keith hogan final year project pdf

______________________________________________

NAME: K. HOGAN

I.D: 11128704

SUPERVISOR: Dr. IAN CLANCY

COURSE: B.Sc. ENERGY

YEAR: FOURTH YEAR

PROJECT TITLE: MODELING

VIBRATIONAL

ENERGY HARVESTING

IN A NONLINEAR

SYSTEM

DATE: 26th March 2015

______________________________________________

11128704 Energy Harvesting

i

Acknowledgements

I would like to acknowledge my supervisor Dr. Ian Clancy for his help and guidance

throughout this project. I also wish to acknowledge my family for their support.


ii

Abstract

The development of efficient methods of powering small scale devices is an on-going

problem. One avenue of research is harvesting electrical energy derived from mechanical

vibrations due to the abundant sources in the environment. This type of energy can produce

the desired voltage to power any low power device or wireless sensor. Currently many of

these devices rely on linear resonators that respond to a narrow band of excitation

frequencies. These linear systems exhibit modes for vibration that do not permit energy

associated with one mode to excite another mode.

This project involves modelling the behaviour of nonlinear oscillators where the potential is

distinct from a linear system’s quadratic potential. The work consisted of creating a nonlinear

harvesting system using MatLAB software and hence seeking an optimum value for this

nonlinearity when driven at various modes. It was discovered that a nonlinear system has an

optimum value for nonlinearity across all frequencies and that there is a limit to the

nonlinearity that can be in a system. The nonlinear properties investigated in this project

compares favourably with what has been found in previous literature.


iii

Table of Contents Acknowledgements ..................................................................................................................... i

Abstract ...................................................................................................................................... ii

Nomenclature ............................................................................................................................ ix

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

2 Literature Review ............................................................................................................... 4

2.1 Energy Harvesters ...................................................................................................... 5

2.1.1 Introduction ........................................................................................................ 5

2.1.2 Energy Harvesting Strategies ............................................................................. 5

2.1.3 Harvesting Energy using the Piezoelectric Effect .............................................. 6

2.1.4 Harvesting Losses & Efficiency......................................................................... 9

2.1.5 Harvester Sizing & Design............................................................................... 10

2.2 Oscillators ................................................................................................................ 11

2.3 Transducers .............................................................................................................. 13

2.4 Nonlinear Systems ................................................................................................... 15

3 Experimental Work .......................................................................................................... 17

3.1 Oscillator Establishment Methods & Equations ...................................................... 17

3.1.1 Euler ................................................................................................................. 17

3.1.2 Euler Cromer .................................................................................................... 18

3.1.3 Oscillator Equations ......................................................................................... 19

3.2 MatLAB ................................................................................................................... 19

3.2.1 Time Selection ................................................................................................. 20

4 Results, Analysis and Discussion ..................................................................................... 23

4.1 System Configuration .............................................................................................. 24

4.1.1 Oscillator Equations ......................................................................................... 24

4.1.2 Applying a Transducer ..................................................................................... 26

4.1.3 Damping the System ........................................................................................ 27

4.1.4 Driving the System ........................................................................................... 28

4.1.5 Nonlinear System ............................................................................................. 30

4.2 Methods .................................................................................................................... 33

4.2.1 Fast Fourier Transform (FFT) .......................................................................... 33

4.2.2 Steady state ...................................................................................................... 35

4.2.3 Total Power ...................................................................................................... 37

4.3 Investigations ........................................................................................................... 38

4.3.1 Random Noise Driving .................................................................................... 38

4.3.2 Varying Amplitude .......................................................................................... 41


iv

4.3.3 Finding the Optimum Nonlinearity .................................................................. 42

4.3.4 Transfer of Energy between two modes ........................................................... 45

4.3.5 Linear vs. Nonlinear ......................................................................................... 49

4.4 Discussion ................................................................................................................ 52

5 Conclusions ...................................................................................................................... 56

6 References ......................................................................................................................... xi

7 Appendix ......................................................................................................................... xiii


v

List of Figures

Figure 1: Simulated deflection of the cantilever in the time domain. The signals are

normalized to the maximum value of amplitude. The nonlinear evolution represents the

stochastic resonance condition. .................................................................................................. 3

Figure 2: Two modes of piezoelectric materials when used for vibration energy harvesting. ... 9

Figure 3: Cantilever & cymbal type ......................................................................................... 10

Figure 4: Graph showing the bistable states of a system ......................................................... 13

Figure 5: Examples of low profile transducer .......................................................................... 14

Figure 6: Schematic representation of piezoelectric energy harvesting ................................... 14

Figure 7: Euler Method showing an increasing sine wave ...................................................... 18

Figure 8: Euler-Cromer Method showing sine waves with consistent peaks .......................... 18

Figure 9: Graph of voltage versus time for a value of N equal to 1000 ................................... 21

Figure 10: Graph of voltage versus time for a value of N equal to 10000 ............................... 21

Figure 11: Graph of voltage versus time for a value of N equal to 100000 ............................. 22

Figure 12: Unchanging sine wave produced from the initial oscillator equations defined in

Equation 8 ................................................................................................................................ 25

Figure 13: Entire frequency range shown with peaks at the modes resulting from the initial

oscillator equations in Equation 8 and by using the parameter values stated in Table 2 ......... 25

Figure 14: Resulting graph from the modified oscillator equations when a transducer is added

as defined in Equation 9 ........................................................................................................... 27

Figure 15: Resulting graph from adding damping to the system as in Equation 10 showing

constant peaks for the sine waves ............................................................................................ 28

Figure 16: Resulting graph of Power(W) versus frequency ( ) when a frequency sweep

is performed as outlined in Equation 11 .................................................................................. 29

Figure 17: Graphs from driving the system with random noise as defined in Equation 12

which results in non-repeatable graphs .................................................................................... 29

Figure 18: Resulting graphs from driving the system at a particular frequency using Equation

13 which presents a peak at one mode which is repeatable ..................................................... 30

Figure 19: Resulting graphs from driving the system at a frequency of 4.1 without

nonlinearity resulting in a total power of 9.1113e-04 W as defined by Equation 14 ............... 31

Figure 20: Resulting graphs from driving the system at a frequency of 4.1 with

nonlinearity resulting in a total power of 8.1379e-04 W as outlined by Equation 14 ............. 31

Figure 21: Resulting graphs from driving the system at a frequency of 4.89 without

nonlinearity resulting in a total power of 0.0023 W using Equation 14 .................................. 32

Figure 22: Resulting graphs from driving the system at a frequency of 4.89 with

nonlinearity resulting in a total power of 0.0046 W as defined by Equation 14 ...................... 32

Figure 23: Graph showing frequency versus time for the system without implementing the

Nyquist Limit resulting in 'mirrored' peaks.............................................................................. 34

Figure 24: Graph showing frequency versus time for the system which implements the

Nyquist Limit resulting in a single set of peaks ....................................................................... 34

Figure 25: Graph showing the steady state occurrence for voltage ......................................... 35

Figure 26: The resulting graph of voltage versus time when the steady state values as defined

in Table 4 are adopted as the initial variables .......................................................................... 36

Figure 27: Result using MatLAB to retrieve the total power generated for a specific

simulation ................................................................................................................................. 37


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Figure 28: Graph showing the relationship between N and Peak 1 when driven with random

noise showing that no obvious correlation is produced ........................................................... 39

Figure 29: Graph showing the relationship between the nonlinearity and Peak 1 when driven

with random noise .................................................................................................................... 40

Figure 30: Graph showing the linear relationship between Amplitude squared and Total

Power when b = 0 .................................................................................................................... 41

Figure 31: Graph showing the nonlinear relationship between Amplitude squared and Total

Power when b = 2 which coincides with work performed by Triplett et al. (Triplett, 2009) ... 42

Figure 32: Graph showing the change in the value of peak 1 as the nonlinearity is changed,

resulting in an optimum of 0 when driven at the lower mode ................................................. 43

Figure 33: Graph showing the change in the value of peak 1 as the nonlinearity is changed,

resulting in an optimum greater than 17.32 which was the max value for which a graph was

returned when driven at the lower mode .................................................................................. 43

Figure 34: Graph showing the change in the value of total power as the nonlinearity is

changed, resulting in an optimum of 0 when driven at the lower mode .................................. 44

Figure 35: Graph showing an optimum value of b = 3 for total power when driven at the

higher mode ............................................................................................................................. 44

Figure 36: Graph showing an optimum value of b = 3 for peak 1 when driven at the higher

mode ......................................................................................................................................... 45

Figure 37: Graph showing the value of total power as the system is driven at various

frequencies with peaks evident at the lower and higher modes ............................................... 46

Figure 38: Graph showing gap in recorded data due to the energy in the peaks overlapping

resulting in just one peak ......................................................................................................... 46

Figure 39: Graph showing the energy in particle 2 which obtains energy from particle 2 in

between the two modes ............................................................................................................ 47

Figure 40: Initial small peak (red) at lower mode and slightly larger driven mode (blue) which

is increased ............................................................................................................................... 47

Figure 41: Driven mode continues to increase resulting in slight increase in lower mode ...... 48

Figure 42: When two peaks are close enough they combine to become one and at their largest

at the lower and higher modes ................................................................................................. 48

Figure 43: Increased driving frequency to higher mode results in corresponding increase in

power........................................................................................................................................ 48

Figure 44: As driving frequency is further increased, the single peak becomes two again as

two modes are resolved ............................................................................................................ 49

Figure 45: Driving frequency is so high that the power of the peaks reduces again ................ 49

Figure 46: Graph showing the effect of varying the nonlinearity and the driving frequency on

Peak 1 ....................................................................................................................................... 50


Peak 2 ....................................................................................................................................... 50


Total Power .............................................................................................................................. 51

Figure 49: MatLAB code created to model the system described in this project implementing

the variables outlined in Table 2 ............................................................................................. xiii

Figure 50: Analysis file which produces the graphs used throughout the project part one .... xiv

Figure 51: Analysis file which produces the graphs used throughout the project part two ..... xv

Figure 52: Graph showing the relationship between N and Peak 2 when driven with random

noise ........................................................................................................................................ xvi


vii

Figure 53: Graph showing the relationship between N and Peak Total Power when driven

with random noise ................................................................................................................... xvi

Figure 54: Graph showing the relationship between b and Peak 2 when driven with random

noise ....................................................................................................................................... xvii

Figure 55: Graph showing the relationship between b and Total Power when driven with

random noise .......................................................................................................................... xvii

List of Tables

Table 1: Sources of energy available in the surrounding which are/can be tapped for

generating electricity. ................................................................................................................. 1

Table 2: A list of the numerous parameter values implemented in this project, their name as

seen in the code, their symbol and the value assigned to each................................................. 23

Table 3: Results used to determine that steady state occurs after 200 seconds........................ 35

Table 4: The steady state values implemented in this project based on the findings of the

steady state investigation ......................................................................................................... 36

Table 5: Resulting values recorded when the system was driven with random noise and the

value of N was varied............................................................................................................... 38

Table 6: Resulting values recorded when the system was driven with random noise and the

value of b was varied ............................................................................................................... 39

Table 7: Table showing how the average and standard deviations for the variables over ten

runs were calculated for the nonlinearity ................................................................................. 41

Table 8: Table showing how the average and standard deviations for the changing N over a

number of runs were calculated ............................................................................................ xviii

Table 9: Table showing how the average and standard deviations for the changing b over ten

runs were calculated ................................................................................................................ xix

Table 10: Data recorded when showing the linear relationship between the Amplitude of

driving squared and Total Power when b = 0 .......................................................................... xx

Table 11: Data recorded when showing the nonlinear relationship between the Amplitude of

driving squared and Total Power when b = 2 ......................................................................... xxi

Table 12: Results from driving the system at the lower mode and changing the nonlinearity xxi

Table 13: Results from driving the system at the higher mode and changing the nonlinearity

