreover, the performance of the employed solar

panel in terms of generating additional power from indoor light sources is experimentally

measured and assessed. Eight harvesters of varying dimensions were constructed using

commercial-off-the-shelf components and experimentally assessed in terms of their dynamics

and power generation characteristics. Following experimental testing, two different analytical

models are developed to represent the solar panels effect as tip masses; the first model

employs a concentrated tip mass representation whereas the second employs a distributed

tip mass representation. Comparison of models to experiments are made and against each

other. Models were then applied to explore and assess the different possible tip mass

configurations providing a fast evaluation of the optimal layouts for the harvesters considered.

3.1. Materials and Methods

3.1.1. Harvester design and realisation

A schematic diagram of the harvester design is provided in Figure 3-1, together with images

of the eight harvesters realized and tested here. The harvesters have been constructed using

a stainless-steel shim core of 0.1 mm thickness by Precision Brand Products, Inc. (Downers

Grove, IL, USA (/precisionbrand.com/); density, 𝜌𝑒 = 7900 kg/m3 ; Young’s modulus, 𝑌𝑒= 180

GPa) which has been covered on both sides with a layer of flexible PVDF piezoelectric

elements in a sandwich arrangement. The PVDF elements used are the LDT2-028K model

from TE Connectivity Ltd. (Schaffhausen, Switzerland (/www.te.com/); density, 𝜌𝑝 = 1780

1 This Chapter is based on the publication: Wang, J., Nabawy, M. R. A., Cioncolini, A. & Revell, A (2019). Solar panels as tip masses in low frequency vibration harvester. Energies, 12(20):3815. (doi: 10.3390/en12203815)

41

kg/m3; Young’s modulus, 𝑌𝑝= 2.3 GPa; piezo strain constant, 𝑑31= 23 × 10−12 C/N; capacitance,

𝐶 = 2.85 nF). Note that each PVDF element comprises a PVDF film covered with silver ink

screen printed electrodes, all contained within a thin plastic coating for protection. The PVDF

elements were attached to the metal shim with small pieces of thin tape to ensure that the

PVDF elements deflection followed closely that of the metal shim. Each harvester was visually

inspected to ensure that PVDF elements were fully attached to the metal shim and closely

followed its motion, without bulging or deformation during deflection. Only a few cm2 of tape

were used for each harvester, so that the added mass of the tape can be neglected, as well

as any other effect of the tape on the mechanical response of harvesters.

The dimensions of one PVDF element are 73 mm × 16 mm × 0.2 mm (total length × width ×

thickness); however, the overhang length of the harvester from the fixed support to the free

tip was varied through changing the clamp position. For example, harvesters H1 and H2 in

Figure 3-1 are nominally identical but differ in the overhang length due to the position of the

clamp. Similarly, for the harvesters H3–H4, H5–H6, and H7–H8. Varying the overhang length

of the harvester via different clamp positions is an effective way to change the active length of

the PVDF elements whilst using off-the-shelf components that come in predetermined size.

The portion of the PVDF element enclosed within the clamp, in fact, does not deform and

therefore does not produce any power. In particular, comparing two different overhang lengths

and two different widths, resulting in a total of four geometric configurations for the harvesters

(H1 through H4 in Figure 3-1(d)). The use of off-the-shelf PVDF elements gave more freedom

in modulating the harvester length as compared to its width, which was constrained to be a

multiple (4 or 7 in the present case) of the width of a single PVDF element.

For each of the baseline geometries employed in this work (H1 through H4 in Figure 3-1(d)),

a corresponding version with two mini flexible solar panels attached at the tip was also

considered, one on each side as shown in Figure 3-1(b) and H5 through H8 in Figure 3-1(e),

thus resulting in a total of eight harvesters realized and tested. The flexible solar panels used

were manufactured by PowerFilm, Inc. (Ames, IA, USA (https://www.powerfilmsolar.com/);

42

density ρs= 1250 kg/m3). Two models were used: the first is the SP3-37 capable of generating

22 mA @ 3 V, and has dimensions of 64 mm × 36.8 mm × 0.22 mm (width × length × thickness),

whereas the second is the MP3-37 that is capable of generating 50 mA @ 3 V, and has

dimensions of 112 mm × 36.8 mm × 0.22 mm. As evident in Figure 3-1, the size of the solar

panels was selected to fit into the present harvesters. The main geometrical details of the

present harvesters are summarized in Table 3-1.

The harvesters were connected to the test apparatus with a custom-made mounting system

that was manufactured from laser cut acrylic, as shown schematically in Figure 3-1(c). The

mounting system was designed to provide a cantilever boundary condition at the harvester

Figure 3-1:The configuration of the harvesters considered in this study: (a) and (b) schematics of the harvester construction; (c) schematic representation of the custom-made mounting system to connect the harvester to the shaker; (d) four harvesters realised without incorporating solar panels; (e) four harvesters with solar panels.

43

base by including C-shape elements to sit between the mounting upper/lower plates and the

harvester. This avoids any direct contact between the mounting system and the PVDF

terminals, while providing a very strong fixation.

Table 3-1: Geometric characteristics of the harvesters considered in this study.

Harvester No of PVDFs Solar Panel Length

(mm)

Width

(mm)

Beam aspect ratio

𝒍

(𝒉𝒆 + 𝟐𝒉𝒑)

𝒍 − 𝒍𝒔

(𝒉𝒆 + 𝟐𝒉𝒑)

H1 8 (4 /side) NA 48 64 96 NA

H2 8 (4 /side) NA 58 64 116 NA

H3 14 (7 /side) NA 48 112 96 NA

H4 14 (7 /side) NA 58 112 116 NA

H5 8 (4 /side) SP3-37 48 64 96 22.4

H6 8 (4 /side) SP3-37 58 64 116 42.4

H7 14 (7 /side) MP3-37 48 112 96 22.4

H8 14 (7 /side) MP3-37 58 112 116 42.4

It may be worth re-iterating that the main objective of the present work was to assess the

feasibility of using solar panels as functional tip masses on PVDF-based piezoelectric

harvesters, and hence was carried out relying entirely on available off-the-shelf components

for the metal shim, the PVDF elements and the solar panels used to realize the harvesters.

This posed limitations on the dimensions of the harvesters that could be realized and explored.

In particular, the width of available solar panels that could allow full coverage of multiple PVDF

elements was the main driver in selecting these two models of solar panel, and hence the two

values of the harvester width investigated in this study. The possibility of having the metal

shim, the PVDF elements and the solar panels custom-made to specific desired dimensions

and material properties will be exploited in a future optimization stage to better explore the

parameter space of the harvesters and to tailor their mechanical properties to the intended

applications, but is not further considered at this stage.

44

3.1.2. Harvester identification

For the purposes of this study, it is essential to identify the mechanical and electrical

characteristics of the harvesters considered, which include: 1) the equivalent tip mass of the

solar panels; 2) the natural frequency of the harvester; 3) the damping ratio of the harvester;

4) the stiffness of the harvester; and 5) the optimum load resistances for the PVDF and solar

elements that maximize the generated electrical power.

As evident in Figure 3-1, the solar panels span over the harvester tip region, and are not actual

concentrated tip masses. This section proposes to transform the distributed mass of the solar

panels into an equivalent concentrated mass located at the tip of the harvester. This was

achieved by first assuming that the solar panels are homogeneous with their center of gravity

located right at the center of the panel. In the case of small angular displacements of the

harvester (which is consistent with the small base excitation acceleration level used in this

study), Rao has shown that an equivalent concentrated tip mass can be estimated by equating

the kinetic energy of the actual mass (in the case the solar panels) to that of an equivalent

mass located at the tip of the harvester [59]. Applying this approach to the configuration led to

the following expression for the equivalent concentrated tip mass, 𝑚𝑡𝑖𝑝,𝑒𝑞:

𝑚𝑡𝑖𝑝,𝑒𝑞 = 𝜌𝐴𝑠𝑙𝑠 (𝑙−(

𝑙𝑠2⁄ )

𝑙)

2

= 2𝜌𝑠𝑏ℎ𝑠𝑙𝑠 (𝑙−(

𝑙𝑠2⁄ )

𝑙)

2

(3.1)

where 𝜌𝐴𝑠 is the mass per unit length of the solar panels, 𝜌𝑠 is the density of the solar panel,

𝑙𝑠 is the length of the solar panel, 𝑙 is the overhang length of the harvester, 𝑏 is the width of

the harvester/solar panel, and ℎ𝑠 is the thickness of the solar panel. The factor two in the

above equation is to account for having two solar panels, one at each side of the harvester.

Note that the squared bracket in Equation 3.1 could be seen as a correction factor through

which the solar panels mass has been converted to an “equivalent concentrated” mass at the

tip. Whilst this approach is simple, particularly when compared with the more complex

approaches for dealing with distributed masses in the literature (e.g. [15][60][61]), it was found

45

to be sufficiently effective in capturing the behavior of the thin, flexible solar panels employed

in this study as will be shown later in Section 3.2. Following Lord Rayleigh, the equivalent

mass of the whole harvester, 𝑀𝑒𝑞, is thus evaluated from:

𝑀𝑒𝑞 = 33

140𝑚ℎ𝑎𝑟𝑣 + 𝑚𝑡𝑖𝑝,𝑒𝑞 (3.2)

where 𝑚ℎ𝑎𝑟𝑣 is the structural mass of the harvester given by:

𝑚ℎ𝑎𝑟𝑣 = 𝜌𝐴ℎ𝑎𝑟𝑣𝑙 = 𝑏(𝜌𝑒ℎ𝑒 + 2𝜌𝑝ℎ𝑝)𝑙 (3.3)

where 𝜌𝐴ℎ𝑎𝑟𝑣 is the mass per unit length of the harvester, 𝜌 denotes density and ℎ denotes

thickness, whereas the subscripts 𝑒 and 𝑝 represent the elastic metal layer and the PVDF

active layer respectively; the factor two in the second term in brackets accounts for having two

layers of PVDFs, one at each side of the metal shim.

Natural frequencies and damping ratios of the harvesters were measured through preliminary

free-decay vibration tests. With the harvesters clamped in a cantilever configuration, the

harvester tip was manually deflected of about 10°- 20° from its rest position, which is small

enough to trigger a mode-1 free-vibration, and then released. The oscillation of the harvester

tip was tracked using a laser vibrometer (model: Polytec PDV-100, resolution up to 0.02 μm/s)

providing displacement time-series of the harvester free-end. The natural frequency was then

determined from the period between peaks on the free-decay amplitude, whereas the damping

ratio was determined from the logarithmic decrement of the free-decay amplitudes of motion.

Once the mode-1 natural frequency, 𝜔1, and equivalent mass, 𝑀𝑒𝑞 , of the harvester were

determined, the harvester’s stiffness, 𝐾 , was then estimated using the classical equation [59]:

𝜔1 = 2𝜋𝑓1 = √𝐾

𝑀𝑒𝑞 . (3.4)

46

Table 3-2 provides the identified values of natural frequency, damping ratio and stiffness for

the eight harvesters considered in this work.

Table 3-2: Mechanical and electric properties of the harvesters considered in this study.

Harvester H1 H2 H3 H4 H5 H6 H7 H8

Natural frequency,

𝑓1 (Hz) 43.5 27.0 43.5 26.3 39.5 23.8 38.5 22.7

Damping ratio,

휁1 0.057 0.056 0.051 0.066 0.049 0.058 0.047 0.066

Stiffness,

𝐾 (N/m) 81.17 37.88 142.1 62.87 97.68 42.93 161.5 68.46

Optimum load resistance

for PVDFs,

𝑅𝑜𝑝𝑡 (kOhm)

161 258 92 152 179 293 104 176

Optimum load resistances that allowed a peak electric power generation were separately determined

for the PVDF elements and the solar panels. For the PVDF elements, theoretical values of the optimum

load resistance at the harvester first resonance frequency were evaluated using the analytical

expression [62]:

𝑅𝑜𝑝𝑡 = 1

𝜔1𝐶𝑒𝑞(

2휁1

√4휁12 + 𝑘31

4) (3.5)

where 𝑅𝑜𝑝𝑡 is the optimum load resistance value and 𝐶𝑒𝑞 is the equivalent internal capacitance

for the PVDF elements. In the present harvester design, the PVDF elements are connected in

parallel, hence the equivalent capacitance, 𝐶𝑒𝑞, is simply the internal capacitance of each

PVDF element (2.85 nF) multiplied by the number of PVDF elements included in the harvester.

Moreover, since the electromechanical coupling coefficient of the PVDF elements 𝑘31 is low

(12%, [22][63]), the bracketed term in Equation 3.5 is almost unity. Values of the optimum

theoretical resistances from Equation. 3.5 for all harvesters are provided in Table 3-2. To

ensure that Equation 3.5 is valid for the current configuration, the optimum load resistance

was also empirically determined. The optimum load resistance values identified

experimentally were very close to those determined from Equation 3.5, with an average

47

difference on the order of ±5-10%. As such, for simplicity and consistency, the identified

theoretical values from Equation 3.5 were used as an acceptable representation of the

optimum load resistance values for the PVDFs.

For the solar panels, the optimum load resistance was determined empirically using a variable

load that was tuned to maximize the power output. The solar panels were illuminated using a

Light-emitting diode (LED) light (model: Vibesta Capra 12 Daylight; 144 bulbs) that was

located perpendicular to the planform of the harvester. Other indoor light sources were not

considered at this stage. The lux illumination from the LED at the position of the solar panel

was measured with a portable light meter (model: ATP DT-1309 USB Logging Light Meter).

Then the optimum load resistance value that would allow a peak electric power generation

was identified (note that the measured power values will be provided later in Section 3.1.7).

Table 3-3 provides the experimental optimum load resistance values measured for the two

models of solar panels adopted in this study.

Table 3-3: Optimum load resistance values for solar panels.

Lux Resistance (kOhm)

PowerFilm SP3-37 PowerFilm MP3-37

500 15.4 2.82

1000 8.8 1.54

1500 7.00 1.10

2000 5.00 0.88

2500 4.70 0.65

3000 3.76 0.57

3500 2.82 0.49

4000 2.82 0.49

4500 2.35 0.41

5000 2.35 0.41

3.1.3. Experimental apparatus

The experimental set-up used to investigate the performance of the present harvesters is

shown in Figure 3-2. The set-up includes a shaker (model: Data Physics - V55) with control

signals provided by a signal generator (model: Tektronix – AFG1022) operated in sine wave

48

mode. The input voltage amplitude driving the shaker was tuned to yield a base acceleration

value of 0.5 g, which was not varied during the tests. Two reasons were behind the selection

of this acceleration amplitude. First, 0.5 g is within the range of acceleration magnitudes for

various vibration sources, as reported in [15], and is therefore representative of actual

harvesting applications. Second, an acceleration of 0.5 g is small enough to model the

harvesters as linear electromechanical systems, thus avoiding piezo-elastic, dissipation and

geometric non-linearities [10]. The base excitation levels were measured using an

accelerometer (model: PCB 336M13) attached and secured to the shaker. For each

experimental run, the harvester tip motion was recorded via a laser vibrometer (model: Polytec

PDV-100).

Each harvester was connected to the identified optimum load resistance. The electric power

generated from the harvesters was collected and processed through LabVIEW 2017 and the

data was gathered through an external DAQ device (model: National Instruments NI-USB-

6225). The data acquisition program was written as a Virtual Instrument (VI) in LabVIEW 2017

using the standard DAQ-mx library. This program gathered, saved and displayed the data with

some processing to allow an immediate impression of the power generated to be seen and to

Figure 3-2: Experimental set-up used in the current study.

49

reaffirm that the load resistance is allowing the highest power output. The sampling rate was

set at 1 kHz to allow sufficient resolution of data through a vibration cycle.

3.2. Considerations for Harvesters with Flexible Solar Panels

3.2.1. Model for an equivalent concentrated tip mass

Piezoelectric cantilever beams have always been attractive systems for analytical modelling

(e.g. see [64]). For harvesting applications, both lumped parameter and distributed parameter

models have been considered. However, lumped parameter models typically need a

correction factor to improve their prediction capabilities when compared to distributed

parameter models (for a comprehensive analysis of this point, the reader is referred to [10]).

As such, distributed parameter models are adopted in this study. For simplicity it was decided

that PVDF elements could be represented as continuous media in the span-wise direction

without significant loss of accuracy, such that a typical bimorph model could be employed to

represent two piezoelectric active layers sandwiching an elastic passive layer [10][65].

Validation results in Section 3.2.2. support the hypothesis that this assumption does not

significantly impact on the model ability to predict the general dynamics of the device.

Distributed parameter models based on the Euler-Bernoulli beam theory are used, implying a

thin beam assumption which is known to be acceptable for the length to thickness ratios used

in the current work, which were in excess of 20 (see Table 3-1) [66]. In this section, the first

modelling approach is considered where the solar panels are represented as an equivalent

concentrated tip mass (evaluated based on Equation 3.6), and hence are of negligible mass

moment of inertia about the tip, see Figure 3-3. This approach, thus, assumes that the solar

panels do not restrain the deformation of the beam section they cover. As such, this approach

better represents thin, flexible panels that do not significantly alter the beam deformation

pattern as is the case for those employed in the experimental part of this study.

50

The characteristic equation for a distributed parameter model of a Euler-Bernoulli beam with

a concentrated tip mass as that shown in Figure 3 is known to be [10]:

1 + cos 𝜆𝑛 cosh 𝜆𝑛 + 𝜆𝑛𝜇(cos 𝜆𝑛 sinh 𝜆𝑛 − sin 𝜆𝑛 cosh 𝜆𝑛) = 0 (3.6)

where 𝜇 =𝑚𝑡𝑖𝑝,𝑒𝑞

𝑚ℎ𝑎𝑟𝑣 ⁄ with 𝑚𝑡𝑖𝑝,𝑒𝑞 and 𝑚ℎ𝑎𝑟𝑣 being evaluated based on Equation 3.1 and

2.3. Hence Equation 3.6 may be solved for the dimensionless nth eigenvalue, 𝜆𝑛. The eigen

function (mode shape) can be obtained as [10]:

𝑋𝑛(𝑥) = 𝐴𝑛 [cos 𝜆𝑛

𝑥

𝑙− cosh 𝜆𝑛

𝑥

𝑙+ 𝜎𝑛 (sin 𝜆𝑛

𝑥

𝑙− sinh 𝜆𝑛

𝑥

𝑙)] (3.7)

where 𝜎𝑛 is obtained from:

𝜎𝑛 =sin 𝜆𝑛 − sinh 𝜆𝑛 + 𝜆𝑛𝜇(cos 𝜆𝑛 − cosh 𝜆𝑛)

cos 𝜆𝑛 + cosh 𝜆𝑛 − 𝜆𝑛𝜇(sin 𝜆𝑛 − sinh 𝜆𝑛) (3.8)

and 𝐴𝑛 is the modal amplitude which can be obtained making use of any of the orthogonality

conditions such as:

∫ 𝑋𝑠(𝑥)

𝑙

0

𝑏(𝜌𝑒ℎ𝑒 + 2𝜌𝑝ℎ𝑝)𝑋𝑟(𝑥)𝑑𝑥 + 𝑋𝑠(𝑙)𝑚𝑡𝑖𝑝,𝑒𝑞𝑋𝑟(𝑙) = 𝛿𝑟𝑠 (3.9)

where 𝛿𝑟𝑠 is the Kronecker delta. The displacement of the harvester is thus obtained from:

Figure 3-3: First modelling approach used in the current study where solar panels are represented as an equivalent concentrated tip mass. Equivalent tip mass is highlighted in blue.

51

𝑤(𝑥, 𝑡) = ∑ 𝜓𝑛𝑋𝑛(𝑥)1

𝜔𝑛2√(1 − (𝜔

𝜔𝑛⁄ )2

)2

+ (2휁𝑛𝜔

𝜔𝑛⁄ )2

A𝑏cos 𝜔𝑡

∞

𝑛=1

(3.10)

where 𝑤 is the displacement along the beam length relative to the base, 𝜔 is the operation

angular frequency, 𝐴𝑏 is the base excitation acceleration amplitude and 𝑡 denotes time. The

expression for 𝜓𝑛 is:

𝜓𝑛 = 𝑏(𝜌𝑒ℎ𝑒 + 2𝜌𝑝ℎ𝑝) ∫ 𝑋𝑛(𝑥)

𝑙

0

𝑑𝑥 + 𝑚𝑡𝑖𝑝,𝑒𝑞𝑋𝑛(𝑙) . (3.11)

Clearly, the interest is in motion around the first mode as well as in the harvester’s tip

displacement as it allows maximum displacement amplitude; as such, Equation 3.10

becomes:

𝑤(𝑙, 𝑡) = 𝜓1𝑋1(𝑙)1

𝜔12√(1 − (𝜔

𝜔1⁄ )2

)2

+ (2휁1𝜔

𝜔1⁄ )2

𝐴𝑏 cos 𝜔𝑡 (3.12)

and hence the tip displacement steady state frequency response transmissibility can be

expressed as:

|𝐻𝑑𝑖𝑠𝑝| =𝑤(𝑙)

𝐴𝑏= 𝜓1𝑋1(𝑙)

1

𝜔12√(1 − (𝜔

𝜔1⁄ )2

)2

+ (2휁1𝜔

𝜔1⁄ )2

. (3.13)

The model employed to evaluate the parallel connection steady state voltage response, 𝑣𝑝, is

based on the bimorph model as [10]:

52

𝑣𝑝

𝑤(𝑥)= |

𝑗(𝜔𝑅𝑜𝑝𝑡𝜅𝑛)

(1 + 𝑗𝜔𝑅𝑜𝑝𝑡𝐶𝑝)𝑋𝑛(𝑥)| (3.14)

where 𝐶𝑝 is the internal capacitance given by:

𝐶𝑝 =2휀𝑏𝑝𝑙

ℎ𝑝 (3.15)

where 𝑏𝑝 is the effective width of the PVDF layer which in this case is the number of PVDF

elements on one side of the harvester multiplied by the effective width of the active part of

each element (12 mm for the LDT2-028K model). The term 𝜅𝑛 is the modal coupling term

which for a bimorph configuration is given by:

𝜅𝑛 = 2𝑑31𝑌𝑝𝑏𝑝

2(ℎ𝑝 + ℎ𝑒) ∫

𝑑2𝑋𝑛(𝑥)

𝑑𝑥2𝑑𝑥

𝑙

0

. (3.16)

The most interest is in the peak steady state voltage as this allows the peak power generation.

This occurs when operating at the first resonance frequency; hence Equation 3.4 becomes:

𝑣𝑝,𝑚𝑎𝑥 = |𝑗(𝜔1𝑅𝑜𝑝𝑡𝜅1)

(1 + 𝑗𝜔1𝑅𝑜𝑝𝑡𝐶𝑝)𝑋1(𝑙)| 𝑤(𝑙) (3.17)

Where 𝑋1(𝑙) is evaluated based on the expression provided in Equation 3.2, and the

root mean square value of the peak power is thus evaluated from:

𝑃𝑚𝑎𝑥 =𝑣𝑝,𝑚𝑎𝑥

2

2𝑅𝑜𝑝𝑡 . (3.18)

53

3.2.2 Concentrated tip mass model vs. experimental measurements

Figure 3-4 provides a comparison between the experimentally measured tip displacement

Frequency Response Function (FRF) and the predicted tip displacement using the

concentrated tip mass model for the eight harvesters (for H1-4, tip mass was set to zero).

Despite of its simplicity, it is evident that the model is capable of capturing the experimental

results with good accuracy for the eight cases considered. Note that, all numerical modelling

was undertaken using MATLAB (version R2018a). The results in Figure 3-4 allow several

observations: First, the harvester length has a clear influence on the tip displacement values:

Longer harvesters (H2 and H4 as well as H6 and H8) have the largest tip displacement

amplitude and largest values for the ratio of tip displacement to beam length. Second,

harvesters with solar panels are capable of delivering higher tip displacement values for all

cases. These two observations have a significant effect on the generated power values as will

be shown later. Third and as expected, Figure 3-4(a) confirms that the resonance frequency

increases with decreasing the length, and that H1 and H3 as well as H2 and H4 have similar

resonance frequency values. This is also evident from Table 3-2 where it could be seen that

these pairs have almost the same resonance frequencies (with the note that if any tiny

difference between the experimental resonance frequency values of these pairs exists, it is

mainly due to minor manufacturing differences). This is expected as apart from the width these

harvester pairs have the same exact configuration. Fourth, comparing Figures 3-4(a) and 3-

4(b), it is evident that the solar panels shift the resonance frequencies to lower values, hence

are capable of modulating the operational frequency of the device. This reduction in the

resonant frequency varied between 14% (for H4 and H8) and 10% (for H1 and H5).

54

Figure 3-5 shows the PVDF power frequency response of the eight harvesters. Some

observations are evident: First, the concentrated tip mass model employed is predicting the

output PVDF power with good accuracy. Second, the PVDF power output from the longer

harvesters is higher which is consistent with the tip displacement comparison discussed

above. Third, the effect of the harvester width is more pronounced in the power results (as

compared to displacement results) where it is clearly evident that wider harvesters are capable

of generating more PVDF power. Finally, comparing Figures 3-5(a) and 3-5(b), it is evident

that harvesters with solar panels are capable of generating more PVDF power confirming the

ability of the employed solar panels to act as effective tip masses. In fact, the increase in PVDF

power due to the presence of solar panels could be up to 54% as when comparing H1 and

H5.

Figure 3-4: Measured and predicted tip displacement output FRFs for the eight harvesters. (a) Harvesters without solar panels (empty circle markers). (b) Harvesters with solar panels (filled circle markers). Markers represent the experimental measurements whereas solid lines represent the equivalent concentrated tip mass model predictions.