............................................................................................................................................... xxii

Table 14: Table showing how the total power and the power of each peak changes as the

driving frequency is changed ................................................................................................ xxiii

Table 15: Data showing the effect of varying the nonlinearity and the driving frequency on

particle 1................................................................................................................................ xxiv


particle 2................................................................................................................................ xxiv


the total power ........................................................................................................................ xxv


viii

List of Equations

Equation 1: Governing electromechanical equations for piezoelectric materials ...................... 8

Equation 2: Transducer efficiency ........................................................................................... 13

Equation 3: Quartic bistable potential as stated by (Gammaitoni, Nonlinear oscillators for

vibration energy harvesting, 2009) where U stands for the potential energy, k is the spring

constant, x is the displacement and b is the nonlinearity. ........................................................ 16

Equation 4: Euler Method derivatives...................................................................................... 17

Equation 5: Euler Method step forward in time ....................................................................... 17

Equation 6: Euler Method second step forward in time ........................................................... 17

Equation 7: Euler Cromer Method ........................................................................................... 18

Equation 8: Oscillation equations for both the new displacement and velocity values ........... 19

Equation 9: Modified oscillator equations to include the transducer terms and .......... 26

Equation 10: Modified oscillator equations to include the damping term ............................ 27

Equation 11: Driving the system method 1 utilising the driving terms and whereby a

sweep of all the frequencies is performed ................................................................................ 28

Equation 12: Driving the system method 2 utilising the driving terms A and rand whereby the

system is driven with random noise ......................................................................................... 29

Equation 13: Driving the system method 3 utilising the driving terms A and whereby the

system is driven at a particular frequency ................................................................................ 30

Equation 14: Modified oscillator equations with added transducer term b .............................. 31

Equation 15: Fourier Transform for both the time domain and the frequency domain . 33

Equation 16: Fast Fourier Transform for both the time domain y(t) and the frequency domain

Y(t) for N datapoints ................................................................................................................ 33

Equation 17: Determining the Nyquist Limit for the project ................................................... 34

Equation 18: Calculating the lower and higher modes ............................................................ 45


ix

Nomenclature

Symbol Name

Amplitude of Driving

Area

Nonlinearity

Capacitance

Compliance

Electric Displacement

Piezoelectric coefficient

Energy

Electric Field

Frequency

Nyquist Limit

Gravity

Timestep

Spring constant

Coupling Constant

Coupling Coefficient

Natural log

Mass

No. of timesteps

Power input

Power output

Resistive Load

Random Number Generator

Strain

Seconds

Time


x

Thickness

Voltage

Velocity

Watts

Displacement

Amplitude

Change in some variable

Damping

Dielectric constant

Efficiency

Pi

Stress

Time Constant

Angular Frequency

Lower mode

Higher mode

Driving Frequency

Frequency Sweep

Phase


1

1 Introduction

Over the last number of years, energy harvesters have become an ever emerging technique to

provide power to autonomous electronic devices. Available energy to power such devices can

come in many forms as outlined in Table 1. (Priya, 2007) Among the most attractive sources

of energy is kinetic energy which covers mechanical vibrations, air flow and human power. In

this paper, the source of vibrations will be discussed and one of the most effective ways of

converting mechanical energy from vibrations into electrical energy has proven to be through

the piezoelectric effect due to the options available with regard piezoelectric materials and the

abundant presence of vibrations.

Table 1: Sources of energy available in the surrounding which are/can be tapped for generating electricity.

Energy Source Energy Form Power Attainable

Human Body

Breathing, blood pressure, exhalation, body

heat, walking, arm motion, finger motion,

jogging, swimming, eating, talking

0.33 – 8.3 W

Vehicles

Aircraft, UAV, helicopter, automobiles,

trains, tires, tracks, peddles, brakes, shock

absorbers, turbines

Cars = 300 W

Trucks, Railcars, Off-road

vehicles = 1-10 kW

Structures

Bridges, roads, tunnels, farm house

structures, control-switch, HVAC systems,

ducts, cleaners, etc.

Up to 85 kW from tall

buildings through the use of

tuned mass dampers (TMDs)

Industrial

Motors, compressors, chillers, pumps, fans,

conveyors, vibrating machinery

Generally in the mW range

Energy harvesting devices can convert ambient energy into electrical energy through many

different forms. It can be described as an operation without any auxiliary energy. Instead of

generating auxiliary energy through an integrated energy source or adding it via an external


2

energy supply, ambient energy that is available in the surrounding environment or that is

acting on the system is converted. Therefore an energy harvester generally speaking is an

energy converter. Energy harvested can be stored in a capacitor, super capacitor or a battery.

If the application needs to provide spikes of energy, capacitors are used. If the application

requires a steady flow of energy then batteries are used.

Present working solutions for converting vibrational energy to electricity are based on linear

systems, i.e., resonant, mechanical oscillators that convert kinetic energy via capacitive,

inductive or piezoelectric methods by tuning their resonant frequency in the spectral region

where most of the energy is available. But, in most cases, the ambient vibrations have their

energy distributed over a wide spectrum of frequencies, mainly at low frequency components

and frequency tuning is not always viable due to geometrical or dynamical constraints. (F.

Cottone, 2009) A system of coupled oscillators will have two normal modes whereby the

particles can oscillate at the same frequency. One mode consists of the two masses vibrating

sinusoidally, in phase, with the same angular frequency √

, and with equal amplitude.

The other will be when the two masses move sinusoidally with angular frequency

√

(where k’ is the spring constant of the central spring) and equal amplitude but the

two masses will be a half-cycle out of phase.

A transducer is a device that converts a signal from one form of energy to another. Any

device which converts energy can be considered a transducer. Priya states that piezoelectric

bimorph transducers have the simplest low frequency resonance structure. They can be easily

mounted in several configurations providing a high degree of adaptability to the available

vibrations. The resonance frequency of the piezoelectric transducer is dependent on the size,

configuration and loading conditions. (Priya, 2007)

Nonlinear systems generate large oscillations over a wider frequency range with respect to the

linear case, thus potentially improving the energy harvested under proper conditions. A


3

particular type of nonlinear system is a bistable system, where two equilibrium positions are

present and the system can rapidly switch between the stable states under proper mechanical

excitations, thereby increasing the velocity and thus, the power converted by the harvester.

The difference in displacement between linear and nonlinear systems is evident in Figure 1

which shows the simulated deflection of the cantilever in the time domain. (M. Ferrari, 2010)

Figure 1: Simulated deflection of the cantilever in the time domain. The signals are normalized to the

maximum value of amplitude. The nonlinear evolution represents the stochastic resonance condition.

Aims & Objectives

Modelling an oscillator with a transducer

Modelling a coupled oscillator

Modelling a coupled oscillator with a transducer

Modelling a nonlinear oscillator

Modelling a nonlinear oscillator with a transducer

Modelling a driven (with broadband spectrum) coupled oscillator with a transducer

Control the transfer of energy between two modes

The report begins with a literature review which looks specifically at energy harvesters,

oscillators, transducers and nonlinear systems. The experimental work and results will follow

this and include the work undertaken with MatLAB. Finally, a discussion and conclusions

will be presented at the end of the project.


4

2 Literature Review

The aim of this literature review is to show the multiple ways in which energy harvesters can

be used to obtain energy from vibrations and to present the means by which this is achieved.

The review looks at the numerous strategies for energy harvesting and more specifically at

energy harvesting from piezoelectric devices, how they work and how power can be harvested

from vibrations with this type of device. The sizing and design of energy harvesters is also

investigated. A study of the workings of an oscillator and the various properties these present

is undertaken. Furthermore, the author will explore the theory and workings of a transducer

and finally finish up with an inspection of nonlinear systems and their characteristics.


5

2.1 Energy Harvesters

2.1.1 Introduction

An energy harvester (also known as an energy scavenger) is a device which derives energy

from external sources such as solar power, thermal energy, wind energy, salinity gradients

and kinetic energy. It captures this energy and stores it for small wireless devices like those

used in wireless sensor networks and wearable technology. The energy harvester can provide

small amounts of power for low-energy electronics. The benefit of energy harvesters is that

the source is present as ambient background and it’s free.

2.1.2 Energy Harvesting Strategies

There are numerous small scale energy sources and methods to accompany these that

generally cannot be scaled up to industrial size. These methods are described in this section

along with some examples where the methods are used.

The thermoelectric effect is the direct conversion of temperature differences across a material

to electric voltage and vice versa. If this difference in temperature is kept constant then this

results in a steady voltage across the thermoelectric crystal. The main requirements for this

type of operation are a heat source and a heat sink. An example of this form of energy

harvesting is in road transport whereby cars and trucks are equipped with thermoelectric

generators (TEGS) which would result in significant fuel savings. The generator can gain

energy from the car running on the road which will reduce fuel consumption.

Piezoelectricity is the electric charge that accumulates in certain solid materials in response to

applied mechanical stress. The word piezoelectricity literally translates as “electricity

resulting from pressure”. Some of the potential sources for this form of energy are human

motion, low-frequency vibrations and acoustic noise. Examples of piezoelectric energy


6

harvesters include: a battery-less remote control which uses the force applied to press a button

to power a wireless radio or infrared signal, piezoelectric floor tiles which harvests energy

generated by the footsteps of crowds to power ticket gates and display systems, and car tyre

pressure sensors which are attached inside the tyres and continuously monitor the pressure

and send this information to the dashboard in the car.

Magnetic induction utilises the small vibrations of a cantilever beam to generate micro

currents. Magnets vibrating on the beam move relative to conductors due to Faraday’s law of

induction. This allows sensors in hard to reach places to generate their own power and send

data to outside receivers.

Another method of harvesting energy from the human body through physical energy is of

note. In the referenced paper by Sue et al. the authors review the multiple types of energy

harvesting currently available through mirco-electromechanical systems or MEMS. The

attainable power is the range of milli-watts at most which results in this device being

restricted. Improving the efficiency can be achieved by matching the frequencies of the

vibration and the micro energy harvester. (Chung-Yang Sue, 2012)

2.1.3 Harvesting Energy using the Piezoelectric Effect

The method chosen in this review is the use of the piezoelectric effect in order to harvest

energy from vibrations. The aim is to produce a model using MatLAB which will show how

energy can be extracted from vibrations in a nonlinear system. The piezoelectric effect uses

the charge that accumulates in certain solid materials in response to an applied mechanical

stress such as the stress on a solid from vibrations. Acquiring the power from a piezoelectric

vibration based energy harvester is subject to many variables and various designs of

harvesters and harvesting circuits are analysed in this section.

Adhikari et al. use a stochastic approach and focus on stack configuration and harvesting

broadband vibration energy. The authors assume that the ambient base excitation is stationary


7

Gaussian white noise which has a constant power spectral density across the frequency range.

The paper looks at the harvesting circuit with and without an inductor. The authors show that

in order to maximise the mean of the harvested power, the mechanical damping in the

harvester should be minimized and the electromechanical coupling should be as large as

possible. For the circuit with an inductor the maximum mean power occurs when the natural

frequency of the electrical circuit is equal to that of the mechanical system. (Adhikari, 2009)

Renno et al. look at the optimization of power acquired from a piezoelectric vibration-based

energy harvester which utilizes a harvesting circuit employing an inductor and a resistive

load. The paper explores the impact of damping on power optimality, an area which will need

to be illustrated using MatLAB for this project. The author also looks at the effect of adding

an inductor to the circuit and it is demonstrated that this addition provides substantial

improvement to the performance of the energy harvesting device. An interesting point made

in the paper is that in another paper by duToit and Wardle, (Noel E. duToit, 2005) was that

coupling results in another optimal frequency at the anti-resonance. The paper accounts for

mechanical damping and demonstrates its qualitative effect on power optimality. In

particular, the authors show that for damping ratios that are below a bifurcation damping

ratio, the power has two maxima (at the resonance and anti-resonance) and one minimum.