55

In order to further assess the concentrated tip mass model capability in predicting the electric

power generation, Figure 3-6(a) shows a comparison between the measured versus predicted

PVDF peak power (i.e. PVDF power at resonance). Here the limitation is assessment to the

PVDF peak power as this is arguably the most important metric for assessment. To provide

an estimate of accuracy in predicting PVDF peak power, Figure 3-6(b) shows the error values

between the concentrated tip mass model predictions and experimental measurements for the

eight harvesters considered in this work. The model typically under-predicts the measured

PVDF peak power with a mean error for the eight harvesters of -4.5%. This is a good

agreement given the assumptions adopted in the model. This discrepancy may be due to the

assumptions adopted in the model, manufacturing imperfections, errors in experimental

measurement, or a combination of these. Nevertheless, this discrepancy is small enough to

be of minor concern for practical applications.

Figure 3-5: Measured and predicted tip displacement output FRFs for the eight harvesters. (a) Harvesters without solar panels (empty circle markers). (b) Harvesters with solar panels (filled circle markers). Markers represent the experimental measurements whereas

56

It is instructive to understand how the length and width affect performance. Note that, in the

current study, the effect of the PVDF elements thickness is not considered as this is

constrained to what was available in the market. Figure 3-6(a) could be used to assess the

variation of PVDF peak power for the different lengths and widths considered. For both

harvester groups (with and without solar panels), the conclusion is the same: to achieve higher

harvested PVDF power values, the harvester should be longer and wider. Moreover, it could

be seen that the length is more influential in achieving higher power values compared to width.

3.3 Considerations for Harvesters with Rigid Solar Panels

3.3.1 Model for a distributed tip mass

In the previous section the strong agreement between experiment and the simple analytical

model demonstrated that the presence of solar panels in this lab experiment may be

considered to have a negligible influence on the deformation, since they are light and flexible.

While these tests are in general limited to the material available from distributors, it is important

to consider how the findings might hold when harvesters are scaled-up or tuned to different

dynamics. In this section, the scenario is considered as when the solar panels have a

Figure 3-6: Measured and predicted power output FRFs for the PVDF elements of the eight harvesters. (a) Harvesters without solar panels (empty circle markers). (b) Harvesters with solar panels (filled circle markers). Markers represent the experimental measurements whereas solid lines represent concentrated tip mass model predictions.

57

significant rigidity which acts to reduce the deformation of the beam. Given that the

piezoelectric layer must deform to generate electricity it is important to consider the case

where the presence of the solar panels prevents deformation at the tip region. Accordingly, a

model is developed where the solar panels and the beam section they cover are represented

as a rigid distributed tip mass as shown in Figure 3-7. This implies that the solar panel entirely

prevents the deformation of the beam for the section it covers.

The model developed here is like that proposed by Kim and Kim in dealing with distributed

masses; however, their model is extended to adopt the new configuration for the harvesters

considered in this study [61]. For this arrangement (shown in Figure 3-7), the tip mass, 𝑀𝑡𝑖𝑝 ,

is evaluated by:

𝑀𝑡𝑖𝑝 = 𝑙𝑠𝑏(2𝜌𝑠ℎ𝑠 + 2𝜌𝑝ℎ𝑝 + 𝜌𝑒ℎ𝑒) (3.19)

and the mass moment of inertia around the tip, 𝐼𝑡𝑖𝑝 , is evaluated as:

𝐼𝑡𝑖𝑝 = 1

12𝑀𝑡𝑖𝑝𝑙𝑠

2 + 2𝜌𝑠𝑙𝑠𝑏ℎ𝑠 ((𝑙𝑠

2)

2

+ (ℎ𝑒

2+ ℎ𝑝 +

ℎ𝑠

2)

2

)

+ 2𝜌𝑝𝑙𝑠𝑏ℎ𝑝 ((𝑙𝑠

2)

2

+ (ℎ𝑒

2+

ℎ𝑝

2)

2

) + 𝜌𝑒𝑙𝑠𝑏ℎ𝑒 (𝑙𝑠

2)

2

.

(3.20)

Note that within this modelling approach, the beam length (i.e. tip location) within the

distributed parameter model is equal to 𝑙 − 𝑙𝑠. It is useful to define the following two non-

dimensional parameters:

Figure 3-7: Second modelling approach used in the current study where solar panels are represented as a distributed tip mass. Tip mass is highlighted in blue.

58

𝜇 =𝑀𝑡𝑖𝑝

𝜌𝐴ℎ𝑎𝑟𝑣(𝑙 − 𝑙𝑠) (3.21)

𝛾 =𝐼𝑡𝑖𝑝

𝜌𝐴ℎ𝑎𝑟𝑣(𝑙 − 𝑙𝑠)3 . (3.22)

The characteristic equation for an Euler-Bernoulli beam with distributed tip mass is thus given

by [61]:

1 + cos 𝜆𝑛 cosh 𝜆𝑛 + 𝜆𝑛𝜇(cos 𝜆𝑛 sinh 𝜆𝑛 − sin 𝜆𝑛 cosh 𝜆𝑛)

− 𝜆𝑛3 𝛾(cosh 𝜆𝑛 sin 𝜆𝑛 + sinh 𝜆𝑛 cos 𝜆𝑛)

+ 𝜆𝑛4 𝜇𝛾(1 − cos 𝜆𝑛 cosh 𝜆𝑛) − 𝜆𝑛

2 𝜇 (𝑙𝑠

𝑙 − 𝑙𝑠) sinh 𝜆𝑛 sin 𝜆𝑛

−1

4𝜆𝑛

4 𝜇2 (𝑙𝑠

𝑙 − 𝑙𝑠)

2

(1 − cos 𝜆𝑛 cosh 𝜆𝑛) = 0 .

(3.23)

The eigen function can be obtained as [61]:

𝑋𝑛(𝑥) = 𝐴𝑛 [cosh 𝜆𝑛

𝑥

𝑙 − 𝑙𝑠− cos 𝜆𝑛

𝑥

𝑙 − 𝑙𝑠

− 𝜎𝑛 (sinh 𝜆𝑛

𝑥

𝑙 − 𝑙𝑠− sin 𝜆𝑛

𝑥

𝑙 − 𝑙𝑠)]

(3.24)

where 𝜎𝑛 is obtained from:

𝜎𝑛 =

sinh 𝜆𝑛 − sin 𝜆𝑛 + 𝜆𝑛𝜇 (cosh 𝜆𝑛 − cos 𝜆𝑛 +𝜆𝑛

2 (𝑙𝑠

𝑙 − 𝑙𝑠) (sinh 𝜆𝑛 + sin 𝜆𝑛))

cosh 𝜆𝑛 + cos 𝜆𝑛 + 𝜆𝑛𝜇 (sinh 𝜆𝑛 − sin 𝜆𝑛 +𝜆𝑛

2 (𝑙𝑠

𝑙 − 𝑙𝑠) (cosh 𝜆𝑛 − cos 𝜆𝑛))

(3.25)

and the modal amplitude, 𝐴𝑛 , is obtained from a corresponding orthogonality condition:

59

∫ 𝑋𝑠(𝑥)

𝑙−𝑙𝑠

0

𝑏(𝜌𝑒ℎ𝑒 + 2𝜌𝑝ℎ𝑝)𝑋𝑟(𝑥)𝑑𝑥 + 𝑋𝑠(𝑙 − 𝑙𝑠)𝑀𝑡𝑖𝑝𝑋𝑟(𝑙 − 𝑙𝑠)

+ 𝑀𝑡𝑖𝑝𝑋𝑠(𝑙 − 𝑙𝑠)𝑙𝑠

2

𝑑𝑋𝑟(𝑥)

𝑑𝑥 |

𝑥=𝑙−𝑙𝑠

+ 𝑀𝑡𝑖𝑝𝑋𝑟(𝑙 − 𝑙𝑠)𝑙𝑠

2

𝑑𝑋𝑠(𝑥)

𝑑𝑥 |

𝑥=𝑙−𝑙𝑠

+ 𝐼𝑡𝑖𝑝

𝑑𝑋𝑠(𝑥)

𝑑𝑥 |

𝑥=𝑙−𝑙𝑠

𝑑𝑋𝑟(𝑥)

𝑑𝑥 |

𝑥=𝑙−𝑙𝑠

= 𝛿𝑟𝑠

(3.26)

where 𝛿𝑟𝑠 is the Kronecker delta. The displacement of the harvester at 𝑥 = 𝑙 − 𝑙𝑠 around the

first mode is thus obtained from:

𝑤(𝑙 − 𝑙𝑠, 𝑡) = 𝜓1𝑋1(𝑙 − 𝑙𝑠)1

𝜔12√(1 − (𝜔

𝜔1⁄ )2

)2

+ (2휁1𝜔

𝜔1⁄ )2

𝐴𝑏 cos 𝜔𝑡 (3.27)

where the expression for 𝜓1 here is given by [61]:

𝜓1 = 𝑏(𝜌𝑒ℎ𝑒 + 2𝜌𝑝ℎ𝑝) ∫ 𝑋1(𝑥)

𝑙−𝑙𝑠

0

𝑑𝑥 + 𝑀𝑡𝑖𝑝𝑋1(𝑙 − 𝑙𝑠)

+ 𝑀𝑡𝑖𝑝

𝑙𝑠

2

𝑑𝑋1(𝑥)

𝑑𝑥 |

𝑥=𝑙−𝑙𝑠

.

(3.28)

Knowing the displacement distribution in the region 0 ≤ 𝑥 ≤ 𝑙 − 𝑙𝑠 allows to define the angle

of the harvester with the horizontal at 𝑥 = 𝑙 − 𝑙𝑠 . This angle is denoted as 𝜃 which once

evaluated could be used to calculate the displacement of the harvester at 𝑥 = 𝑙:

60

𝑤(𝑙, 𝑡) = 𝑤(𝑙 − 𝑙𝑠, 𝑡) + 𝑙𝑠 sin 𝜃. (3.29)

This means that harvester is straight within the region 𝑙 − 𝑙𝑠 < 𝑥 ≤ 𝑙.

The parallel connection steady state voltage response, 𝑣𝑝, for the distributed tip mass model

could also be evaluated based on Equation 3.14 which when operating at the first resonance

frequency takes the form:

𝑣𝑝,𝑚𝑎𝑥 = |𝑗(𝜔1𝑅𝑜𝑝𝑡𝜅1)

(1 + 𝑗𝜔1𝑅𝑜𝑝𝑡𝐶𝑝)𝑋1(𝑙 − 𝑙𝑠)| 𝑤(𝑙 − 𝑙𝑠) (3.30)

where 𝐶𝑝 is the internal capacitance given by Equation 3.15 and the modal coupling term, 𝜅1,

for the distributed tip mass model is given by:

𝜅1 = 2𝑑31𝑌𝑝𝑏𝑝

2(ℎ𝑝 + ℎ𝑒) ∫

𝑑2𝑋1(𝑥)

𝑑𝑥2𝑑𝑥

𝑙

0

= 2𝑑31𝑌𝑝𝑏𝑝

2(ℎ𝑝 + ℎ𝑒) ∫

𝑑2𝑋1(𝑥)

𝑑𝑥2𝑑𝑥

𝑙−𝑙𝑠

0

(3.31)

where the mode shape / eigen function in the above expressions is evaluated based on

Equation 3.24. Note that here 𝜅𝑛 is only evaluated over the region 0 ≤ 𝑥 ≤ 𝑙 − 𝑙𝑠 as beyond

this region the second derivative in the above expression vanishes. This implies that the

rigidity of the beam underneath the solar panels prevents any electric power generation from

this part of the PVDF layers. Consequently, PVDF power generation from this model is

expected to be lower than that from the concentrated tip mass model which allows the whole

beam length to deform and contribute to the power generation. Once the voltage is evaluated,

the root mean square value of the peak power could be evaluated from Equation 3.18.

61

3.3.2 Flexible vs. rigid solar panels

Figure 3-8 provides a comparison between the predictions from the two modelling approaches

employed in this study for the harvesters with solar panels. The comparison of the tip

displacements in Figure 3-8(a) indicates that both model results are close, with the rigid tip

mass model predictions reaching slightly higher values. The increase in the peak tip

displacement predicted from the rigid tip mass model is similar for the all harvesters with a

value of 6.6%. The differences between the two configurations become more pronounced

when comparing the power results, Figure 3-8(b). As expected, the rigid tip mass model

predicts lower power values than the flexible tip mass model, on account of the restricted

region which is free to deform. However, the penalty due to restricted deformation at the tip is

notably small. Comparing peak PVDF power results from both models, it was found that the

reduction due to the rigid tip mass was 28% in cases H5 and H7 (shorter harvesters) and

18.5% in cases H6 and H8 (longer harvesters). For a fixed length of solar panel, the loss in

performance for the longer harvesters was lower as in these cases a longer portion of PVDF

remains uncovered by the solar panels and thus free to deform.

Figure 3-8: Flexible vs. rigid tip model predictions for harvesters with solar panels. (a) Tip displacement frequency response function. (b) PVDF power frequency response function.

62

3.4 Optimum Tip Mass Configuration

The developed theoretical models could be used to understand the effect of varying the solar

panels configuration on the generated PVDF peak power (i.e. power value at resonance),

Figure 3-9. Note that, in this demonstration, the beam rigidity is evaluated based on the

expression for a composite beam in a bimorph configuration [65]. The x-axis in Figure 3-9

represents the solar panels coverage (i.e. 𝑙𝑠/𝑙) whereas the y-axis represents the ratio of the

mass per unit length of the solar panels to the mass per unit length of the harvester (see

Equation 3.1 and 3.3), hence providing an indication of the solar panels relative weight. The

contours shown in Figure 3-9 represent the peak power normalized by the maximum peak

power value of both plots.