Beyond the bifurcation damping ratio, the power exhibits only one maximum. The other case

analysed by this paper involves employing an optimal inductor in the circuit which can

substantially enhance the harvested power. The adding of an inductor allows for tuning the

energy harvesting device to scavenge the optimal power for a broad range of excitation

frequencies. It is also critical to maintain an optimal strain rate in order to maintain optimal

power for any excitation frequency. The power optimization problem corresponds to the rate

of strain of the mechanical element and is not related to the magnitude of the strain itself.

(Renno, 2009)

Heung Soo Kim et al. review some key ideas and performances of piezoelectric energy

harvesting from vibrations. It states that piezoelectric materials have high energy conversion


8

ability from mechanical vibration. Energy harvesting is defined here as capturing minute

amounts of energy from one or more surrounding energy sources, accumulating them and

storing them for later use. Piezoelectric materials can be split into two types, piezoceramics

and piezoploymers. Piezoceramics have large electro-mechanical coupling constants and

provide a high energy conversion rate but they are brittle to use with a generic transducer.

Piezopolymers have smaller electro-mechanical coupling constants compared to

piezoceramics but they are flexible. In another paper by Priya (Priya, 2007), the author

calculated that the energy density of piezoelectric energy harvesting devices is 3-5 times

higher than electrostatic or electromagnetic devices. In order to utilize the piezoelectric

energy harvester, power production must be designed with a rectifier. Circuit layouts are

described by the author in this paper. Fatigue and crack of the energy harvesting devices due

to harsh vibrations and shocks can be critical in real applications. Thus, development of

flexible and resilient piezoelectric materials is necessary. (Heung Soo Kim, 2011)

According to Zuo et al., piezoelectric material is one of the most widely used smart materials.

It can generate a voltage or charge on its surface when a pressure or force is exerted on it.

The governing electromechanical equations for piezoelectric materials can be expressed by

Equation 1.

Equation 1: Governing electromechanical equations for piezoelectric materials

[

] [

] [ ]

where S and are strain and stress, respectively; c is compliance; D is electric displacement

(charge per unit area); E is electric field (volts per unit length); d is piezoelectric coefficient;

and is dielectric constant. When used as an energy harvester, the piezoelectric material can

work in (seen in piezoelectric film) or (seen in piezoelectric stack) mode as shown in

Figure 2, where is the thickness of piezoelectric materials or the distance between

electrodes in the polarization direction and is the area of conductive electrodes.


9

Figure 2: Two modes of piezoelectric materials when used for vibration energy harvesting.

Because piezoelectric materials have a relatively small strain, it prevents their direct

application in large amplitude vibrations. This is also one of the benefits of piezoelectric

materials such as in situations where small deformation is preferred. (Lei Zuo, 2013)

2.1.4 Harvesting Losses & Efficiency

Efficiency is a greater concern in every operating machine than it ever was and identifying the

losses which occur in a machine is crucial from an efficiency improvement point of view. In

vibration energy harvesters, improving efficiency requires employing more efficient

transducers, motion mechanisms and electronic circuits. A fundamental challenge is that

large-scale vibration is very irregular at time-varying frequency and at low, alternating

velocities, which makes efficient and reliable energy conversion difficult and limits the

options for efficient power take-off technology. Further research into the novel mechanical

motion rectifier that converts irregular oscillatory vibration into regular unidirectional rotation

needs to be carried out. (Lei Zuo, 2013)

Umeda et al. have analysed the electric energy generated from a piezoelectric diaphragm

structure which is vibrated by dropping a steel ball. The authors found the efficiency of the

system when the steel ball did not bounce off the diaphragm was computed to be 52%.

Further study showed that efficiency increased with an increase in the mechanical quality

factor, the coupling factor and a decrease in the dielectric loss factor. (M. Umeda, 1997)


10

Priya also speaks of a quantitative model created for energy conversion efficiency which

provides a method for evaluating the material and transducer design in the resonance

condition. It was found that increasing the effective mass by a factor of 2 results in an

increase in efficiency of 8% and decreasing the damping by a factor of 2 increases efficiency

by 17%. (Priya, 2007)

A paper on wideband vibrations by Ferrari et al. investigates a nonlinear bistable piezoelectric

converter for power harvesting from ambient vibrations. The authors found that the nonlinear

converter performed better under wideband excitation with respect to a linear system,

increasing the useful converter bandwidth and power output level. They also noted an 88%

increase in the rms voltage when run over a wider bandwidth which is consistent with a 250%

increase in the power delivered to the load. (M. Ferrari, 2010)

2.1.5 Harvester Sizing & Design

A cantilever type vibration energy harvester has a simple structure and can produce a large

deformation under vibration. It uses axially compressed piezoelectric bimorph in order to

decrease resonance frequency by up to 24%. Power output is around 65-90% of the nominal

value at frequencies 19-24% below the unloaded resonance frequency. (Leland, 2006)

Figure 3: Cantilever & cymbal type

Cymbal structure can produce a large in-plane strain under a transvers external force, which is

beneficial for micro energy harvesting. Stack type piezoelectric transducers can produce a

large amount of electrical energy since it uses mode of piezoelectric materials and has a


11

large capacitance due to multi-stacking of piezoelectric layers. Shell structure devices can

generate a larger strain than a flat plate device so it can improve the efficiency of piezoelectric

energy harvesting. (Heung Soo Kim, 2011)

Piezoelectric bimorph bender transducers have been implemented by Messineo et al. instead

of the stack structure outlined above. The authors investigate the use of an elastic element

which can alter the natural frequency of the mechanism and a tuning mass which can more

accurately find the natural frequency of the device. Two different configurations are realized

which results in a voltage difference of 100V. (Antonio Messineo, 2012)

2.2 Oscillators

The term vibration is sometimes used more narrowly to mean a mechanical oscillation but is

sometimes used as a synonym of oscillation. This section will look at the operation of

oscillators and how they can be utilised to harvest energy. Devices which use oscillators

include clocks, watches and radios as well as the energy harvester being modelled in this

project.

Oscillators are based on the principle of oscillation, a change between two positions which

changes periodically with time. Thus, oscillations are said to be periodic, and display

periodic motion. In a pendulum, energy moves between potential energy and kinetic energy.

When the pendulum is at one end of its swing, it has all potential energy and it is ready to fall.

When the pendulum is in the middle of its cycle, all of the potential energy is converted into

kinetic energy and the pendulum is moving at a high velocity. As the pendulum moves

toward the other end of its swing, all the kinetic energy turns back into potential energy. This

movement of energy between the two forms is what causes the oscillation. Without a driving

force, any oscillator will eventually stop due to friction. In a pendulum clock, a spring


12

provides the energy to keep the pendulum moving, giving it a small amount of energy on each

swing to make up for the energy lost due to friction.

When an oscillator is in its resting position, it is said to be in a state of equilibrium. The

oscillator will remain in this position until it is disturbed or, as Isaac Newton puts it in

Newton’s 1st law of motion, an external force is applied to it. If such a force is applied to the

oscillator, it will begin oscillating or vibrating. The oscillator has forced vibration and it will

swing or vibrate back and forth as friction or other forces then slow the oscillator down until

it is in its equilibrium state again. The displacement from its equilibrium position is

dependent on the magnitude of the force applied. A driving force would be required to keep

the oscillator moving back and forth.

Vibration is often undesirable as it wastes energy and creates unwanted sound. Examples of

these losses in energy can be found in the vibrational motions of engines, electric motors, or

any mechanical device in operation. Such vibrations can be caused by imbalances in the

rotating parts, uneven friction, the meshing of gear teeth, etc. Careful designs usually

minimize unwanted vibrations.

Triplett et al. demonstrate an energy harvesting system consisting of an oscillating mass

subject to excitation of the supporting base which develops an electric charge. (Triplett,

2009) Another paper by Vocca et al. consider a piezoelectric cantilever subjected to magnetic

repulsion forces which oscillates as the cantilever experiences vibrations from the

environment. Three different driving scenarios are tested for this investigation as well as

driving the system with white noise. (Vocca, 2012)

A bistable system such as the one implemented by Ferrari et al. is one where two equilibrium

states are present. Under some mechanical excitation, the system can quickly switch between

the states. If the difference in energy (y-axis) between the states 1 and 2 is small in Figure 4

below, then the system only needs small excitation and coincidentally, a large difference

needs a large excitation. (M. Ferrari, 2010)


13

Figure 4: Graph showing the bistable states of a system

2.3 Transducers

A transducer is a device that converts a signal from one form of energy to another.

Transducer efficiency is defined as the ratio of the power output to power input. If P

represents the power input and Q represents the power output, then the efficiency is given in

Equation 2. No transducer is 100 per cent efficient as some power is always lost in the

conversion process. This loss is usually in the form of heat.

Equation 2: Transducer efficiency

In piezoelectric harvesting applications, low profile transducers are beneficial due to their

light weight, flexibility, easy mounting, large response and low frequency operation. Figure 5

shows some promising low profile transducers. The author states the components of each of

these devices, how they operate and which types of energy sources they harvest energy from.

(Priya, 2007)


14

Figure 5: Examples of low profile transducer

The general principle for converting mechanical low frequency AC stress into electrical

energy by means of a piezoelectric electric transducer is shown schematically in Figure 6.

Figure 6: Schematic representation of piezoelectric energy harvesting

There are three primary steps which are outlined in the above schematic: (1) trapping the

mechanical AC stress from an available source, (2) converting the mechanical energy into

electrical energy with a piezoelectric transducer and (3) processing and storing the generated

electrical energy.

Some properties of a transducer are spoken of by Stephen in his paper on energy harvesting

from ambient vibration in which he states that the ideal electromechanical transducer should

have a high flux density and a very small internal resistance so that the maximum power can


15

be transferred to the load. A spring mass damper dependent on a sinusoidal force applied

directly to the mass is the energy harvesting device used here. (Stephen, 2006)

2.4 Nonlinear Systems

Cottone et al. demonstrate that a bistable oscillator can provide better performances compared

to a linear oscillator in terms of energy extracted from a generic wide spectrum vibration.

The author states that a nonlinear oscillator can present a wide spectral response, larger than a

linear one, and can be operated in such a way that its frequency response matched more

closely what is available in the environment. (F. Cottone, 2009)

Gammaitoni et al. acknowledge the difficulties which arise both because the tuning of the

oscillators is constrained by geometrical factors and because the energy spectra of the

available vibration are commonly spread in a wide frequency range, with the prevalence of

low frequency components. A way of overcoming this problem by considering non-linear

oscillators instead of linear was outlined in another paper by Cottone et al. (F. Cottone, 2009)

In this paper, the authors prove that a bistable oscillator can outperform a linear oscillator in

terms of the energy extracted from a wide spectrum of vibration. Gammaitoni at al. choose to

focus on a piezoelectric energy harvesting device, but state that most of the considerations

presented in the paper are applicable to other energy conversion mechanisms based on

dynamical oscillators. The authors have demonstrated that the nonlinear dynamical properties

of a noise activated energy harvesting device can play a favourable role in enhancing the

performances in terms of power produced (proportional to ). The paper sets out an

equation which characterises a quartic bistable potential. By finding the derivative of this

potential, the nonlinear term can then be attained. (Gammaitoni, Nonlinear oscillators

for vibration energy harvesting, 2009) This term should exhibit some optimum property

whereby the maximum power can be extracted from the system. The equation is similar to

that presented by Cottone et al. in a paper on nonlinear kinetic energy harvesting whereby


16

varying the distance between the magnetic tip mass and another mass with an opposing

magnetization. (Cottone, 2011)

Equation 3: Quartic bistable potential as stated by (Gammaitoni, Nonlinear oscillators for vibration energy

harvesting, 2009) where U stands for the potential energy, k is the spring constant, x is the displacement

and b is the nonlinearity.

The reasons for implementing a nonlinear system over a linear one are numerous. A paper by

Quinn et al. concludes that a properly tuned linear attachment can harvest maximum power

from a single stationary frequency but when frequency mistunings arise, a nonlinear energy

harvester provides more power. The nonlinear energy harvester is therefore particularly

effective when no dominant frequency can be determined. (Quinn, 2011) Secondly,

Gammaitoni et al. demonstrates in a separate paper that nonlinear properties in an energy

harvesting device can improve the performance of the system in terms of power produced.

(Gammaitoni, The benefits of noise and nonlinearity: Extracting energy from random

vibrations, 2010) The superior performance of the nonlinear harvester is again outlined by

Vocca et al. who test the system against three real world scenarios as well as in the presence

of white noise. (Vocca, 2012) Further work into energy harvesting from wideband vibrations

indicate that a nonlinear bistable oscillator again outperforms its linear counterpart. (M.

Ferrari, 2010)


17

3 Experimental Work

This section will outline the methods implemented in the project in order to model the

coupled oscillator system. The fundamental equations chosen for the work will be defined

here.

3.1 Oscillator Establishment Methods & Equations

In order to solve differential equations on a computer, numerical methods need to be used and

the method chosen is the Euler-Cromer method which stems from the Euler method. This is a

fairly simple approach for second order, ordinary differential equations.

3.1.1 Euler

Shorthand notation for time derivatives are used where

Equation 4: Euler Method derivatives

The Euler method is the simplest numerical method to solve differential equations. If time is

divided into small time steps , and we take a step forward in time we get

Equation 5: Euler Method step forward in time

And at another step forward in time, we need to know the first derivative at that later time

which is approximately

Equation 6: Euler Method second step forward in time


18

Figure 7: Euler Method showing an increasing sine wave

3.1.2 Euler Cromer

A simple modification gives the Euler Cromer method where the approximate first derivative

at this later time is used. The new value of is used to find the new value of x.

Equation 7: Euler Cromer Method

Figure 8: Euler-Cromer Method showing sine waves with consistent peaks

0 1 2 3 4 5 6 7 8 9 10-1.5

-1

-0.5

0

0.5

1

1.5

time (seconds)

theta

(ra

dia

ns)

0 1 2 3 4 5 6 7 8 9 10-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

time (seconds)

theta

(ra

dia

ns)


19

3.1.3 Oscillator Equations

The basic equations for an oscillator are given here which are the foundations on which this

project is built on. Further developments to these equations are stated in the relevant sections.

In Equation 8, h represents the change in time, represents the initial displacement and

the new displacement. The same format occurs for v, the velocity, which represents the first

derivative of x. is the angular frequency of the sine wave, is defined by the equation

,

which is the ratio of the spring constant to the mass of the oscillating mass, and t is the time

(which is defined by the time step, h, multiplied by the number of time steps, N, in the code).

Subscripts of 1 and 2 represent the particles 1 and 2 respectively which oscillate in the

system.

Equation 8: Oscillation equations for both the new displacement and velocity values

(

)

(

)

3.2 MatLAB

The experimental apparatus for this project consists solely of MatLAB software and all work

is completed using a computer. The experimental work for the project consists of creating

models of oscillatory systems using MatLAB. Being competent in operating different

functions relevant to this project and using them to create the code for modelling these

systems provided many challenges which needed to be overcome. The software is used for

technical computing and it incorporates computation, programming and modelling in a user

friendly manner. The interactive system is used to create a model of an oscillatory system

with a number of variables which can then be altered to perform a number of analytical

investigations.


20

In MatLAB, code is created to run the above equations and the pseudo code to summarize this

is as follows:

Initialise Variables

Open file to write to

o Begin time loop

Iterate equations of motion

Print to file

o Close time loop

Close file

Re-Open file to read

o Read data

o Plot graphs

Close file

3.2.1 Time Selection

Choosing a suitable time for the system requires some initial experimentation as there is the

risk that too small a time step will result in not seeing the full picture and a time step too large

may make it difficult to see how the system oscillates and when steady state occurs. The

initial variable of time is defined by the time step multiplied by the number of time steps.

By selecting 0.04 for and 10000 for , we obtain a clear-cut representation of the

oscillations. Decreasing by a factor of 10 whilst holding constant, or by doing the

opposite and decreasing by a factor of 10 and leaving constant, both result in the same

outcome due to their multiplicative relationship. An example is shown in the following

figures whereby values for of 1000, 10000 and 100000 were used. It is evident that using

= 10000 results in a graph in which the oscillations, their peak values and the time at which

steady state occurs can be easily read.


21

Figure 9: Graph of voltage versus time for a value of N equal to 1000

Figure 9 above shows values for voltage up to a time value of 40 seconds. It is not clear from

this graph if steady state is occurring. Therefore, changing N to a value of 10000 below gives

a clearer representation of steady state.



22


The above graph for N equal to 100000 clearly shows steady state but the waves are packed

so close together that it is difficult to determine where the steady state starts. Because of this,

a value of N equal to 10000 will be used in this project.


23

4 Results, Analysis and Discussion

This section will present the results of the initial set up of the system, the investigations

partaken to demonstrate and discover the various properties of the system and produce some

items for discussion.

Unless stated in the various sections, the parameters used for the project to run the system are

as defined in Table 2. These values can be altered and updated in certain sections so as to

perform some investigations and attain results and will be stated as such. The actual code

generated with MatLAB to obtain these results is presented in the appendix in Figure 49.

Table 2: A list of the numerous parameter values implemented in this project, their name as seen in the

code, their symbol and the value assigned to each

Description Codename Symbol Value

No. of data points npoints

250

Start point for time time t 0

Time step h h 0.04

No. of time steps N N 10000

Oscillator 1 initial displacement x1old 0

Oscillator 2 initial displacement x2old 0

Oscillator 1 initial velocity v1old 0

Oscillator 2 initial velocity v2old 0

Angular frequency omega 4

K/m OMEGA 2

Initial voltage Vold 0

Coupling constant Kc 0.5

Time constant tau=RC tau 2

Coupling coefficient Kv 0.5

Amplitude of driving amp 0.1

Damping gamma 0.1

Non-linearity term b 3

Driving frequency omegast 4.1

Total Power TotalPower

0


24

4.1 System Configuration

4.1.1 Oscillator Equations

The equations defined in Section 3.1.3 are implemented in MatLAB in order to model the

coupled oscillators. The basic equations take into account the displacement, velocity, angular

velocity and

. The equations below have been reproduced from Section 3.1.3, Equation 8.

(

)

(

)

The variables in the equations above are all given specific names and initialised at the start of

the code. As the configuration runs through the loop, the new values found for the

displacement, velocity and voltage are then used as the old values for the next loop in time.

This model returns a constant sine wave as seen in Figure 12 below for voltage with

invariable values for the max and min of this wave. The model also returns a total power

value of the entire frequency range as seen in Figure 13.


25

Figure 12: Unchanging sine wave produced from the initial oscillator equations defined in Equation 8

Figure 13: Entire frequency range shown with peaks at the modes resulting from the initial oscillator

equations in Equation 8 and by using the parameter values stated in Table 2

In Figure 13 there are two modes resolved but as the second mode is of much smaller a value

to the first mode, it is difficult to make it out from this figure and work in later sections will

provide a clearer representation of the system.


26

4.1.2 Applying a Transducer

As stated in Section 2, the transducer converts a signal from one form of energy to another.

In this case, we use it to convert the mechanical vibrations of the oscillator into an electrical

signal. This can also be referred to as the piezoelectric effect. This is achieved by

multiplying the voltage of the system by the coupling coefficient and subtracting the

product from the equation for the velocity of particle one. When calculating the new voltage

value, another change has to be made whereby the old voltage value is multiplied by the

inverse of and subtracted from the equation, in which is the time constant related to the

coupling capacitance C and the resistive load R where . in the new voltage

equation accounts for the coupling constant. The modified equations are as seen below.

Equation 9: Modified oscillator equations to include the transducer terms and

(

)

( )

The values chosen for , and are all similar to those chosen in the paper by Gammaitoni,

where = 0.5, = 0.5 and = 2. (Gammaitoni, Nonlinear oscillators for vibration energy

harvesting, 2009) In this paper, the author uses a value of = 10 but on investigation this

doesn’t represent a large difference in final values. The total power of the system increases

slightly by less than 1.4% so using a value of 2 is deemed acceptable. The resultant graph for

the voltage is shown in Figure 14. It is evident that the system is losing voltage as time passes

due to the transducer being added.


27

Figure 14: Resulting graph from the modified oscillator equations when a transducer is added as defined in

Equation 9

4.1.3 Damping the System

Damping is the effect on an oscillatory system of reducing or restricting the oscillations. This

effect needs to be included in the system for it to have a more uniform value. This is

replicated in MatLAB by adding a damping term which provides the opposing force. This

then reduces the oscillations of the system to some constant value.

Without damping, the voltage can be seen to oscillate freely as seen in Figure 14, but with the

damping term added below of = 0.1, a steadier flow is witnessed as in Figure 15. The only

downside of this effect is that the constant value is lower than the value seen at the peaks

above. The updated equations for the new values of the velocities of particles one and two

can be seen below.

Equation 10: Modified oscillator equations to include the damping term

(

) ( )

(

) ( )


28

Figure 15: Resulting graph from adding damping to the system as in Equation 10 showing constant peaks

for the sine waves

4.1.4 Driving the System

The system can be expressed as a function of either time or frequency and in order to find

optimum frequency values with regard the corresponding power output, a number of methods

can be exerted. The first of these is to perform a frequency sweep which will run the system

at a number of intervals, defined in the code, and return power values for those specific

frequencies. A loop of the chosen frequencies is created in order to do this. represents the

amplitude of the driving and equates to the frequency sweep values. The resultant

graph can then be used to find optimum values for the driving frequency based on the initial

variables used.

Equation 11: Driving the system method 1 utilising the driving terms and whereby a sweep of all

the frequencies is performed

(

) ( )


29

Figure 16: Resulting graph of Power(W) versus frequency ( ) when a frequency sweep is performed as

outlined in Equation 11

The second method is to drive the system using a random number generator. This utilises the

‘rand’ function in MatLAB which drives the system completely at random. The resultant

graphs are then difficult to analyse as each one is completely different as seen in the Figure

17.

Equation 12: Driving the system method 2 utilising the driving terms A and rand whereby the system is

driven with random noise

(

) ( )

Figure 17: Graphs from driving the system with random noise as defined in Equation 12 which results in

non-repeatable graphs

The third method is to set the system to run at a particular frequency, which can be the

frequency at which the most power is obtained. This method is the final method used which

allows for calculation of other variables such as the nonlinearity and amplitude of the system.


30

Equation 13: Driving the system method 3 utilising the driving terms A and whereby the system is

driven at a particular frequency

(

) ( )

This driving term is therefore defined by the amplitude of the driving multiplied by the sin of

the product of the driving frequency and time. By knowing the optimum driving frequency

and then driving the system at this frequency, we can then obtain the largest amount of power

from the system.

Figure 18: Resulting graphs from driving the system at a particular frequency using Equation 13 which

presents a peak at one mode which is repeatable

4.1.5 Nonlinear System

As stated in Section 2.4, Gammaitoni et al. present equations which can be used to simulate

nonlinearity in the system. The term used to define this nonlinearity is stated to be .

The main addition of the nonlinearity is essentially to extract the most energy from the system

across the entire frequency range. When driven at the lower mode of 4.1, the system presents

results which indicate that a nonlinearity value of 0 zero returns the most power. Yet, when


31

driven at the higher mode of 4.89, this is not the case, rather an optimum value of b can be

found. Further investigation into this optimum value is performed in Section 4.3.3.