Figure 3-9 shows that for both models increasing the tip mass ratio will always improve the

power output. Results for the optimum solar panels coverage (i.e. 𝑙𝑠/𝑙) are more interesting

and differ depending on the model employed. For the concentrated tip mass model, the

optimum coverage does not depend on the tip mass ratio and has a constant value of two

thirds of the harvester length. This result is expected since if Equation 3.1 is differentiated with

respect to coverage and equated to zero, the resulting quadratic equation has only one

feasible root of 2/3. On the other hand, the optimum coverage from the distributed tip mass

Figure 3-9: Flexible vs. rigid tip model predictions for harvesters with solar panels. (a) Tip displacement frequency response function. (b) PVDF power frequency response function.

63

model is function of the tip mass ratio: it starts at zero for zero tip mass and increases with the

increase of the tip mass ratio. Remarkably, it has an asymptotic value of 0.5 for significantly

high tip mass ratios. Hence, values for optimum coverage form the distributed tip mass model

are always lower than the optimum coverage from the concentrated tip mass model. Note that

Figure 3-9 (b) shows that for high values of coverage, the distributed tip mass model could not

provide a solution (white area in Figure 3-9 (b)) which is a limitation of the model. However,

this happens only for values of coverage and tip mass ratio that are far from optimum or

practical configurations of interest, hence this model limitation is considered insignificant.

3.5 Solar Power

The previous sections showed how solar panels can act as an effective tip mass to affect the

performance of the harvesters through decreasing the operational frequency and increasing

the PVDF peak power. Whilst these are useful outcomes consistent with their role as a tip

mass, a major contribution from the solar panels in terms of power characteristics could be

probably attributed to the fact that these panels provide another source of harvesting energy

from available light sources. Figure 3-10 shows the solar power values measured

experimentally for the two models of solar panels tested in this study when employing the

optimum resistance values identified in Section 3.1.2 Note that the solar power values shown

in Figure 3-10 are for only one panel (in the present experimental setup, only one solar panel

could be illuminated); hence with two solar panels attached to each harvester the power will

theoretically be doubled if subjected to the same light exposure. It is important to mention that

the power generated from the solar panels is a DC power; however, due to the harvester

motion this could change dynamically. Nevertheless, because the tip displacements

experienced in this study are relatively low, the effect of the motion from the harvester was

negligible as it hardly affected the degree to which the solar panel is incident to the light source,

hence a constant power was still generated.

64

The solar power trends shown in Figure 3-10 were found to follow a near linear variation (at

least within the range of Lux measured here). As such, fitting relations were produced for both

models and are shown in Figure 3-10. It could be seen that the power generated from the

larger panel (MP3-37) is 1.5 times more than the smaller panel (SP3-37). Nevertheless, if

power density is concerned, the larger panel is 1.75 larger in volume, so the smaller panel is

more favorable from a power density point of view. It may be useful to note that the lux level

for full daylight is around 10,000 lux, for overcast day is around 1,000 lux and for a very dark

day is around 100 lux. This helps when assessing the expected amount of power generation

under various lighting conditions. As such, for a typical overcast day the SP3-37 should

produce 600 μW per panel whereas the MP3-37 would produce 900 μW per panel. This is

much higher magnitude compared to the power obtained from the PVDF elements hence

allowing a significant boost to the total power generation from the device.

It should be noted that the solar panels employed in this study can in principle be expected to

achieve better performance in outdoor applications, where the lighting conditions (intensity

and wavelength spectrum) would be more favorable with respect to the indoor LED light

illumination considered here. The performance of solar panels in indoor applications is known

to depend on the light source used [67][68]. Minnaert and Veelaert [68] showed that LED

Figure 3-11: Experimentally measured power from a single solar panel in milli-Watt versus light lux. Markers represent the experimentally measured values whereas lines represent curve fitting for the measured data.

65

light illumination, as compared with other light sources, leads to low performance for

photovoltaics, with a decrease in performance ranging from a quarter for amorphous silicon

up to two thirds for crystalline silicon cells. They also indicate that, under LED light illumination,

amorphous silicon cells should perform significantly better with respect to other photovoltaics.

The solar panels used here are thin-film amorphous silicon photovoltaics, and should therefore

perform reasonably well under LED light illumination. As a matter of fact, according to the solar

panel manufacturer (PowerFilm), amorphous silicon has its peak quantum efficiency within a

range of wavelengths that matches well the range of wavelengths emitted by LED lights.

Finally, it is worth highlighting that despite the relatively high power output from the solar

panels, these panels would have their own limitations when harvesting energy within more

realistic conditions. For example, operating in dirty environment could cause the panel to

become obscured, and hence affect the amount of energy harvested. The intermittent nature

of most light sources, notably daylight, is clearly another limitation. A possible solution would

be to integrate the solar panels with energy storage devices, such as batteries and

supercapacitors, and use the stored energy when needed.

66

4. Systematic Experimental Study on Planform Geometry

and Excitation Effects2

The previous chapter has touched on the effect that harvester planform can have in influencing

the dynamics and power generation. However, the employed harvesters where not spanning

a wide range of length and width values that would allow solid conclusions. As such, this

chapter will provide a more focused investigation into the effect of planform geometry on the

harvesting performance and how the planform effects depend on the excitation level.

Moreover, this investigation will enable an improved understanding of the non-linear effects

that kick in when planform geometries are varied particularly when considering harvesters with

long lengths.

4.1 Materials and Methods

4.1.1 Harvesters design and realization

As schematically shown in Figure 4-1(a), the present harvesters are cantilever bimorphs that

include a passive elastic layer and two active piezoelectric layers in a sandwich arrangement.

The elastic layer is a stainless-steel shim by Precision Brand (/precisionbrand.com/; density:

𝜌𝑒 = 7900 kg/m3; Young’s modulus: 𝑌𝑒 = 180 GPa; thickness: ℎ𝑒 = 0.1 mm), whilst the two

active piezoelectric layers are made from discrete PVDF elements that are bonded to the core

metal shim using double-sided adhesive tape by Tesa (/www.tesa.com/en/; density: 𝜌𝑏 = 1100

kg/m3; thickness: ℎ𝑏 = 0.1 mm). The PVDF elements employed (Figure 4-1(b)) are from TE

Connectivity-model DT4-028K (/www.te.com/; density: 𝜌𝑝 = 2280 kg/m3; Young’s modulus: 𝑌𝑝

= 2.8 GPa; piezo strain constant: 𝑑31 = 23x10-12 C/N; capacitance: 𝐶 = 11.00 nF; thickness:

ℎ𝑝 = 0.064 mm). Each element comprises a PVDF film covered with silver ink screen-printed

electrodes, all contained within a thin plastic coating for protection. Note that these PVDF

2 Elements of this chapter is currently under review as a paper submitted to the Journal of Intelligent Materials Systems and Structures.

67

elements are so flexible that a more rigid elastic layer is strictly required as a passive substrate

to provide a meaningful structure. For testing, the harvesters were connected to the shaker

(described later) using a laser-cut acrylic clamp designed and manufactured in-house (Figure

4-1(c)).

The PVDF elements used are the longest that could be found commercially available off-the-

shelf and come with dimensions of 171 mm × 22 mm × 0.064 mm (length × width × thickness;

see Figure 4-1(b)). In order to produce harvesters of different lengths, the PVDF elements

were cut down to the desired length, taking care to avoid conduction of internal piezoelectric

layers from exposed edges. Each element was cut using a new scalpel to avoid contamination

which could result in damaging the PVDF element. Overall, eight harvesters were produced:

four with a length of 78 mm (harvesters H1-H4, Figure 4-1(d)), and four with a length of 155

mm (harvesters H5-H8, Figure 4-1(e)). The main geometric characteristics of the eight

harvesters are summarized in Table 4-1. The width was adjusted by including additional PVDF

elements, up to four, resulting in a range of values (22 mm, 44 mm, 66 mm, and 88 mm). Note

that a fraction of the harvester length is inactive since its rests within the clamp and cannot

deform, i.e. does not produce any power. After each PVDF element was individually tested to

ensure expected operation, elements were connected in parallel to supply a load resistor.

Following common practice, the value of the load resistance (provided in Table 4-1) was

empirically determined to maximize the power output of each harvester, i.e. conducting an

experimental power scan with different resistance values and identifying the optimum

resistance value that would allow maximum power generation.

68

Figure 4-1: The harvesters considered in this study: (a) design schematics (the example shown includes four PVDF elements on each side); (b) one PVDF element; (c) schematic representation of the custom-made mounting system to connect the harvester to the shaker (the upper plate is removed to show internal details); (d) harvesters H1-H4 with length of 78 mm; (e) harvesters H5-H8 with length of 155 mm.

69

Table 4-1: Characteristics of the harvesters considered in this study.

Harvester Length, 𝒍

(mm)

Width, 𝒃

(mm)

Aspect Ratio, 𝒍 𝒃⁄

(-)

Second Moment of

Area, 𝟏 𝟑⁄ 𝒃𝒍𝟑 (cm4)

Optimum load resistance, 𝑹𝒐𝒑𝒕

(kOhm)

H1 78 22 3.55 348 1000

H2 78 44 1.77 696 600

H3 78 66 1.18 1044 400

H4 78 88 0.89 1392 250

H5 155 22 7.05 2731 2400

H6 155 44 3.52 5462 1200

H7 155 66 2.35 8192 800

H8 155 88 1.76 10923 600

4.1.2 Experimental set-up

The experimental setup is shown in Figure 4-2 and includes a signal generator (by Tektronix,

model AFG1022), a power amplifier (by Data Physics, model PA300E), and a shaker (by Data

Physics, model V55). The acceleration transmitted from the shaker to the harvesters was

monitored with an accelerometer (by PCB Piezotronics, model PCB 336M13) attached to the

shaker close to the base excitation point of the harvesters. A laser vibrometer (by Polytec,

model PDV-100) was used to measure the tip displacement of the harvesters. When the tip

velocities exceeded the maximum resolution of the vibrometer (500 mm/s), the tip

displacement was measured optically with a high-speed camera (by Phantom, model v310,

equipped with a Nikon AF Micro-Nikkor 60mm f/2.8D lens). The camera was operated at 3,200

frames per second with 1280x800 spatial resolution and an exposure time of 310 μs. The

videos were postprocessed with the free, open source video analysis software Tracker version

5.0.7 (https://physlets.org/tracker/). The power output from the harvesters as well as the

accelerometer and vibrometer signals were collected through an external DAQ device (by

National Instruments, model NI-USB-6225) and processed through LabVIEW 2017. The

sampling rate was set at 1 kHz to allow sufficient resolution of data through a vibration cycle.

The data acquisition program, which was written as a Virtual Instrument (VI) in LabVIEW 2017

70

using the standard DAQ-mx library, gathered, saved and displayed the data in real time during

the tests.

The tests were conducted under single-frequency excitation for frequencies in the range of 3-

18 Hz (low enough to excite a mode-1 vibration in the harvesters) and base acceleration levels

ranging from 0.2 g up to 0.6 g. As highlighted in the introduction, vibration sources cover

frequencies of 0-200 Hz and accelerations from 0.01 g up to 1.2 g [15]. Even though the

energy spectrum of these sources is typically broadband, the vibration energy is frequently

concentrated on a few, rather narrow peaks [69]. As such, the results presented here can be

considered to be generally relevant to energy harvesting from low-frequency ambient

vibrations.

Figure 4-2: Experimental set-up used to test the harvesters.

71

4.1.3 Harvesters dynamic properties

A series of static deflection tests was conducted to measure the elastic restoring force, while

both forced and free vibration tests were made to determine the damping ratio of the

harvesters. These results are provided in Figure 4-3 as a function of the tip displacement,

which is the perpendicular displacement of the free edge of the harvester with respect to its

rest position. Note that, due to the different lengths, the range of tip displacements tested for

the short harvesters is around a quarter of that for the longer harvesters. Static deflection tests

were conducted by applying sets of known masses at the tip of the harvester and measuring

the deflection caused by the corresponding load. Forced vibration tests were conducted to

determine the mode-1 damping ratio of the short harvesters (H1-H4) based on an evaluation

of the harmonic vibration response [70]:

휁1 =(𝜔ℎ𝑝2−𝜔ℎ𝑝1)

𝜔1, (4.1)

where 𝜔ℎ𝑝1 and 𝜔ℎ𝑝2 are the half-power point frequencies where the response amplitude is

1/√2 of the peak amplitude. Forced vibration tests were made at 0.2 g, 0.3 g, 0.4 g, 0.5 g,

and 0.6 g excitation levels. This study is unable to apply the same method to characterise

damping for set of long harvesters (H5-8), since the signal generator used had a minimum

resolution of 0.5 Hz. For this case, the damping ratio is estimated based on the classical

logarithmic decrement deduced from free vibration tests in stagnant air [71]. During the free

vibration tests the PVDF elements were connected to the resistive load, so that the damping

ratio values provided in Figure 4-2 represent the total damping of the harvesters (inclusive of

the structural damping, the fluid damping, and the electrical damping).