Equation 14: Modified oscillator equations with added transducer term b

(

) ( ) (

)

(

) ( )

Figure 19: Resulting graphs from driving the system at a frequency of 4.1 without nonlinearity resulting

in a total power of 9.1113e-04 W as defined by Equation 14

Figure 20: Resulting graphs from driving the system at a frequency of 4.1 with nonlinearity resulting in

a total power of 8.1379e-04 W as outlined by Equation 14


32

The above graphs show that when driven at the lower mode, the total power for the system is

larger without any nonlinearity. The total power without the nonlinearity is 0.00091113 W

and with the nonlinearity as seen in Figure 20 is 0.00081379 W, a decrease of 0.00009734 W

or approximately 10.7%. When driven at the higher mode as seen below in Figure 21 &

Figure 22, more power is attained when the nonlinearity is added.

Figure 21: Resulting graphs from driving the system at a frequency of 4.89 without nonlinearity

resulting in a total power of 0.0023 W using Equation 14

Figure 22: Resulting graphs from driving the system at a frequency of 4.89 with nonlinearity resulting in

a total power of 0.0046 W as defined by Equation 14

Here the value for total power without the nonlinearity is 0.0023 W and with the nonlinearity

is 0.0046 W, therefore we see an increase of twice the power.


33

4.2 Methods

4.2.1 Fast Fourier Transform (FFT)

All waveforms can be characterised by simple sinusoidal waveforms (or sinusoids). This is,

in essence, what is defined by the Fourier transform. It was shown by Joseph Fourier neatly

200 years ago that virtually any signal can be represented this way. The equation for this

definition is conveniently expressed as an integral over frequency, as shown below, where

is the amplitude, is the frequency, is the phase of the th sine wave component, is

the time domain and is the frequency domain. (Giordano, 1997)

Equation 15: Fourier Transform for both the time domain and the frequency domain

∫

∫

In order to compute a Fourier transform, we use what is called the Fast Fourier Transform

(FFT). The governing equations for this are below where is the number of data points.

(Giordano, 1997)

Equation 16: Fast Fourier Transform for both the time domain y(t) and the frequency domain Y(t) for N

datapoints

∑

∑

The FFT allows us to transform a function of time into a function of frequency and vice versa.

In this project, it is used to plot power against frequency so that optimum values can be found

for the driving frequency, i.e., where the highest value for power is found.


34

With respect to the frequencies associated with , it is noted that . The highest

frequency Fourier component is . If the signal is measured by time intervals spaced by

, which is true in this case, then the spectral components that can be recovered with a

Fourier transform are those with frequencies below

. A simple calculation

gives a Nyquist limit of 78.539 radians per second which correlates with the point in Figure

24 where the graph ends.

Equation 17: Determining the Nyquist Limit for the project

Figure 23: Graph showing frequency versus time for the system without implementing the Nyquist Limit

resulting in 'mirrored' peaks

Figure 24: Graph showing frequency versus time for the system which implements the Nyquist Limit

resulting in a single set of peaks


35

4.2.2 Steady state

Steady state is defined as when a system is unchanging with time. This is determined in the

final model to be after 200 seconds at which point the difference between two successive

peaks is less than 0.0001 or less than 0.354% which can be seen in following figure and table

using the variable of voltage (Vnew) as an example. The initial values in this case were set to

zero.

Figure 25: Graph showing the steady state occurrence for voltage

Table 3: Results used to determine that steady state occurs after 200 seconds

Time (s) Vnew (V) Difference (V) % Difference

200.28 0.042801

201.8 0.042953 0.000152 0.353875166

203.32 0.042956 0.000003 0.00698389

204.88 0.042871 -0.000085 -0.198269226

206.4 0.042977 0.000106 0.246643554

207.92 0.042989 0.000012 0.027914118

209.48 0.042905 -0.000084 -0.195781377


36

This process can be repeated for the values of the displacements and velocities for both

particles which results in similar graphs as the systems reaches steady state after 200 seconds.

These steady state values for x1new, x2new, v1new, v2new and Vnew can then be set as the

initial values for the system. The values for each variable at steady state are given below in

Table 4 and a figure of the new voltage, as shown above with initial values at zero, is now

plotted against time with the steady state values.

Table 4: The steady state values implemented in this project based on the findings of the steady state

investigation

Particle 1 New

Steady State

Displacement

Particle 2 New

Steady State

Displacement

Particle 1 New

Steady State

Velocity

Particle 1 New

Steady State

Velocity

New Steady

State Voltage

0.085 0.085 0.350 0.310 0.0428

Figure 26: The resulting graph of voltage versus time when the steady state values as defined in Table 4 are

adopted as the initial variables


37

4.2.3 Total Power

In order to calculate the total power of each simulation, the MatLAB software is utilised to

record changes in the total power from each loop. This is achieved by initialising a variable

for total power at the beginning of the code to zero, adding on the power generated during a

loop and then setting the new initial value as the previous initial value plus the change. At the

end of the simulation the value for total power will be an addition of all the power generated

by the system divided by the number of iterations. This method is similar to that

implemented for other variable such as the displacement, velocity and voltage.

The pseudo code for the implantation of this function is as follows.

Initialize variable to zero

o Begin if loop

Add new values to variable

o Close if loop

Divide variable by number of iterations

By simply typing the variable into MatLAB’s command window and hitting enter, a value for

the variable is returned as shown in

Figure 27: Result using MatLAB to retrieve the total power generated for a specific simulation


38

4.3 Investigations

4.3.1 Random Noise Driving

System is driven with random noise and because of this, each run is different, which results in

different values recorded for peaks 1 and 2 and the total power. In order to obtain some

average value for each variable, the system is run a number of times and an average and a

standard deviation is taken for each value.

The nonlinearity term b and the time value N were both varied and ran a number of times

resulting in different values for both the average and standard deviation and there exists no

apparent relationship between the increase in the constant values b and N and the returned

values for peak 1 (P1), peak 2 (P2) and the Total Power (TP).

Table 5: Resulting values recorded when the system was driven with random noise and the value of N was

varied

400 5.991465 0.553856 -0.590850 1.128154 0.120583 0.553984 -0.590620

4000 8.294050 0.347925 -1.055768 0.301767 -1.198099 0.193903 -1.640398

400000 12.899220 0.187922 -1.671729 0.446893 -0.805437 0.066385 -2.712291

4000000 15.201805 0.541996 -0.612496 0.632278 -0.458426 0.033138 -3.407062

The above table shows the final figures obtained using the average ( ) and standard

deviation ( ) function in excel and are then used to create the graphs below which do not

show any clear relationship between the increase in N and the returned values. The natural

log ( ) of is plotted against the natural log ( ) of the standard deviation ( ) of the

variable divided by the average ( ) of the variable as shown in Table 5. If the investigation

is re run, a completely new set of values will be obtained due to the fact that this process in

not repeatable. The full results are placed in the appendix in Table 9.


39

Figure 28: Graph showing the relationship between N and Peak 1 when driven with random noise showing

that no obvious correlation is produced

Figure 28 shows the lack of a correlation between the variables and this can be explained by

the randomness of the system. Joining the dots in the above figure is done so as to show the

order in which the data was collected and possible correlation which may exist. Figure 52

and Figure 53 in the appendix show the same graph as above but for peak 2 and the total

power instead. Neither graph produces any clear correlation.

When this is repeated for changing values of nonlinearity , once again there is no obvious

relationship between the two sets of data. The headings in the table are as described earlier.

Table 6: Resulting values recorded when the system was driven with random noise and the value of b was

varied

0.000100 -9.210340 0.650389 -0.430185 0.664757 -0.408334 0.241434 -1.421161

0.000010 -11.512925 0.494939 -0.703322 0.542375 -0.611798 0.313952 -1.158514

0.000001 -13.815511 0.495332 -0.702527 0.570319 -0.561560 0.154996 -1.864356

Table 6 above shows the final figures obtained for changing b using the same method as with

the N values earlier. The resulting graphs again do not show any obvious correlation between

the change in b and the returned values for peak 1 (P1), peak 2 (P2) and the Total Power (TP).

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0 5 10 15 20

ln(sdP1/avgP1)

ln (N)

Graph showing the relationship between N and Peak 1 when driven with random noise

ln(sdP1/avgP1)


40

Figure 29: Graph showing the relationship between the nonlinearity and Peak 1 when driven with random

noise

Again the graph shows the lack of similarities between the variables and this can be explained

by the fact that this is not a repeatable system. Figure 54 & Figure 55 in the appendix show

similar graphs for peak 2 and total power. These figures also lack any evident correlation.

Table 7 gives an example of the results from running the system 10 times whilst keeping the

variables constant each time. The full results are placed in the appendix in Table 9. As is

evident in the table, the results obtained for the peaks and the total power vary each time. An

average and a standard deviation was calculated and then used to plot the graphs.

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

-15 -10 -5 0

ln(sdP1/avgP1)

ln(b)

Graph showing the relationship between b and Peak 1 when driven with random noise

ln(sdP1/avgP1)


41

Table 7: Table showing how the average and standard deviations for the variables over ten runs were

calculated for the nonlinearity

Amplitude of

Driving (A) Nonlinearity (b)

Peak 1

(Power) (W)

Peak 2

(Power) (W)

Total Power

(W)

0.1 0.0001 1 0.55 6.45E-07

0.1 0.0001 2.2 2.3 1.23E-06

0.1 0.0001 1.55 2.1 8.73E-07

0.1 0.0001 1.6 5.6 1.34E-06

0.1 0.0001 1.6 1.35 9.54E-07

0.1 0.0001 0.86 1.97 8.17E-07

0.1 0.0001 2.2 2.6 1.11E-06

0.1 0.0001 1.25 1.16 8.95E-07

0.1 0.0001 5.7 4.5 1.51E-06

0.1 0.0001 2.2 0.85 9.62E-07

Average (avg) 2.016 2.298 1.03412E-06

Standard Deviation (sd) 1.311184198 1.527611207 2.49672E-07

4.3.2 Varying Amplitude

An investigation into the relationship between the amplitude squared and the total power for a

fixed value of b results in a linear relationship as shown in the graph below. In this case, b is

set to 0 with values of amplitude ranging from 0 to 20. When repeated with a value of b = 2

the graph does not feature such a linear relationship due to this nonlinearity in the system.

Figure 30: Graph showing the linear relationship between Amplitude squared and Total Power when b = 0

0

5

10

15

20

25

30

35

40

0 100 200 300 400 500

Total Power (W)

Amplitude of driving squared

Relationship between Amplitude of driving squared and Total Power when b = 0


42

Figure 31: Graph showing the nonlinear relationship between Amplitude squared and Total Power when b

= 2 which coincides with work performed by Triplett et al. (Triplett, 2009)

The graph above shows how the introduction of some nonlinearity to the system changes the

relationship between the amplitude squared and the total power.

4.3.3 Finding the Optimum Nonlinearity

This section exhausts an investigation into finding an optimum value for nonlinearity.

Multitudes of numbers were run for b whilst holding all other values constant in the system.

Initially the system was driven at the lower mode of aiming to find an optimum

value for b. From the resulting graphs it is evident that there is a different optimum for each

of the two peaks. The peak at the lower mode (P1) has an optimum value of b at zero

whereas the higher mode (P2) has the largest power when b is at its largest. No data was

returned for values of b greater than 17.32 meaning that the nonlinearity was too large. This

finding of a max value for the nonlinearity coincides with the conclusions drawn by Triplett et

al. who conclude that choosing a nonlinearity value too large can result in less power in the

system. (Triplett, 2009) Table 12 & Table 13 showing the results of driving the system at

both the higher and lower modes are located in the appendix.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 1 2 3 4 5 6

Total Power (W)

Amplitude of driving squared

Relationship between Amplitude of driving squared and Total Power when b = 2


43

Figure 32: Graph showing the change in the value of peak 1 as the nonlinearity is changed, resulting in an

optimum of 0 when driven at the lower mode

Figure 33: Graph showing the change in the value of peak 1 as the nonlinearity is changed, resulting in an

optimum greater than 17.32 which was the max value for which a graph was returned when driven at the

lower mode

0

1000

2000

3000

4000

5000

6000

0 5 10 15 20

Peak 1 (W)

Nonlinearity

Value of Peak 1 vs change in nonlinearity

P1

0

1

2

3

4

5

6

7

0 5 10 15 20

Peak 2 (W)

Nonlinearity


P2


44

Figure 34: Graph showing the change in the value of total power as the nonlinearity is changed, resulting in

an optimum of 0 when driven at the lower mode

When the investigation is repeated at the higher mode of , we see a different

outcome. This time only one peak is discovered which sits at the driving frequency and a

distinct optimum value for b is found. A value for b of 3 is acquired for both graphs of peak 2

and total power. This is our optimum nonlinearity value.

Figure 35: Graph showing an optimum value of b = 3 for total power when driven at the higher mode

0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

0.0009

0.001

0 5 10 15 20

Total Power (W)

Nonlinearity

Value of Total Power vs change in nonlinearity

TotalPower

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

0.005

0 5 10 15 20

Total Power (W)

Nonlinearity

Value of Total Power vs change in nonlinearity


45

Figure 36: Graph showing an optimum value of b = 3 for peak 1 when driven at the higher mode

4.3.4 Transfer of Energy between two modes

This section explores how changing the driving frequency affects the system. In order to do

so, the nonlinearity is set to zero and all other variables are kept constant so that the results

can be compared.

As can be seen in Figure 37, the total value has recorded peaks at both 4 and 4.9. These are

the lower and higher modes in the system. Knowing the values for and to be 4 and 2

respectively, we can calculate the higher mode having chosen 4 as the lower mode. The

completed equations below show how this is achieved and the data collected can be seen in

the appendix in Table 14.

Equation 18: Calculating the lower and higher modes

√

√

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

0 5 10 15 20

Peak 2 (W)

Nonlinearity



46

Figure 37: Graph showing the value of total power as the system is driven at various frequencies with peaks

evident at the lower and higher modes

Gap due to peaks overlapping are evident for particle one below in Figure 38. At the modes,

the two particles overlap giving the high value, and because the peaks are close together to

allow for this transfer of energy, there is only one peak evident in between the modes.

Figure 38: Graph showing gap in recorded data due to the energy in the peaks overlapping resulting in just

one peak

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0 1 2 3 4 5 6

TotalPower (W)

omegast (driving frequency) (𝜔)

Value of TotalPower vs change in driving frequency

TotalPower

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 1 2 3 4 5 6

Particle 1 (W)

Driving frequency (𝜔)

Value of max peak of particle 1 vs change in driving frequency


47

Figure 39: Graph showing the energy in particle 2 which obtains energy from particle 2 in between the two

modes

As it is hard to understand form the above graphs how the energy transfers between the

modes; a step by step guide is shown below of how the author visualizes the system response.

Initially there is a small peak (red) at the lower mode and a slightly larger peak at the driving

frequency (blue). As the driving frequency is increased (moving along the x-axis), both peaks

increase but not by the same magnitude. The driving frequency harbours more of the power.

Figure 40: Initial small peak (red) at lower mode and slightly larger driven mode (blue) which is increased

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

0 1 2 3 4 5 6

Particle 2 (W)

Driving frequency)(𝜔)

Value of max peak of particle 2 vs change in driving frequency


48

Figure 41: Driven mode continues to increase resulting in slight increase in lower mode

Figure 42: When two peaks are close enough they combine to become one and at their largest at the lower

and higher modes

When the driving frequency gets close to the lower mode of 4, the two peaks merge to

become one large peak (purple). At the modes, these peaks are at a maximum and in between

the modes it still holds a large value.

Figure 43: Increased driving frequency to higher mode results in corresponding increase in power


49

Figure 44: As driving frequency is further increased, the single peak becomes two again as two modes are

resolved

Figure 45: Driving frequency is so high that the power of the peaks reduces again

When the driving frequency is such a value above the higher mode, two peaks are again

evident as the process continues. The opposite of the initial few steps can now be seen until

both peaks become very small. An excel table showing how the total power and the power of

each peaks changes as the driving frequency is changed is located in the appendix in Table

14.

4.3.5 Linear vs. Nonlinear

The investigation procedure in Section 4.3.4 above is repeated here with a number of different

nonlinearity values. It has already been found that a nonlinearity value of 3 results in the

most power being recorded for the system when run at the higher mode and a nonlinearity of

zero when run at the lower mode. How the nonlinearity affects the system at other driving

frequencies is not known and this is an investigation to see what effect various nonlinearity

values have at various driving frequencies. An optimum value for b can then be found for the

entire system.


50

As in Section 4.3.4 above, a gap is recorded in the values for Peak 1 where the two modes

become one for some frequencies and as such, the resulting graphs have a gap in them. Due

to the values being very small, the difference between the nonlinearities is even smaller as

seen in Figure 46. Initially, a nonlinearity of 6 provides the most power but at a driving

frequency of 3, this value of 6 provides the least power. When the driving frequency is 5.5,

the nonlinearity of 6 again achieves the highest power.

Figure 46: Graph showing the effect of varying the nonlinearity and the driving frequency on Peak 1

Figure 47: Graph showing the effect of varying the nonlinearity and the driving frequency on Peak 2

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 1 2 3 4 5 6

Peak 1 (W)

Driving Frequency (w)

Graph showing the effect of varying the nonlinearity and the driving frequency on Peak 1

b = 0

b = 0.5

b = 1.5

b = 3

b = 4.5

b = 6

0

10000

20000

30000

40000

50000

60000

0 1 2 3 4 5 6

Peak 2 (W)


Graph showing the effect of varying the nonlinearity and the driving frequency on Peak 2

b = 0

b = 0.5

b = 1.5

b = 3

b = 4.5

b = 6


51

When power for peak 2 is recorded, a clear optimum lies with b = 3 as was discovered in the

earlier section. An interesting point to note is that a nonlinearity of 6 results in the least

power which corresponds to the findings by Triplett & Quinn in their work on nonlinearity.

(Triplett, 2009) Figure 48 below presents the same findings for total power with 3 being the

optimum value for b.

Figure 48: Graph showing the effect of varying the nonlinearity and the driving frequency on Total Power

Table 15, Table 16 & Table 17 in the appendix show the data recorded for the above graphs.

-0.001

0

0.001

0.002

0.003

0.004

0.005

0.006

0 1 2 3 4 5 6

Total Power (W)


Graph showing the effect of varying the nonlinearity and the driving frequency on Total Power

b = 0

b = 0.5

b = 1.5

b = 3

b = 4.5

b = 6


52

4.4 Discussion

The main aim of this project has been to model coupled oscillators in a nonlinear system. A

number of studies have been carried out on both coupled oscillator systems and nonlinear

system but to the knowledge of this author, there is no published work detailing a nonlinear

coupled oscillator system present. Gammaitoni et al., (Gammaitoni, Nonlinear oscillators for

vibration energy harvesting, 2009) and Triplett et al., (Triplett, 2009) both investigated the

nonlinear features of an energy harvester and their observations correlate well with the work

in this project.

The results of this project have been outlined in detail in Section 4. The oscillator equations

derived by Gammaitoni et al. were implemented using the Euler Cromer method which

resulted in constant sine waves for the oscillators. When the power is plotted against

frequency, dominant peaks can be found at two frequencies; the lower and higher mode.

Further alterations to these initial equations include a transducer, driving the system, damping

the system and making the system nonlinear as stated in the aims of the project. The

transducer term is pivotal in being able to extract energy from the system when implemented

in a practical sense. Gammaitoni et al presents us with this term in his work on nonlinear

oscillators. (Gammaitoni, Nonlinear oscillators for vibration energy harvesting, 2009) The

inclusion of this term therefore is necessary for this reason and the effect it has on the system

is a loss in power. As time passes, the system loses power as seen in the work.

The damping of the system is there to negate the effect of the transducer term by outputting a

constant sine wave as discussed by McInnes et al. (C.R. McInnesa, 2008) This unfortunately

reduces the magnitude of the power recorded and therefore finding an optimum value for this

damping or rather, one which perfectly represents the practical application of this

phenomenon.


53

Driving the system has been presented in similar ways by Ferrari (M. Ferrari, 2010) and

Cottone (Cottone, 2011) amongst others and here in this project, the author has experimented

with three possible ways of this driving. Firstly a frequency sweep was performed whereby a

range of frequency values were chosen through which the system was run. This results in a

single graph showing peaks at certain frequencies. If ran across a large range of frequencies,

the process proved time consuming and hence, when a general idea of where the peaks can be

seen is discovered, this frequency sweep can be refined to a smaller range thereby reducing

the time necessary for the system to run. The second method is to drive the system

completely at random. This method replicates running the system with white noise which

represents the effects of the environment on vibrational energy harvesters as outlined by

Adhikari who specifically investigates the mean harvested power due to stationary Gaussian

white noise base excitation. (Adhikari, 2009) The final method is to drive the system at

individual frequencies. The benefit of this method is that by knowing the lower and higher

modes, the system can then be run at these modes and therefore allows for obtaining the

maximum power from the system. This is a repeatable method unlike when running the

system at random. It is clear from the resulting figures that the peaks are more clearly defined

than that of the frequency sweep and total power values for individual frequencies can then be

calculated.

A number of research papers have been found which consistently state that a nonlinear system

outperform its linear counterpart. The governing equations which were implemented in this

project were presented by Gammaitoni et al. with other papers also utilising these equations

making the work performed here consistent with other works by Heung Soo Kim (Heung Soo

Kim, 2011), Ferrari (M. Ferrari, 2010), Vocca (Vocca, 2012) and Cottone (F. Cottone, 2009).

The steady state solution found in this project after a chosen time results in an initially higher

voltage before levelling off at a similar value as seen previously. Optimizing the steady state

value in this system could possibly improve on the energy conversion efficiency as stated by


54

Shu and Lien when they investigate the efficiency of energy conversion in a piezoelectric

power harvesting system. (Y C Shu, 2006)

The figures created throughout this project would not have been possible without creating a

total power variable in the code. This was crucial in the investigations in Section 4.2.3.

Driving the system with random noise, as stated earlier, produced unrepeatable simulations

which were presented in this paper. The investigation was performed for varying values of

both the number of time steps and the nonlinearity so as to confirm that unrepeatable

simulations were not just the case for one variable.

The linear relationship found between the amplitude of driving squared and the total power

when the nonlinearity value was at zero is as expected as is the nonlinear relationship

recorded when nonlinearity was introduced. There is an opportunity for further investigation

as to the limit of power for the system which can be obtained when the amplitude of driving

squared is increased. From the resulting figures it is evident that there is a max power which

can be attained and therefore a max amplitude of driving can be found be continuing on this

investigation for larger values. Work by Marzencki et al. reports that when the amplitude of

vibration increases, the nonlinear term becomes dominant. (Marcin Marzencki, 2009) Hence,

varying the value of the nonlinearity as well as the amplitude and recording results for the

total power can produce an optimum value for a specific system.

Investigation into finding an optimum nonlinearity value resulted in a few different scenarios.

The fact that no data was produced when the system was run with nonlinearity values greater

than 17.32 when run at both the lower and higher mode meant that the system coincides with

the findings of Triplett et al. (Triplett, 2009) In their paper, they state that choosing a

nonlinearity value too large can result in a reduction of power in the system. The optimum

for the lower mode peak 1 and total power resided at zero since these were the main power

consumers in this mode, whereas peak 2 wanted a larger nonlinearity value which makes

sense since peak 2 resides at the higher mode. A definite optimum is seen when driven at the


55

higher mode as both peak 2 and total power result in a max value of power at a nonlinearity

value of 3.