72

As is evident in the top panels of Figure 4-3, the elastic restoring force of the harvesters is

linear for small tip displacements, becoming non-linear for larger displacements. This is

unsurprising since cantilevers, in fact elastic structures in general, are well known to behave

linearly only in the limit of small displacements. As can be noted from the bottom panels of

Figure 4-3, the damping ratios of the short harvesters exhibit no appreciable variation with the

tip displacement. However, since the range of points considered in this test is relatively narrow,

it is not be able to extrapolate beyond them. This is not the case for the long harvesters where

the damping ratios show clear variation with the tip displacement, so that the resulting damping

force is non-linear, as is expected for large amplitudes [72]. In particular, as tip displacement

Figure 4-3: Measured elastic restoring force (top) and damping ratio (bottom) of the harvesters (the continuous fitting lines are included to help visualize the trend in the data).

73

increases, the damping ratio of the long and narrow harvesters (widths 22-66 mm) is first

observed to increase, before slowly saturating towards a constant value: a trend consistent

with available observations for wing sections [73]. Also note that, the damping ratio results for

both sets of harvesters share a common feature that is the damping ratio increases with the

width of the harvester, indicating an increased contribution of air-damping as the surface area

increases. Damping ratios measured during preliminary tests (not documented here) for the

metal substrate alone, without the PVDF elements, were significantly lower than the values

reported in Figure 4-3 (damping ratios on the order of 0.02 for a tip displacement of 10 mm),

indicating that the inclusion of the PVDF elements and the bonding layer significantly increase

the structural damping of the harvesters.

4.1.4 Linear electro-mechanical model

The demonstration in section 4.1.3 was mainly targeted at confirming that for large tip

displacement the elastic restoring force and the damping of the present harvesters become

non-linear. Nevertheless, a remarkable behaviour that is evident from Figure 4-3 is that the

variation of both elastic restoring force and damping ratio with displacement amplitude is

smooth and gradual. This, in turn, can complicate the assessment/visualisation of non-linear

effects as they are expected to appear gradually and not in an abrupt fashion. Because simple

inspection of the measured results would not reveal any sharp discontinuities or transitions,

the ability to detect whether non-linearity kicks in (just from visual observation of the measured

trends alone) could be easily hindered. A simple linear electro-mechanical model of the

present harvesters was therefore developed to help quantification of the presence of non-

linear effects. Schematically, a good agreement between measurements and linear model

predictions would indicate that the excitation level is small to introduce significant non-linear

effects, so that the response of the harvesters is approximately linear. A disagreement

between measurements and linear model predictions, on the other hand, would indicate the

contrary and can be traced back to non-linear effects in the response of the harvesters. This

74

way, the comparison between measured data and linear model predictions can help assess

the influence of non-linear effects in the measured response of the harvesters.

The model adopted here is a distributed parameter linear model based on the Euler-Bernoulli

beam theory, which is applicable in the context because the present harvesters qualify as

slender structures (the length to thickness ratio is well in excess of 20) [65][74]. For simplicity,

the piezoelectric layers are represented as continuous media in the span-wise direction, an

approximation considered acceptable for the present scope, and tested by the authors in a

previous contribution [75]. The characteristic equation for a cantilever beam is well known as

[65][74]:

1 + cos 𝜆𝑛 cosh 𝜆𝑛 = 0 , (4.2)

which can be solved for the dimensionless nth eigenvalue, 𝜆𝑛 . The corresponding

eigenfunction (mode shape) is:

𝑋𝑛(𝑥) = 𝐴𝑛 [cos 𝜆𝑛

𝑥

𝑙− cosh 𝜆𝑛

𝑥

𝑙+ 𝜎𝑛 (sin 𝜆𝑛

𝑥

𝑙− sinh 𝜆𝑛

𝑥

𝑙)] , (4.3)

where 𝜎𝑛 is obtained from:

𝜎𝑛 =sin 𝜆𝑛 − sinh 𝜆𝑛

cos 𝜆𝑛 + cosh 𝜆𝑛, (4.4)

and 𝐴𝑛 is the modal amplitude, which can be obtained using an orthogonality condition such

as:

∫ 𝑋𝑠(𝑥)

𝑙

0

𝑏(𝜌𝑒ℎ𝑒 + 2𝜌𝑏ℎ𝑏 + 2𝜌𝑝ℎ𝑝)𝑋𝑟(𝑥)𝑑𝑥 = 𝛿𝑟𝑠 , (4.5)

where 𝛿𝑟𝑠 is the Kronecker delta. The displacement with respect to base, 𝑤 , along the

harvester length 𝑥 at time 𝑡 is thus obtained from:

75

𝑤(𝑥, 𝑡) = ∑ 𝜓𝑋(𝑥)1

𝜔𝑛2√(1 − (𝜔

𝜔𝑛⁄ )2

)2

+ (2휁𝑛𝜔

𝜔𝑛⁄ )2

A𝑏cos 𝜔𝑡,

∞

𝑛=1

(4.6)

where 𝜔 is the operation angular frequency and A𝑏 is the base excitation acceleration

amplitude. The expression for 𝜓 specific to the current harvester configuration could be

shown to take the form:

𝜓 = 𝑏(𝜌𝑒ℎ𝑒 + 2𝜌𝑏ℎ𝑏 + 2𝜌𝑝ℎ𝑝) ∫ 𝑋(𝑥)

𝑙

0

𝑑𝑥. (4.7)

Given that the interest here is in the tip displacement of the harvesters around the first mode,

Equation (4.6) simplifies to:

𝑤(𝑙, 𝑡) = 𝜓𝑋(𝑙)1

𝜔12√(1 − (𝜔

𝜔1⁄ )2

)2

+ (2휁1𝜔

𝜔1⁄ )2

𝐴𝑏 cos 𝜔𝑡, (4.8)

The parallel connection steady state voltage response, 𝑣𝑝, can be evaluated based on an

expression presented by the authors in [75]:

𝑣𝑝 = |𝑗(𝜔𝑅𝑜𝑝𝑡𝜅𝑛)

(1 + 𝑗𝜔𝑅𝑜𝑝𝑡𝐶𝑝)𝑋𝑛(𝑙)| 𝑤(𝑙), (4.9)

where 𝜅𝑛 is the modal coupling term, which for the current harvester configuration of this study

could be evaluated as:

𝜅𝑛 = 2𝑑31𝑌𝑝𝑏𝑝

2(ℎ𝑝 + 2ℎ𝑏 + ℎ𝑒)

𝑑𝑋𝑛(𝑥)

𝑑𝑥 |

𝑥=𝑙, (4.10)

and 𝐶𝑝 is the internal capacitance given by:

𝐶𝑝 =2휀𝑏𝑝𝑙

ℎ𝑝, (4.11)

76

where 휀 is the permittivity, and 𝑏𝑝 is the effective width of the PVDF layer which in this case is

the number of PVDF elements on one side of the harvester multiplied by the effective width of

the active part of each element (19 mm for the DT4-028K model). The root mean square (RMS)

value of the peak power could thus be evaluated from:

𝑃 =𝑣𝑝

2

2𝑅𝑜𝑝𝑡. (4.12)

4.2 Results and Discussion

4.2.1 Dynamics and power generation measurements

Tip displacement and RMS power output measurements are shown in Figure 4-4, for all

harvesters, plotted as function of the frequency of the excitation, for different amplitudes of the

level of excitation. The predictions of the linear electro-mechanical model are also included in

Figure 4-4. As can be noticed, the mode-1 resonance frequency of the short harvesters (H1-

H4) is around 13-15 Hz, whilst that of the longer harvesters (H5-H8) is around 3-4 Hz. As

expected, when the excitation frequency approaches the resonance frequency of the

harvester the tip displacement increases, reaching a maximum that is proportional to the

amplitude of the excitation. The power output follows a trend similar to that observed for the

tip displacement, consistently indicating that a larger tip displacement leads to larger

deformation and therefore a larger strain, which yields a higher power. Even though the tip

displacements do not show significant dependence on the width of the harvesters, the power

clearly varies with the number of PVDF elements embedded within the harvesters, and

therefore is related to the harvester width (more discussions on this point will follow later).

For the short harvesters (H1-H4), the agreement between measurements and linear model

prediction is rather good, indicating that non-linear effects are negligible. For these harvesters,

in fact, the tip displacement never exceeds 13 mm (i.e. maximum tip displacement to length

ratio of 17%), which is small enough to approximate the elastic restoring force in Figure 4-2

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as linear. As shown in Figure 4-3 the damping ratios are rather large (0.05-0.06), and this is

reflected in the rather broad appearance of the peaks in Figure 4-4. Though not ideal for power

generation, broad peaks yield a more robust harvester design for final applications: frequency

matching between harvester and host structure is easier, and the power penalty from

frequency mismatch becomes less severe.

For the long harvesters (H5-H8), the linear model predictions significantly overpredict the

measurements, indicating that non-linear effects are generally present. For these harvesters,

the tip displacement reaches as high as 55 mm (i.e. maximum tip displacement to length ratio

of 35%), which is large enough to make the non-linear effects apparent. The over prediction

of the linear model indicates that non-linear effects reduce the displacement amplitude and

are therefore not beneficial for power generation. This is hardly surprising, as increasing the

stiffness and the damping of any mechanical system would reduce, for a given excitation, the

vibration amplitude. These results indicate that the intrinsic non-linearity of the present

harvesters is not beneficial for energy harvesting. This, however, should not be taken as an

indication that non-linear effects are necessarily detrimental. Quite the opposite: non-linear

effects can be exploited to improve the performance of energy harvesters [27], notably to

broaden the frequency bandwidth or increase the response amplitude [76][77][78].

78

4.2.2 Planform effects

Peak power levels measured for all harvesters are presented in Figure 4-5 as a function of the

width of the harvester, with excitation amplitude as parameter. Note that, for confirmation,

peak power refers to the power value at the first resonance frequency measured with optimum

Figure 4-5: Tip displacement and RMS power measurements versus excitation frequency. Markers represent measured responses whereas continuous lines are the linear model predictions. For the long harvesters (length = 155 mm), the theoretical model predictions are greyed out for better visualisation of the experimental results as theoretical results are just included to assess the presence of non-linear effects.

79

load resistance. At low excitation amplitude (0.2 g-0.4 g) a maximum is observed in the trend

of the peak power versus harvester width, after which the power decreases. At high excitation

amplitude (0.5 g-0.6 g), the trend increases monotonically throughout the range of values

tested, though in most cases the trend appears to be saturating. Since an indefinite growth of

the peak power with harvester width does not seem physically plausible (significant increase

in width to length ratio will ultimately prevent reasonable harvester deformation), the observed

trend in Figure 4-5 suggests that the peak power maximum gradually shifts to higher width

values (i.e. to the right in Figures 4-5(a) and 4-5(b)) as the excitation level is gradually

increased, and is therefore no longer observed in the present data in Figure 4-5 for high

excitations (0.5 g-0.6 g). This indicates that, for a given excitation amplitude, there is an

optimum harvester width that maximizes the power output. This optimum harvester width

depends on the excitation amplitude, so that narrower harvesters are more suited for small

excitations whilst for large excitations wider harvesters perform better.

It is notable that at low excitation amplitude (0.2 g) the maximum peak power of the long

harvesters (length of 155 mm) is approximately twice as large as that of the short ones (length

of 78 mm). This is no longer the case at higher excitations: at 0.6 g excitation amplitude the

Figure 4-6: Peak power as a function of harvester width for different base excitation levels; (a) set of short harvesters (H1-H4); (b) set of long harvesters (H5-H8). Dashed lines are added to help visualize the trends but do not imply a linear variation between measurement points.

80

peak power of long and short harvesters is comparable in magnitude, see Figure 4-6(a). This

indicates that, for a given excitation, a longer cantilever is beneficial for energy harvesting only

as long as it responds linearly. If non-linear effects come into play, then a shorter cantilever

should be preferred. This is further clarified in Figure 4-6(b) through the power density trends

where it is evident that, as the excitation level increases, the short harvesters could achieve

power density values more than twice that of the long harvesters. Note that in this

demonstration, the peak power density has been calculated by considering only the active

volume of the harvester as common practice used, which is the volume of the PVDF layers.

To conclude, for applications with high vibration excitation levels, having two short harvesters

that respond linearly would be far more efficient than having just one long harvester that

operates non-linearly: higher power output from the same harvester total volume.

4.2.3 Aspect ratio vs second moment of area

To inform design optimization studies, it is convenient to group the dimensions and the shape

of the harvesters into a single parameter that captures the geometric effect on the power

generation. The aspect ratio is considered, defined as the ratio of the harvester length to its

width (values provided in Table 4-1). Note that the present definition of the aspect ratio differs

Figure 4-7: Relative performance of the two sets of harvesters; (a) quotient of peak power of long harvesters divided by that of short harvesters; (b) quotient of peak power density of short harvesters divided by that of long harvesters. Dashed lines are added to help visualize the trends.

81

from the normal practice within vibrations literature, where the same term is used to denote

the length to thickness ratio. The peak power and the peak power density are presented as

functions of the aspect ratio in Figure 4-7.