The transfer of energy between modes is graphically explained in the respective section

showing how the energy transfers as the driving frequency changes as it is not entirely clear

from the recorded graphs. Gaps occur due to peaks overlapping in one of the graphs and large

power values occur due to the said peaks overlapping in the other. It is then clear that the

various driving frequencies investigated have different effects on the system. Further

investigation could be in finding optimum values for the modes so that the most power can be

attained, i.e., should the modes be further apart or closer together, and what effect does this

have on the stability of the system?

Investigating the effect that nonlinearity has on the system is also quite interesting. The work

shows that having no nonlinearity at all in the system can actually produce more power

compared to some nonlinearity values. The figures in Section 4.3.5 display this by showing

that values of nonlinearity of 4.5 and 6 actually result in less power. Triplett et al. conclude

that the above is true which prompts the thinking that even though nonlinear systems

outperform linear ones, the value for nonlinearity still has to be carefully chosen to suit the

particular system or a better tuned linear counterpart may actually work better. (Triplett,

2009)


56

5 Conclusions

In conclusion, the concept of a coupled oscillator in a nonlinear system has been modelled

and investigated based on the equations consistently found in a number of studies. The

expressions implemented here are useful in quantifying the harvested power under both

random vibration and vibration at specific frequencies. The system employed here has

presented results which indicate that a nonlinear system can outperform its linear counterpart

provided the magnitude of the nonlinearity is carefully chosen. When driven at specific

frequencies, the total power of the system can be easily identified and the transfer of energy

between the lower and higher modes is evident. The study finds optimum values for

nonlinearity based on the variables chosen and that there is in fact a limit to the magnitude of

the nonlinearity which can exist within a structure. The amplitude of the driving and the total

power attained have a proportional relationship in a linear system but are not proportional in a

nonlinear system.

In the future, further work into the finding of an optimum value for damping in the system

could prove helpful as outlined in the discussion. Also, work towards optimum steady state

values can improve the energy conversion efficiency when the system is employed in a

piezoelectric energy harvesting system. Furthermore, efforts to obtain the limit for the

amplitude of driving at which the power is at a maximum can again improve the power

harvested by the system. The transfer of energy between the lower and higher modes for this

specific system is defined in this paper but further work can be to seek optimum values for

these modes so that the total power of the system is at a maximum.


xi

6 References

Adhikari, F. I. (2009). Piezoelectric energy harvesting from broadband random vibrations.

Smart Materials And Structures, 7.

Antonio Messineo, A. A. (2012). Piezoelectric Bender Transducers for Energy Harvesting

Applications. ICAEE, 39-44.

C.R. McInnesa, D. G. (2008). Enhanced vibrational energy harvesting using nonlinear

stochastic resonance. Journal of Sound and Vibration, 655–662.

Chung-Yang Sue, N.-C. T. (2012). Human powered MEMS-based energy harvest devices.

Applied Energy, 390-403.

Cottone, M. N. (2011). Nonlinear Kinetic Energy Harvesting. The European Future

Technologies Conference and Exhibition 2011, 190-191.

F. Cottone, H. V. (2009). Nonlinear Energy Harvesting. PHYSICAL REVIEW LETTERS,

080601-(1-3).

Gammaitoni, N. V. (2009). Nonlinear oscillators for vibration energy harvesting. Applied

Physics Letters, 3.

Gammaitoni, N. V. (2010). The benefits of noise and nonlinearity: Extracting energy from

random vibrations. Chemical Physics, 435-438.

Giordano, N. J. (1997). Computational Physics. New Jersey: Prentice Hall.

Heung Soo Kim, J.-h. K. (2011). A Review of Piezoelectric Energy Harvesting Based on

Vibration. International Journal of Precision Engineering and Manufacturing, 1129 -

1141.

Lei Zuo, X. T. (2013). Large-scale vibration energy harvesting. Intelligent Material Systems

and Structures, 1405 - 1430.

Leland, W. (2006). Resonance tuning of piezoelectric vibration energy scavenging generators

using compressive axial preload. Smart Materials and Structures, 1413-1420.

M. Ferrari, V. F. (2010). Improved energy harvesting from wideband vibrations by nonlinear

piezoelectric converters. Sensors and Actuators A: Physical, 425–431.

M. Umeda, K. N. (1997). Energy storage characteristics of a piezo-generator using impact

induced vibration. Japanese Journal of Applied Physics, 3146-3151, 3267-3273.

Marcin Marzencki, M. D. (2009). MEMS Vibration Energy Harvesting Devices With Passive

Resonance Frequency Adaptation Capability. JOURNAL OF

MICROELECTROMECHANICAL SYSTEMS, 1444-1453.

Noel E. duToit, B. L.-G. (2005). Design Considerations for MEMS-Scale Piezoelectric

Mechanical Vibration Energy Harvesters. Integrated Ferroelectrics, 121-160.


xii

Priya, S. (2007). Advances in energy harvesting using low profile piezoelectric transducers.

Journal of Electroceramics, 165-182.

Quinn, T. B. (2011). Comparing Linear and Essentially Nonlinear Vibration-Based Energy

Harvesting. Journal of Vibration and Acoustics, 011001-1 - 011001-8.

Renno, D. I. (2009). On the optimal energy harvesting from a vibration source. Journal of

Sound and Vibration, 386 - 405.

Stephen. (2006). On energy harvesting from ambient vibration. JOURNAL OF SOUND AND

VIBRATION, 409-425.

Triplett, Q. (2009). The Effect of Non-linear Piezoelectric Coupling on Vibration-based

Energy Harvesting. Journal of Intelligent Material Systems and Structures, 1959-

1966.

Vocca, N. T. (2012). Kinetic energy harvesting with bistable oscillators. Applied Energy, 771-

776.

Y C Shu, I. C. (2006). Efficiency of energy conversion for a piezoelectric power harvesting

system. JOURNAL OF MICROMECHANICS AND MICROENGINEERING, 2429–

2438.


xiii

7 Appendix

Figure 49: MatLAB code created to model the system described in this project implementing the variables

outlined in Table 2


xiv

Figure 50: Analysis file which produces the graphs used throughout the project part one


xv

Figure 51: Analysis file which produces the graphs used throughout the project part two


xvi

Figure 52: Graph showing the relationship between N and Peak 2 when driven with random noise

Figure 53: Graph showing the relationship between N and Peak Total Power when driven with random

noise

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0 5 10 15 20

ln(sdP2/avgP2)

ln(N)

Graph showing the relationship between the number of timesteps N and Peak 2 when driven with random noise

ln(sdP2/avgP2)

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0 5 10 15 20

ln(sdTP/avgTP)

ln(N)

Graph showing the relationship between the number of timesteps N and Peak Total Power when driven with random noise

ln(sdTP/avgTP)


xvii

Figure 54: Graph showing the relationship between b and Peak 2 when driven with random noise

Figure 55: Graph showing the relationship between b and Total Power when driven with random noise

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

-15 -10 -5 0

ln(sdP2/avgP2)

ln(b)

Graph showing the relationship between the nonlinearity b and Peak 2 when driven with random noise

ln(sdP2/avgP2)

-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

-15 -10 -5 0

ln(sdTP/avgTP)

ln(b)

Graph showing the relationship between the nonlinearity b and Total Power when driven with random noise

ln(sdTP/avgTP)


xviii

Table 8: Table showing how the average and standard deviations for the changing N over a number of runs

were calculated

No. of Data

Points (N)

Amplitude of

Driving (A)

Peak 1

(Power) (W)

Peak 2

(Power) (W)

Total Power

(W)

1000 0.1 0.175 0.137 2.88E-06

1000 0.1 0.0458 0.0245 8.23E-07

1000 0.1 0.057 0.013 7.89E-07

1000 0.1 0.108 0.005 1.01E-06

1000 0.1 0.054 0.034 1.59E-06

Average (avg) 0.08796 0.0427 1.41885E-06

Standard

Deviation (sd) 0.048717208 0.048172191 7.86019E-07

10000 0.1 2.1 1.4 1.10E-06

10000 0.1 4.57 0.9 1.41E-06

10000 0.1 2.35 1.6 1.13E-06

10000 0.1 1.94 1.49 7.63E-07

10000 0.1 2.75 2.35 1.31E-06

Average (avg) 2.742 1.548 1.14416E-06

Standard

Deviation (sd) 0.954010482 0.467135955 2.21855E-07

100000 0.1 4.2 0.8 9.12E-07

100000 0.1 3.6 2.25 1.01E-06

100000 0.1 3.45 1.2 1.06E-06

100000 0.1 2.3 0.8 1.08E-06

100000 0.1 3.1 1 9.32E-07

Average (avg) 3.33 1.21 9.9814E-07

Standard

Deviation (sd) 0.625779514 0.540740233 6.62611E-08

1000000 0.1 0.97 1.83 1.09E-06

1000000 0.1 1.95 0.75 1.08E-06

1000000 0.1 1.45 1.3 1.01E-06

1000000 0.1 1.68 0.36 1.06E-06

1000000 0.1 2.9 2.2 1.02E-06

1000000 0.1 5.7 3.2 1.04E-06

1000000 0.1 1.4 3.35 1.04E-06

1000000 0.1 4.56 0.5 1.02E-06

1000000 0.1 2.51 0.85 1.06E-06

1000000 0.1 2.05 1.6 9.63E-07

Average (avg) 2.517 1.594 1.04E-06

Standard

Deviation (sd) 1.364205063 1.007851179 3.44204E-08


xix

Table 9: Table showing how the average and standard deviations for the changing b over ten runs were

calculated

Amplitude of

Driving (A)

Nonlinearity

(b)

Peak 1

(Power) (W)

Peak 2

(Power) (W) Total Power (W)

0.1 0.0001 1 0.55 6.45E-07

0.1 0.0001 2.2 2.3 1.23E-06

0.1 0.0001 1.55 2.1 8.73E-07

0.1 0.0001 1.6 5.6 1.34E-06

0.1 0.0001 1.6 1.35 9.54E-07

0.1 0.0001 0.86 1.97 8.17E-07

0.1 0.0001 2.2 2.6 1.11E-06

0.1 0.0001 1.25 1.16 8.95E-07

0.1 0.0001 5.7 4.5 1.51E-06

0.1 0.0001 2.2 0.85 9.62E-07

Average (avg) 2.016 2.298 1.03412E-06

Standard

Deviation (sd) 1.311184198 1.527611207 2.49672E-07

0.1 0.00001 1.5 1.13 8.73E-07

0.1 0.00001 3.02 3.35 1.69E-06

0.1 0.00001 2.25 1.15 1.21E-06

0.1 0.00001 2.3 1.7 1.08E-06

0.1 0.00001 1.2 1.45 7.35E-07

0.1 0.00001 0.8 0.98 6.17E-07

0.1 0.00001 1.9 1.27 1.03E-06

0.1 0.00001 1.15 0.35 6.55E-07

0.1 0.00001 3.7 1.15 1.43E-06

0.1 0.00001 4.3 1.1 1.30E-06

Average (avg) 2.212 1.363 1.06E-06

Standard

Deviation (sd) 1.094804092 0.739257059 3.32785E-07

0.1 0.000001 2.75 1.2 1.10E-06

0.1 0.000001 5.1 1.2 1.36E-06

0.1 0.000001 1.4 1.4 9.54E-07

0.1 0.000001 1.35 3.5 1.06E-06

0.1 0.000001 1.25 3.45 1.13E-06

0.1 0.000001 3.9 1.2 1.28E-06

0.1 0.000001 2.35 1.2 8.92E-07

0.1 0.000001 1.7 1 8.99E-07

0.1 0.000001 4.15 0.8 1.36E-06

0.1 0.000001 1.96 1.43 9.71E-07

Average (avg) 2.591 1.638 1.10E-06

Standard

Deviation (sd) 1.283405236 0.934181995 1.70505E-07


xx

Table 10: Data recorded when showing the linear relationship between the Amplitude of driving squared

and Total Power when b = 0

Driving

Frequency

Nonlinearity

(b)

Amplitude

of Driving

(A)

Amplitude

of Driving

Squared

Peak 1

(Power)

(W)

Frequency

Peak 1

Total

Power (W)

4.1 0 0 0 0.0205 4.0518 4.5104E-09

4.1 0 0.1 0.01 5229.214 4.1461 0.00091198

4.1 0 0.2 0.04 20917 4.1461 0.0036

4.1 0 0.3 0.09 47060 4.14605 0.0082

4.1 0 0.4 0.16 83664 4.14605 0.0146

4.1 0 0.5 0.25 130725 4.14605 0.0228

4.1 0 1 1 522900 4.14605 0.0911

4.1 0 1.5 2.25 1176500 4.14605 0.205

4.1 0 2 4 2091600 4.14605 0.3645

4.1 0 2.5 6.25 3268100 4.14605 0.5695

4.1 0 3 9 4706000 4.14605 0.82

4.1 0 3.5 12.25 6405000 4.14605 1.1162

4.1 0 4 16 8366300 4.14605 1.4578

4.1 0 4.5 20.25 10590000 4.14605 1.8451

4.1 0 5 25 13072500 4.14605 2.2779

4.1 0 6 36 18824000 4.14605 3.2801

4.1 0 7 49 25625000 4.14605 4.4646

4.1 0 8 64 33465000 4.14605 5.8313

4.1 0 9 81 42355000 4.14605 7.3803

4.1 0 10 100 52290000 4.14605 9.1114

4.1 0 15 225 117650000 4.14605 20.5006

4.1 0 20 400 209150000 4.14605 36.4455


xxi

Table 11: Data recorded when showing the nonlinear relationship between the Amplitude of driving

squared and Total Power when b = 2

Driving

Frequency

Nonlinearity

(b)

Amplitude

of Driving

(A)

Peak 1

(Power)

(W)

Frequency

Peak 1

Peak 2

(Power)

(W)

Frequency

Peak 2

Total

Power

(W)

TotalPower

4.1 2 0 0 0.0209 4.0518 0.0000565 4.9313 4.5932E-09

4.1 2 0.1 0.01 4960.3 4.14605 0.85 8.229 0.00086327

4.1 2 0.2 0.04 17425 4.14605 8.2 8.229 0.003

4.1 2 0.3 0.09 33780 4.14605 28.5 8.229 0.0059

4.1 2 0.4 0.16 51770 4.14605 64 8.229 0.009

4.1 2 0.5 0.25 70637 4.14605 117 8.229 0.0123

4.1 2 1 1 168500 4.14605 635 8.229 0.0294

4.1 2 1.5 2.25 270000 4.14605 1600 8.229 0.0472

4.1 2 2 4 375000 4.14605 3065 8.229 0.0657

4.1 2 2.25 5.0625 428000 4.14605 3975 8.229 0.075

Table 12: Results from driving the system at the lower mode and changing the nonlinearity

Driving

Frequency

Nonlinearity

(b)

Amplitude

of Driving

(A)

Peak 1

(Power)

(W)

Frequency

Peak 1

Peak 2

(Power)

(W)

Frequency

Peak 2

Total

Power (W)

4.1 0 0.1 5229.2 4.146

0.00091198

4.1 0.1 0.1 5228.47 4.1461

0.00091185

4.1 0.2 0.1 5226.27 4.1461

0.00091145

4.1 0.3 0.1 5222.6 4.1461

0.00091079

4.1 0.4 0.1 5217.515 4.1461

0.00090987

4.1 0.5 0.1 5211 4.1461

0.00090868

4.1 1 0.1 5157.66 4.14605

0.00089899

4.1 1.5 0.1 5072.57 4.14605

0.00088353

4.1 2 0.1 4960.3 4.14605

0.00086327

4.1 2.5 0.1 4826.034 4.14605

0.00083944

4.1 3 0.1 4675.625 4.14605 1.4 8.2293 0.00081339

4.1 3.5 0.1 4515.811 4.14605 1.7 8.2293 0.00078636

4.1 4 0.1 4353.75 4.14605 2 8.2293 0.00075922

4.1 4.5 0.1 4195.3 4.14605 2.3 8.2293 0.00073239

4.1 4.75 0.1 4118.6 4.14605 2.5 8.2293 0.00071912

4.1 5 0.1 4043.58 4.14605 2.64 8.2293 0.00070594

4.1 6 0.1 3756.07 4.14605 3.16 8.2293 0.00065431

4.1 7 0.1 3481.396 4.14605 3.61 8.2293 0.0006065

4.1 8 0.1 3233.32 4.14605 4.017 8.2293 0.00056461

4.1 9 0.1 3020.914 4.14605 4.371 8.2293 0.00052694

4.1 10 0.1 2826.13 4.14605 4.66 8.2293 0.00049245

4.1 15 0.1 2115.25 4.14605 5.71 8.2293 0.00036922

4.1 17.32 0.1 1892.5 4.14605 6.044 8.2293 0.00033035


xxii

Table 13: Results from driving the system at the higher mode and changing the nonlinearity

Driving

Frequency

Nonlinearity

(b)

Amplitude

of Driving

(A)

Peak 1

(Power)

(W)

Frequency

Peak 1

Peak 2

(Power)

(W)

Frequency

Peak 2

Total

Power (W)

4.89 0 0.1

20605 4.9313 0.0023

4.89 0.1 0.1

20618 4.9313 0.0023

4.89 0.2 0.1

20649.5 4.9313 0.0023

4.89 0.3 0.1

20703 4.9313 0.0023

4.89 0.4 0.1

20779 4.9313 0.0023

4.89 0.5 0.1

20878 4.9313 0.0023

4.89 1 0.1

21773 4.9313 0.0024

4.89 1.5 0.1

23624 4.9313 0.0027

4.89 2 0.1

27500 4.9313 0.0031

4.89 2.5 0.1

35357 4.9313 0.004

4.89 3 0.1

40350 4.9313 0.0046

4.89 3.5 0.1

37125 4.9313 0.0042

4.89 4 0.1

31875 4.9313 0.0036

4.89 4.5 0.1

27463.5 4.9313 0.0031

4.89 5 0.1

23847.5 4.9313 0.0027

4.89 6 0.1

18397.5 4.9313 0.0021

4.89 7 0.1

14677 4.9313 0.0016

4.89 8 0.1

12023 4.9313 0.0013

4.89 9 0.1

10065 4.9313 0.0011

4.89 10 0.1

8574.35 4.9313 0.00096149

4.89 15 0.1

4588.7 4.9313 0.00051462

4.89 17.54 0.1

3594.8 4.9313 0.00040326


xxiii

Table 14: Table showing how the total power and the power of each peak changes as the driving frequency

is changed

Driving

Frequency

Nonlinearity

(b)

Amplitude

of Driving

(A)

Peak 1

(Power)

(W)

Frequency

Peak 1

Peak 2

(Power)

(W)

Frequency

Peak 2

Total Power

(W)

0 0 0.1 5.53E-05 4.931 0.02055 4.0518 4.5104E-09

0.1 0 0.1 0.02047 4.0518 0.055 0.125 9.4814E-09

0.2 0 0.1 0.02053 4.0518 0.157 0.22 2.372E-08

0.5 0 0.1 0.02035 4.0518 0.8751 0.534 7.5991E-08

1 0 0.1 0.02 4.0518 1.5239 1.0365 1.3756E-07

1.5 0 0.1 0.02 4.0518 2.0357 1.539 2.033E-07

2 0 0.1 0.0211 4.0518 2.8206 2.0416 3.2704E-07

2.5 0 0.1 0.024 4.0518 4.407 2.5442 6.3162E-07

3 0 0.1 0.031 4.0518 8.9253 3.047 1.6954E-06

3.5 0 0.1 0.0808 4.08325 59.56 3.5178 9.1474E-06

3.75 0 0.1 0.6625 4.08327 317.73 3.769 0.00004314

3.8 0 0.1 853.27 3.832 0.000068321

3.85 0 0.1

789.615 3.8948 0.00011986

3.9 0 0.1

2817.5 3.9262 0.00024671

3.95 0 0.1

7062.77 3.989 0.00068725

4 0 0.1

29716 4.0204 0.0035

4.1 0 0.1

5229.2 4.146 0.00091198

4.15 0 0.1

4631 4.1775 0.00091198

4.2 0 0.1

2168.93 4.2403 0.0002271

4.25 0 0.1

1411.6 4.2717 0.0001561

4.3 0 0.1

1450.3 4.3345 0.00011981

4.35 0 0.1

521.68 4.3659 0.000099659

4.4 0 0.1

1073.96 4.4287 0.000087932

4.45 0 0.1

720.08 4.4916 0.000082201

4.5 0 0.1

790.73 4.4523 0.00008052

4.55 0 0.1

973 4.5858 0.000083017

4.6 0 0.1

450.13 4.6172 0.000090657

4.65 0 0.1

1307.1 4.68 0.00010521

4.7 0 0.1

1051.02 4.7429 0.00013272

4.75 0 0.1

1971.16 4.7743 0.00018729

4.8 0 0.1

3536.49 4.8371 0.00031519

4.85 0 0.1

5110.08 4.8685 0.00073963

4.89 0 0.1

20607 4.9313 0.0023

4.9 0 0.1

40519.6 4.9313 0.0032

4.95 0 0.1

12876 4.9941 0.0018

5 0 0.1 0.3253 4.05185 4159.525 5.0255 0.00037359

5.05 0 0.1 0.15895 4.0519 1431.68 5.0884 0.00013494

5.1 0 0.1 0.0763 4.0205 498.98 5.1198 0.000064188

5.15 0 0.1 0.0271 4.0518 443.3477 5.1826 0.000035631

5.2 0 0.1 0.0165 3.9262 134.126 5.2454 0.000021699


xxiv

5.25 0 0.1 0.03823 4.0518 164.9063 5.2768 0.00001416

5.3 0 0.1 0.0316 4.518 96.008 5.3396 9.6805E-06

5.35 0 0.1 0.0327 4.052 59.0657 5.3711 6.8649E-06

5.4 0 0.1 0.02625 4.052 61.6 5.434 5.0328E-06

5.5 0 0.1 0.02437 4.052 34.74 5.5281 2.8872E-06

Table 15: Data showing the effect of varying the nonlinearity and the driving frequency on particle 1

Particle 1

Nonlinearity 0 0.5 1.5 3 4.5 6

Driving

Frequency

0 0.02055 0.0206 0.0208 0.0212 0.0216 0.02195

3 0.031 0.0313 0.0317 0.03255 0.033 0.0305

4

4.5

4.9

5.5 0.02437 0.0244 0.0248 0.0258 0.027 0.02735

Table 16: Data showing the effect of varying the nonlinearity and the driving frequency on particle 2

Particle 2


Driving

Frequency

0

3

8.9253 8.9253 8.9267 8.93196 8.942 8.9547

4

29716 30170 43635 44690 27200 18717

4.5

790.73 790.74 790.77 790.91 791 790.92

4.9

40519.6 41265 48470 51913 33318 22666

5.5

34.74 34.737 34.7342 34.72 34.7 34.6624


xxv

Table 17: Data showing the effect of varying the nonlinearity and the driving frequency on the total power

Total

Power


Driving

Frequency

0

4.5104E-09 4.5295E-09 4.5708E-09 4.6421E-09 4.7269E-09 4.8148E-09

3

1.6954E-06 1.6955E-06 1.6956E-06 1.6959E-06 1.6961E-06 1.6959E-06

4

0.0035 0.0037 0.0053 0.0054 0.0033 0.0023

4.5

0.00008052 0.000080521 0.000080526 0.000080545 0.000080573 0.000080595

4.9

0.0032 0.0033 0.0039 0.0042 0.0027 0.0018

5.5

2.8872E-06 2.8872E-06 0.000002887 2.8861E-06 2.8843E-06 2.8815E-06

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