Even though the data points in Figure 4-7 are, to some extent, stratified as a function of the

excitation amplitude, the trends are not very clear, indicating that the aspect ratio does not

capture geometry effects that well with the present harvesters. Interestingly, the aspect ratio

does capture the geometry effects in the case of PVDF-based inverted flag energy harvesters

under wind excitation [28][29][40][79][80] surprising as the aspect ratio is considered a

fundamental parameter in describing the interaction of 3D bodies (e.g. wings, flags, etc.) with

fluid flows. However, for base excitation the aspect ratio is not a key parameter within the

governing Equation, hence the unclear trend in Figure 4-7 is not surprising.

A more relevant geometric parameter is the second moment of area of the harvester,

calculated relative to the clamped axis of the cantilever. As is well known, the second moment

of area is a dimensional parameter that measures how the points of an area are distributed in

space with regard to an arbitrary axis. In the present case, the second moment of area is

simply (values provided in Table 4-1):

Figure 4-8: Variation of the peak power (left) and the peak power density (right) with harvester aspect ratio. Open circles represent the set of short harvesters whereas filled circles represent the set of long harvesters.

82

𝐽 =1

3 𝑏 𝑙3 (4.13)

where 𝑏 and 𝑙 are the cantilever width and length, respectively. Note that the second moment

of area increases gradually from harvester H1 to harvester H8 and allows discriminating

between harvesters H1 and H6 (same aspect ratio of about 3.5) and harvesters H2 and H8

(same aspect ratio of about 1.8). The peak power and the peak power density are presented

as functions of the second moment of area in Figure 4-8.

The data points in Figure 4-8 appear rather well separated and stratified as a function of the

excitation amplitude, and the features previously discussed are clearly captured. In particular,

for a given length, the existence of an optimum harvester width that maximizes the peak power

output is clearly recognizable because the trends in Figure 4-8 (left) mirror those observed in

Figure 4-5, and the power penalty arising from non-linear response at high excitation

amplitude is evident in Figure 4-8 (right) from the drop in peak power density of the long

harvesters in comparison with the short ones. Interestingly, there is a single value of the

second moment of area (H2 in this case) that provides a clear maximum for the power density.

Within the limits of the present study, therefore, the second moment of the area seems to

Figure 4-9: Variation of the peak power (left) and the peak power density (right) with harvester second moment of area. Open circles represent the set of short harvesters whereas filled circles represent the set of long harvesters

83

capture geometric effects quite well. The usefulness of the second moment of the area as a

geometric parameter has also been highlighted by Li et al. while analysing PVDF-based

cantilever harvesters under vortex shedding excitation [81].

84

5. Long-term Power Degradation Testing of Piezoelectric

Vibration Energy Harvesters for Low Frequency

Applications3

Given the current limited understanding of how piezoelectric bending harvesters perform when

operated for prolonged durations, this chapter aims to show how the power output from these

piezoelectric energy harvesting devices changes over time. This investigation is carried out

whilst focusing on the effect that the inclusion of a tip mass has on performance degradation.

This chapter adopts a purely experimental approach and provides results for the degradation

performance for three of the most used piezoelectric materials that can be adopted to realise

low-frequency vibration energy harvesting devices. The chapter also sheds the light on how

tuning the input frequency signal to match the changing device natural frequency could affect

performance.

5.1. Materials and Methods

5.1.1. Harvesting materials and configurations

The construction of the active and passive layers of the harvesting devices employed in this

study are schematically presented in Figure 5-1. Here, the three off-the-shelf piezoelectric

options is used: PVDF, MFC, and QP. Even though these piezoelectric alternatives are

significantly different in terms of mechanical and electrical properties, they are all designed for

energy harvesting, and are therefore comparable in this respect. The first harvester built is

based on polyvinylidene fluoride (PVDF) provided from TE Connectivity (/www.te.com/ ; model:

LDT2-028K/L; density: 𝜌 = 1780 kg/m3; Young’s modulus: 𝑌 = 2.3 GPa; piezo strain constant:

𝑑31 = 23x10-12 C/N; capacitance: 𝐶 = 2.85 nF). The second harvester built is based on Macro

Fiber Composite (MFC) provided from Smart Material Corp.(/www.smart-material.com/; model:

M5628-P2; density: 𝜌 = 5440 kg/m3; Young’s modulus: 𝑌 = 15.86 GPa; piezo strain constant:

3 Elements of this chapter is currently under review as a paper submitted to Engineering Research Express.

85

𝑑31 = -170x10-12 C/N; capacitance: 𝐶 = 113 nF). The third harvester built is based on lead

zirconate titanate PZT-5J wafer QuickPackTM (QP) from Mide Technology Corp. (/piezo.com/;

model: S128-J1FR-1808YB; density: 𝜌 = 7800 kg/m3; Young’s modulus: 𝑌 = 51 GPa; piezo

strain constant: 𝑑31 = -210x10-12 C/N; capacitance: 𝐶 = 100 nF). PVDF is a highly flexible

piezo-polymer, MFC is a moderately flexible piezo-ceramic, whereas QP is a realtively rigid

piezo-ceramic. This difference in rigidity between MFC and QP is in that MFC is composed of

piezoelectric fibers whereas QP is composed of piezoelectric sheets. PVDF has silver ink

screen printed electrodes, MFC has interdigitated patterned electrodes, and QP has copper

electrodes. There is quite a recognisable difference in the mechanical properties of the three

materials with PVDF having the lowest density and Young’s modulus whereas QP has the

highest density and Young’s Modulus. This translates to a difference in rigidity giving PVDF

the potential to realise harvesting devices for lower frequency applications (due to the

expected lower natural frequency), and vice versa with respect to QP harvesters. Additionally,

the electric properties of the three materials are quite different with MFC and QP having much

higher values for capacitance and piezo strain constant in 3-1 mode. This leads to differences

in the ability to generate electric power when exposed to vibration in the 3-1 mode.

The QP device used as part of these tests comes readily assembled from the manufacturer

and involves one piezo layer. As such, only one layer of MFC and PVDF is considered when

manufacturing these harvesting devices to ensure some sort of consistency. Because QP has

relatively higher stiffness there was no need to further stiffen it with a passive elastic layer. On

the other hand, both MFC and PVDF required stiffening to allow meaningful structures. Hence,

both MFC and PVDF were bonded to a passive elastic layer made of stainless steel provided

by Precision Brand (/precisionbrand.com/; density: 𝜌𝑒 = 7900 kg/m3; Young’s modulus: 𝑌𝑒 =

180 GPa; thickness: ℎ𝑒 = 0.1 mm). The elastic layer was attached to both the MFC and PVDF

using a very thin layer of epoxy of negligible thickness but sufficient to ensure effective

bonding. As shown in Figure 1, the final configuration for the MFC and PVDF harvesters is

essentially a unimorph configuration (one active piezo layer and one passive elastic layer), a

86

configuration widely employed in piezoelectric devices [65], [74]. On the other hand, the QP

harvester has also one active piezo layer but wrapped in a passive epoxy resin, and no added

passive elastic layer. Note that epoxy is not equally distributed on both sides of the active

layer. In fact, there is more epoxy to one side than the other allowing the QP harvester to have

strong resemblance to a unimporh configuration. Even though the QP harvester is not, strictly

speaking, of unimorph design, all harvesters considered here have a similar configuration (i.e.

one active piezo layer and a passive substrate), similar enough to allow a meaningful

comparison.

Figure 5-1: Schematic diagram of the configuration of the passive and active layers construction for the three harvesting devices employed in the current study. (a) PVDF; (b) MFC; and (c) QP. Thicknesses are not drawn to scale and are exaggerated for better visualization of the components. In MFC, polyimide film contains interdigitated patterned electrodes but are not shown in the side view for simplicity. Also, layers of structural epoxy sandwich the PZT-5A1 fibers as shown but structural epoxy additionally exists between fibers.

87

The developed devices were clamped in a custom-made laser-cut acrylic housing (see Figure

5-2) to allow rigid attachment to the shaking device. Note that the portion of the harvester

enclosed within the clamp is strictly fixed, does not deform, and therefore does not produce

any power. To investigate the effect of tip mass inclusion, set amounts of mass were added

to the tip of each device. The value of tip mass to be attached was evaluated based on the

required ratio of the tip mass to the equivalent mass of the harvester beam. The equivalent

mass of the harvester beam was evaluated based on the Rayleigh-Ritz approach in which the

total mass of the cantilever beam is scaled by a factor of 33/140. This is essentially an

idealisation of the harvester beam as a mass/spring system that allows easy

representation/modulation of the device natural frequency due to the addition of a tip mass. In

fact, this method is well adopted by manufacturers such as Mide Technology Corp.

(manufacturers for QPs) who provide applicable data up to a tip mass ratio of 50 [82]. In this

study, tip mass ratios of 10, 20 and 30 were chosen to cover a range of values that may be

used in realistic operations.

88

Metallic tip masses with the required weight were firmly attached to both sides of the device

in a symmetric fashion. Each tip mass was ensured to extend along the entire width of the

beam. The centre line of each tip mass (on each side) was set to be coincident with the tip

edge of the beam. High-strength adhesive tape was used to ensure tip masses fixation during

testing. Figure 5-2 shows a selection of devices with and without affixed tip masses.

5.1.2. Experimental setup

Figure 5-3 shows the experimental setup used during testing. A signal generator (by Tektronix,

model AFG1022) was used to produce a sinusoidal signal sent to the shaker (by Data Physics,

model V55) via an amplifier (by Data Physics, model PA300E). Each energy harvester tested

was firmly fixed to the shaker unit and was excited at its base. An accelerometer (by PCB

Piezotronics, model PCB 336M13) was fixed close to the base excitation point of the

harvesters to monitor the vibrational acceleration where this signal was amplified and then

Figure 5-2: Harvesting devices attached to a custom-made acrylic clamp. (a) Devices with no tip mass; (b) and (c) example devices with tip mass set at different angles for better visualisation.

89

measured. Each harvesting device was connected to its corresponding optimum load

resistance value. A thermocouple (by RS Components, model RS PRO Type K) was used to

monitor the ambient temperature in the testing environment. The temperature and

accelerometer readings as well as the voltage across the resistance were all recorded using

a Data Acquisition unit (by National Instruments, model NI-USB-6225). This unit in turn fed

the values to a computer running LabVIEW 2017 for real time results monitoring and recording.

The power from the device was calculated in LabVIEW using the measured voltage together

with the load resistance value. Finally, a cooler unit was used to avoid overheating the shaker

unit during the long operational times.

5.1.3. Testing conditions

The conditions for testing were designed to match, as much as possible, realistic scenarios. It

is well documented that a frequency range of up to 200 Hz with an acceleration up to 1.5 g

represent the operational range for most sources of ambient vibration [15]. Without some sort

of active control, energy harvesting devices are typically designed for operation at targeted

frequencies. Indeed, PVDF, MFC, and QP harvesters each have different mechanical

properties making them suitable for harvesting energy at different frequencies. With no tip

mass, the length dimension was used to tune the first natural frequency at 50, 110, and 150

Hz, for the PVDF, MFC, and QP harvesters, respectively. Note that, the PVDF harvester is

designed for the lowest frequency because it has the lowest stiffness (highest flexibility),

whereas the QP harvester is designed for the highest frequency because it has the highest

Figure 5-4: Experimental Setup used in the current study.

90

stiffness (lowest flexibility). The values of 50, 110 and 150 Hz represent good base line values

which, with the addition of tip masses, will be lowered 10−40 Hz: a range representative of

low-frequency ambient vibrations. Table tab1 shows the dimensions of the realised harvesting

devices for testing. As explained in Section 5.1.1, tip masses were varied to achieve tip mass

ratios of 10, 20, and 30. To produce a meaningful and easy to follow designation for each test

case, the tested devices are named as Material–Tip Mass Ratio. For example, PVDF-20 thus

indicates the test case for the harvester comprising PVDF and having a tip mass ratio of 20.

Table 5-1: Geometric characteristics of the developed harvesting devices to achieve desired natural frequency.

PVDF MFC QP

Device

(including

passive and

active layers)

Length (mm) 52 46 47

Width (mm) 17 32 26

Thickness (mm) 0.42 0.52 0.69

Volume (mm3) 0.38 0.75 0.82

Piezo

Element

(active layer

only)

Length (mm) 50 44 39

Width (mm) 13 28 21

Thickness (mm) 0.040 0.30 0.69

Volume (mm3) 0.027 0.38 0.56

Natural frequency (Hz)

(without tip mass) 50 110 150

The decision was made to keep the base excitation acceleration level constant at 1g during

the tests. Two reasons were behind this selection: firstly, it is a representative value for typical

excitation level from ambient vibrations; and secondly, this allows all measured quantities to

be regarded as per unit g excitation. Table 5-2 shows the standard deviation values of the

excitation level from the target mean 1g value for the tests conducted. The mean and

maximum standard deviation values for the nine tests is 1.5% and 4.7%, respectively. These

values confirm that the excitation acceleration remained constant at 1g to within a few percent

throughout the tests.

91

Table 5-2 provides the settings for the set of tests conducted. In this table, the value of the

optimum resistance that would allow the maximum power generation is provided. This value

was empirically determined through conducting an experimental power scan with different

resistance values and identifying the optimum resistance value that would allow maximum

power generation. At the start of each test the natural frequency and optimum load resistance

of each harvester were measured. As discussed later, this was repeated after the testing was

concluded to assess any variation of the natural frequency and optimum load resistance that

may have arisen from change in device properties due to prolonged operation. Note that the

natural frequency and optimum load resistance values included in Table 5-2 are those

measured at the start of the tests. Finally, it is important to stress that, for each test case, a

brand-new piezoelectric layer/material was used to ensure no previous hysteresis, memory,

or fatigue effects were present.

Table 5-2: Testing conditions for the current experiment.

Excitation Standard

Deviation

(%)

Harvester Natural

Frequency

(Hz)

Optimum Load

Resistance

(kOhm)

Total Run Time

(hours)

0.32 PVDF-10 17 2300 50

0.52 PVDF-20 11 1700 76

4.7 PVDF-30 10 2000 84

0.61 MFC-10 24 34 35

0.76 MFC-20 20 37 42

0.31 MFC-30 12 43 70

0.59 QP-10 40 11 21

1.2 QP-20 30 12 28

4.4 QP-30 25 15 34

The testing of the harvesters was performed under single-frequency excitation. At the

beginning of each test the excitation frequency of the shaker was tuned to the resonant

frequency of the harvester in Table 5-2 and was not changed afterwards. To thoroughly

examine long-term effects, instead of running the tests for a specific length of time, the tests

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were run for a predefined number of vibrational cycles to allow better comparison between

devices. Tests were run for three million cycles representing a good compromise between

being sufficient timing to identify if degradation effects exist and allowing good use of testing

time. Note that this represents a significant increase of testing times compared to some of the

previous studies that conducted test on the order of thousand cycles, e.g. [55] . Tables 5-2

provides the corresponding total run time for each test case. As can be seen, the PVFD-30

test case required the longest testing time of 3.5 days, whilst the testing of the harvesters QP-

10 lasted for 21 hours. The total cumulative testing time for all harvesters was 440 hours,

which corresponds to around 18.3 continuous operation days.

As noted by Kim et al. [83], a change in ambient temperature can cause a change in power

output of a piezoelectric device. In particular, they found that the power output from PZT

decreased with increasing temperature, and that the temperature effect becomes most

impactful when over 40°C approximately. The ambient temperature was recorded during the

present tests, and the results are provided in Table 5-3.

Table 5-3: Temperature variation (in °C) during testing.

Tip Mass Ratio PVDF MFC QP

10 27.7±0.7 23.7±0.3 25.3±0.4

20 27.0±0.4 25.1±0.5 25.2±0.3

30 32.0±1.2 26.5±0.3 25.0±0.4

As can be seen, the ambient temperature during each test remained constant to within 1.2°C.

Any effects of the operating temperature on the present harvesters’ performance, therefore,

should be minimal and negligible.

5.1.4. Data processing

The power and acceleration signals from each test were recorded every 1000 cycles. For the

test with the greatest vibration frequency (40 Hz) this corresponds to a sampling frequency of

0.04 Hz. For the test with the lowest vibration frequency (10 Hz) this corresponds to a sampling

frequency of 0.01 Hz. Since the main focus here is the long-term variation of the power, and

93

not its short-term fluctuation, the recorder time-series were smoothed to remove the short-

term fluctuations and better expose the underlying long-term trends. Operatively, this was

achieved using MATLAB built-in functions (the function ‘smoothdata’, implemented with

default settings). A representative example of a power time-series smoothing is provided in

Figure 5-4.

To better compare data from the different harvesting devices, the recorded power signals were

normalised with respect to the maximum power value measured during the corresponding test:

𝑃𝑛𝑜𝑟𝑚,1(𝑡) =𝑃(𝑡)

𝑚𝑎𝑥(𝑃(𝑡)) (5.1)

This definition is used to compare the trends rather than the amplitudes of the harvested power.

This is deemed useful as the expected power amplitudes will vary significantly between the

different devices due to their significantly different mechanical and electrical properties

discussed in Section 5.1.1.

Figure 5-5: An example of power time-series smoothing (case of MFC-10): the red line is the original time-series, whilst the blue line is the smoothed time-series.

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5.2. Results and Discussions

5.2.1. Power degradation

Piezoelectric vibration energy harvesters, as noted previously, are electromechanical systems;

so that any degradation in their performance can be attributed to a degradation of their

mechanical properties, a degradation of their electrical properties, or a combination of both. In

all the tests, natural frequency and optimum load resistance were measured before and after

each run, so that their overall variation during the tests can be assessed. The change in natural

frequency can be regarded as indicative of the change in mechanical properties of the

harvester. The optimum load resistance, on the other hand, depends on the mechanical

(frequency and damping) and electrical (capacitance) properties of the harvester, so that any

variation can capture both mechanical and electrical degradation effects. Even though it is

clearly not possible to separate the effects, monitoring the variations of natural frequency and

optimum load resistance can provide useful indications on the variation of the underlying

mechanical and/or electrical properties of the harvesters. In what follows, therefore, the

recorded changes in natural frequency and optimum load resistance are provided together

with the power time-series to help interpret the observed trends. Figure 5-5 shows the change

in natural frequency and optimum load resistance measured at the end of each test as a

percentage of the initial values measured at the start of the test.

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As can be seen in Figure 5-5, the PVDF harvesters do not show any detectable variation in

natural frequency, whereas a decrease in natural frequency on the order of 3-9% is observed

with the MFC-10, MFC-20, QP-20 and QP-30 harvesters. On the other hand, all harvesters

except the MFC-10 experience a variation in optimum load resistance, which increases in all

cases except for the harvester PVDF-30.

Figure 5-6 shows the output power harvested during the tests, whilst summarizing power

figures are provided in Table 5-4 and Figure 5-7. As can be noted, the harvested power levels

from PVDF cases is the lowest (on the order of 0.1 mW), the harvested power levels from

MFC cases is about an order of magnitude higher (on the order of 1 mW), and the harvested

power levels from QP cases is another order of magnitude higher (on the order of 10 mW).

This is expected due to the differences in mechanical and electrical characteristics of the

harvesters discussed in Section 5.1.1.

Figure 5-6: Change in (a) natural frequency, and (b) optimum load resistance for the test cases considered in this study.

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Table 5-4: Variations of the harvested power.

Test Power average

over test

(mW)

Power value at

start of test

(mW)

Power value at

end of test

(mW)

% change (start

to end)

PVDF-10 0.047 0.053 0.044 -19%

PVDF-20 0.076 0.077 0.079 1.9%

PVDF-30 0.078 0.070 0.081 17%

MFC-10 0.55 0.59 0.54 -7.7%

MFC-20 0.65 1.2 0.57 -53%

MFC-30 3.3 3.2 3.4 5.3%

QP-10 12 13 11 -11%

QP-20 11 14 9.6 -31%

QP-30 8.0 13 5.7 -56%

Figure 5-8: Power output (top) and normalized power output (bottom) recorded during testing for the present harvesters.

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For the PVDF harvesters, the variation in power output is evident although rather limited in

magnitude (within ±19%). Since for the PVDF harvesters there is no significant variation in

natural frequency, the observed variation of power output can be traced back to the variation

in optimum load resistance. The harvester PVDF-20 experiences the lowest variation in load

resistance (+6%), and correspondingly presents the more stable power output during the test.

The harvesters PVDF-10 and PVDF-30 experience larger variations in load resistance (+17%

and -20%, respectively), and their power output profiles correspondingly show more

pronounced variations. The power trends are opposite: decreasing for PVDF-10 and

increasing for PVDF-30, and this may be the consequence of the opposite variation of the

corresponding load resistance, which increases for PVDF-10 and decreases for PVDF-1-30.

For the PVDF harvesters, therefore, the observed variation in power output seems to reflect

the variation in the optimum load resistance.

For the MFC harvesters, the most pronounced variation in power output is observed for MFC-

20. For this harvester, there is a variation in both the natural frequency (-5%) and optimum

load resistance (+5%). For the harvester MFC-10 there is a variation in natural frequency (-

4%) but no detectable variation in optimum load resistance, and the corresponding power

output variation is quite limited. Similarly, for the harvester MFC-30 there is a variation in

optimum load resistance (+7%) but no detectable variation in natural frequency, and the power

output variation is again quite limited. For the MFC harvesters, therefore, the observed

variation in power output seems to reflect the variations in the natural frequency and in the

optimum load resistance: a larger cumulative variation in natural frequency and optimum load

resistance yields a larger variation in power output.

For the QP-10 harvester there is a variation in optimum load resistance (+10%) but no

detectable variation in natural frequency, for the QP-20 harvester there is a variation in both

the natural frequency (-3%) and optimum load resistance (+34%), and for the QP-30 harvester

there is a more pronounced variation in both the natural frequency (-9%) and optimum load

resistance (+37%). The variation in power output of the QP harvesters is milder for QP-10,

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more pronounced for QP-20 and even more so for QP-30, indicating that a larger cumulative

variation in natural frequency and optimum load resistance yields a larger variation in power

output.

In conclusion, therefore, for all harvesters the variation in power output can be traced back to

the variations in natural frequency and optimum load resistance. As can be noted in Table 5-

4 and Figure 5-7, the power output in most tested cases decreases with time, consistently

indicating a degradation in performance as consequence of the degradation in mechanical

and/or piezoelectric properties. Importantly, by comparing the average power values it is

evident that increasing the tip mass does not necessarily increases the average power output.

This is particularly clear for the QP harvesters and suggests that a larger tip mass might

exacerbate the degradation of the mechanical and/or piezoelectric properties of the harvester,

therefore reducing the power gain that would in principle be expected from having a bigger tip

mass.

Figure 5-9: Comparison of initial and final power output from the harvesters. (a) Normal scale, and (b) log-scale to better compare orders of magnitude.

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5.2.2. Power density

Given that the sizes of the employed devices have significant differences, power generation

as an absolute measure does not represent a fair comparison metric. As such, power density

is defined as the averaged output power over the testing period divided by either the total

volume of the harvesting device or by only the piezoelectric layer volume (also see Table 5-

1). Another important metric introduced here is the power density to material cost ratio, a

metric that could be used to assess power generation cost of the three material options

presented in this study. The power density to material cost ratio was obtained by dividing the

power density by the cost of a single unit. Note that, the prices used to compare the devices

were acquired directly from the manufacturers based on 2019 pricing, and were taken from

the cost of a single harvesting device; any materials used in making the device other than the

active piezoelectric layer, such as the steel shim and tip masses, were not accounted for in

estimating the cost. Using both power density and power density to cost ratio is believed to be

very helpful in deciding which devices to use for a given need. Here, the comparisons provided

consider degradation effects, thus providing a more realistic comparison of devices’

performance. The results are provided in Table 5-5 and Figure 5-8.

Table 5-5: Comparison of power density and power density to cost ratio.

Test Power Density

(mW/cm3)

Power Density to cost ratio (mW/cm3$)

Based on total

harvester volume

Based on piezo

layer volume

Based on total

harvester volume

Based on piezo

layer volume

PVDF-10 0.12 1.7 0.011 0.16

PVDF-20 0.21 3.0 0.021 0.30

PVDF-30 0.22 3.1 0.021 0.31

MFC-10 0.73 1.4 0.0069 0.014

MFC-20 0.81 1.6 0.0077 0.015

MFC-30 4.5 9.0 0.043 0.086

QP-10 14 20 0.12 0.17

QP-20 12 17 0.098 0.14

QP-30 7.0 10 0.058 0.084

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As can be seen from Table 5-5 and Figure 5-8, whilst the highest power density is achieved

with the QP harvesters, the highest power density to cost ratio is achieved with the PVDF

harvesters, indicating that these latter are cost-effective despite the lower power output. Within

the tests, degradation effects were clearly seen for all QP cases. Obviously, QP tests showed

that as tip mass ratio increased the power density and power density to cost ratio decreased

due to increasing degradation effects. Despite this, QP showed the best power density for

given tip mass ratios, Figure 5-8. MFC cases showed greater power density values based on

the total volume definition when compared to the PVDF cases even for the MFC-20 case

which showed significant degradation. Nevertheless, PVDF cases showed much better power

Figure 5-10: Comparison of the (a) power density, and (b) power density to cost ratio for tests conducted. (c) and (d) are the same plots as (a) and (b) but on a log-scale for better visualisation of the orders of magnitude.

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density values when evaluated based on the volume of the piezoelectric layer only. Neither

the MFC nor the PVDF harvesters were able to show a clear superiority in power density when

based on the active layer volume, with the PVDF-10, and PVDF-20 cases showing better

values whereas the MFC-30 case showing higher power density compared to its

corresponding PVDF case. MFC cases showed the least values of power density to cost ratio,

indicating they are the least favourable choice when cost is considered.

5.2.3. Active tuning of natural frequency

Since the power degradation can be traced back to the variations of natural frequency and

optimum load resistance, an obvious compensation strategy would be to actively tune the

natural frequency and load resistance during operation. A preliminary assessment of the

feasibility and potential of active frequency tuning during operation is presented below. Figure

5-9 shows the excitation frequency and power output of harvester QP-30, which, as previously

discussed, was tested under single-frequency constant excitation (note: this is the same result

shown in Figure 5-6). Also included in Figure 5-9 are the measurements obtained for QP-30

when tested whilst actively tuning the excitation frequency to match the changing natural

frequency. In this latter test, both the power output and the natural frequency were measured

periodically. When a shift of at least 1 Hz was measured for the natural frequency, the input

frequency to the shaker was manually adjusted to compensate for the measured shift (manual

adjustments are highlighted with a red marker in Figure 5-9).

As can be noted, at first the active frequency tuning is beneficial in terms of harvested power.

This is no longer the case towards the end of the test, however, and the harvester broke at

about 200 min. Even though the results in Figure 5-9 are no more than a preliminary proof of

concept, they seem to indicate that active frequency tuning may be beneficial for energy

harvesting in the short term, whereas in the long term it may exacerbate the mechanical

degradation of the harvester thereby shortening its mechanical life. Clearly, active frequency

(and electrical load) tuning should be properly investigated, but this goes beyond the scope of

this study.

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Figure 5-11: Results for active natural frequency tuning case of the QP-30 device showing change in natural frequency, and power output with reference to corresponding initial measured value. Red markers denote instants where tuning of the input signal was conducted to match the shift in natural frequency. Due to the variation in input signal frequency, x-axis is shown in time rather than vibration cycle.

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6. Conclusions and Future Work

This chapter will summary the main conclusions from Chapters 3, 4, and 5, as well as providing

recommendations for future directions of research.

6.1. Solar Panels as Tip Masses (Chapter 3)

Chapter 3 managed to show the effective use of solar panels as active energy harvesting tip

masses for PVDF-based vibration energy harvesters suitable for low frequency applications.

A total of eight PVDF energy harvesters have been realized allowing a repetitive assessment

of the role of solar panels, together with providing preliminary insights into the effect of

changing the length and width on the harvesters’ performance. The dynamics of the harvesters

was assessed through measuring the frequency response of the tip displacement whereas

the PVDF power generation was assessed through measuring the power frequency response.

Peak power generation from the solar panels employed in this work over a range of practical

light illuminations was also measured and reported. Two electromechanical distributed-

parameter models have been developed. The first models the solar panels as a concentrated

tip mass and is more representative of thin, flexible solar panels. Indeed, the model

demonstrated good predictive capabilities vs measured dynamics and PVDF power

generation responses from harvesters tested in this work with flexible solar panels. The

second model represented the solar panels as rigid distributed tip masses and was developed

to investigate the potential loss of power by restricting tip motion, since for up-scaling, larger

solar panels may be rigid.

The experimental results obtained confirmed the ability of the solar panels employed in this

study to modulate the operation frequency and increase the power generation from the PVDF

elements. Moreover, solar panels were capable of generating additional harvested power from

ambient lighting that can significantly boost the total power generation from the harvester. The

theoretical models employed were instrumental in exploring how tip mass configurations could

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be optimised to improve harvesting performance. In terms of the optimum length of the solar

panel, the concentrated tip mass model showed that having flexible solar panels that are two

thirds of the total length will maximise the power output irrespective to the tip mass ratio. The

distributed tip mass model, on the other hand, showed that the optimal length for rigid solar

panels length increases as the tip mass ratio increases, approaching a value that is half of the

total length for significantly high tip mass ratios.

In summary, solar panels offer a potential design improvement allowing better performance

both for the dynamics and power generation performances of PVDF-based energy harvesters.

This funding will be contributed to further development of hybrid, multi-source energy

harvester field. There are increasing number of publications of hybrid, multi-source energy

harvesters in recent years, such as study of piezoelectric-electromagnetic based hybrid

harvester by Fan et al. [84]. In their design, a cuboidal magnet applied as tip mass, and

resulted efficiency improvement up to 14% compared to signal piezoelectric part [84]. Similar

intergraded structure also conducted by Hu et al., it reports greatly power output with harvest

energy from vibration and magnetic simultaneously [85]. Lu et al. released a design of an

umbrella that collecting energy by using photovoltaic, piezoelectric, electromagnetic, and radio

frequency [87]. Novel harvester, in future, can be developed integrated on many material and

structure, whereas solar panel with piezoelectric is just one combination among all the

possible configurations. With more attentions received on hybridization and mature research

on single energy harvester, the feasibility of multi-source harvester will be improved,

particularly benefit industrial applications, such as self-power sensing systems of self-charging

energy stores [88].However, it is recommended to carry out further future investigations, since

the study was limited to available off-the-shelf components in the market. The possibility of

having the metal shim, the PVDF elements and the solar panels custom-made to specific

desired dimensions and material properties will be useful in future optimization to better

understand the parameter space and to tailor their mechanical properties to the intended

applications.

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6.2. Planform Geometry and Excitation Effects (Chapter 4)

A systematic experimental investigation into the effect of planform geometry and excitation

level is conducted on the energy harvesting performance of PVDF-based low frequency

vibration energy harvesters. Eight PVDF-based energy harvesters (with two different lengths

and four different widths) have been realized and tested over a sufficient range of excitation

frequencies around the first resonance bending frequency, and this was repeated for different

base excitation acceleration levels ranging from 0.2g up to 0.6g. All harvesters had the same

cross sectional geometric and material properties. This provided sufficient data to provide

preliminary insights into the planform and excitation effects on tip displacement dynamics and

power generation performance. A linear Euler-Bernoulli cantilever beam model was also

developed for the harvester configuration employed for use as an identification tool of the

presence / absence of non-linear effects within the measured responses.

A number of useful insights were obtained / confirmed. It was found that the short set of

harvesters has the tip displacement and power responses behaving in a linear fashion with

the system identification experiments showing nearly constant damping values that are

relatively larger in value (compared to the longer harvester set) leading to broader frequency

response functions. The longer set of harvesters, on the other hand, showed displacement

and power responses that behaved in a non-linear fashion, and this was attributed mainly to

the experimentally identified non-linear variations of the elastic restoring force and damping

ratios at the much higher tip displacement values experienced by these harvesters. For both

sets of harvesters, it was confirmed that the displacement response is independent of width

whereas the power values have clear dependency on the width value. The results showed

that there is a width value that allows a maximum peak power generation and that this

maximum gradually shifts to higher width values as the excitation level is gradually increased.

It was also shown that longer harvesters operating at high excitation levels experience

significant non-linear responses which hinder the full exploitation of their active volume leading

to power density values that can deteriorate to up to half of the power density values of the

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corresponding shorter harvester operating in the linear domain. Finally, it was shown that the

second moment of could be used as an independent variable that can be used to optimize for

the power density performance.

In summary, the link between the planform geometry, excitation level, and harvesting

performance metrics (displacement and power) is assessed. However, a non-linear Euler-

Bernoulli cantilever beam model to model the cases where the harvesters were behaving in a

non-linear fashion is currently missing. Cantilever-based harvesters are characterized by its

high flexibility and wide range of deformation, from small up to large values relative to its length.

As will be shown in this study, this enables a more comprehensive analysis of the planform

and excitation effects, whilst considering both linear and non-linear responses. Jiang et al.

fabricated and tested a harvester by laminating one PVDF layer with a polyester layer [89].

Rammohan et al. constructed an array of three bimorph harvesters, each comprising a copper

foil between two PVDF layers [90]. Song et al. reported a bimorph harvester comprising two

PVDF films bonded together with an adhesive layer and a load mass at the free end of the

beam [91]. Tsukamoto et al. tested a bimorph harvester comprising a flexible 3D meshed-core

elastic layer sandwiched between two PVDF layers [92]. Chandwani et al. investigated multi-

band harvesters, measuring an average power of 6μW for the frequency band 21-35Hz and

an average 100 power of 7.7μW for the frequency band 45-60Hz [93]. As prior investigations,

it clearly demonstrate the potential of PVDF-based vibration energy harvesters of cantilever

design for low frequency applications. A systematic study of the influence of planform

geometry and excitation input remains absent from the literature, to demonstrate in detail how

these factors affect the dynamics and power generation. As such, this is an important direction

to be pursued in future work. Developing such model will enable better evaluation for the

boundary between linear and non-linear behaviour and will provide a useful tool for sizing this

class of harvesters against different objectives suitable for different applications. In fact,

studies of non-linear behaviours of similar harvester configurations are limited for different

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base excitation range, especially within the range of 0.2 - 0.6 g covered in this study, providing

an interesting research orientation and motivation to tackle this issue in the future.

6.3. Long-term Power Degradation Effects (Chapter 5)

The power degradation performance over prolonged operation of piezoelectric vibration

energy harvesters realized is analysed and compared using PVDF, MFC, and QP: three of

the most frequently used piezoelectric materials in energy harvesting applications. Overall,

the nine Piezoelectric vibration energy harvesters were realised and tested: three different

piezoelectric options (PVDF, MFC, and QP) and three different tip mass ratios (tip mass ratios

of 10, 20, and 30). The harvesters, unimorphs of cantilever beam geometry configuration,

were tested under single-frequency excitation for three million vibration cycles, observing

power degradation that ranged from a few percentages up to 60%. During testing, the

excitation amplitude was kept constant at 1g, whilst tip masses were added to the harvester

so that the vibration frequencies varied within 10−40 Hz: a range representative of low-

frequency vibration applications. The observed variations in harvested power reflect the

variations in natural frequency and optimum load resistance of the harvester, thereby linking

the degradation in performance to the degradation in mechanical and/or electrical properties

of the harvester. Tip masses have a direct effect on the mechanical and/or piezoelectric

properties of the harvester, therefore affecting the power gain that would in principle be

expected from having a bigger tip mass. Whilst the highest power density is achieved with the

QP harvesters, the highest power density to cost ratio is achieved with the PVDF harvesters,

indicating that these latter are cost-effective despite the lower power output.

Overall, experimental studies have shown that piezoelectric devices do show degradation

under different loading conditions. This is same output as report from Pillatsch et al. [51],

Selten et al. [52] and Soadano [58, 59], which are more detailed reviewed in section 2 of this

report. The mechanism reasons behind the degradation can be mechanical (stiffness) and

electrical (impedance) properties based on the study from Pillatsch et al. [51], which request

108

further study, but out of scope at this stage. Although, this study systemically showed the

performance degradation within three of the main commercially available piezoelectric devices,

there are other piezoelectric materials that could be investigated in the future. Especially, the

further investigation is necessary to conduct with characteristics of material-wise, in order to

give comprehensive explanation of degradation. Corresponding with the study of Tai and Kim

[48] and Cain et al [49], another future work direction is to attempt to develop useful models

to predict the degradation performance for this class of harvesters. Although, this study

systemically showed the performance degradation within three of the main commercially

available piezoelectric devices, there are other piezoelectric materials that could be

investigated in the future. Another future work direction is to attempt to develop useful models

to predict the degradation performance for this class of harvesters. This is believed to be a

very useful asset for designers and users when deciding on the most suitable

configuration/material for different applications. Finally, future research should also conduct

further investigations on the highlighted potential strategies to mitigate power degradation of

these harvesters through active tuning of the natural frequency and/or electrical load during

operation.

6.4. Summary

Piezoelectric energy harvester is explored during a style of fields within the past decade, as

well as wearable devices, medical implants, vehicles, and wireless device networks. For the

practical power scavenge from surrounding environment, targeting resonance frequency and

expending the frequency range are significant criteria as vibration energy harvester. This

made supply of energy is accessible in most of the industry plants, machinery, vehicles of

every kind, and machinery. The amount of generated power from these structures powerfully

depends on the various factors together with the applied load, the frequency of vibration, the

geometric options, and also the boundary conditions. As vibration based, cantilever energy

harvesters were first studied to power tire pressure monitoring system and assembled on the

tire to harvest mechanical energy [94]. Following the research effort, the further application of

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cantilever harvester was proposed as energy harvester from such as tread wall, building &

bridge oscillations [94]. One classic design for harvester energy from tire was made of

piezoelectric based plate loading with tip mass [94]. Inspiring by the findings of this study,

optimal planform geometry and solar panel as active tip mass potentially to be applied and will

improve the performance ideally. Finally, the third important outcome indicated the direct effect

on the mechanical and piezoelectric properties of harvester. So, it is highly suggested

researchers consider potential power degradation effect while investing of tip mass, which is

critical for realization of harvester in real world.

Piezoelectric energy harvesters represent a viable and well-proven solution to convert ambient

vibrations into useful electric power for modern life applications, particularly as promising

solution for widespread use of wireless sensors in remote locations. With a large amount of

studies has focused on improving power output from these device design, there are many

directions to investigate. In this MPhil study, three novel topics are conducted; 1) assess the

inclusion of solar panels as active tip masses on the dynamics and power generation

performance of cantilevered PVDF (polyvinylidene fluoride)-based vibration energy harvesters;

2) a systematic investigation of planform geometry and excitation level effects on the dynamics

and power generation characteristics of PVDF-based cantilevered vibration energy harvesters;

3) experimentally investigate how piezoelectric vibration energy harvesters degrade during

long-term operation in realistic harvesting conditions; Correspondingly, a number of funding

are obtained as discussed above. All the insights of this study are significant important for low

frequent vibrations applications and can open door for further investigation and uptake of thoes

metric / aspects when assessing harvester performance.

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