Vibration based electrical energy harvesting

UNIVERSITA’ DEGLI STUDI DI PERUGIA

DOTTORATO DI RICERCA IN FISICA

XX CICLO

NNNNonlonlonlonlinear Piezoelectricinear Piezoelectricinear Piezoelectricinear Piezoelectric Generators Generators Generators Generators

for Vibration Energy Harvestingfor Vibration Energy Harvestingfor Vibration Energy Harvestingfor Vibration Energy Harvesting

Francesco Cottone

Relatore Coordinatore

prof. Luca Gammaitoni prof. Maurizio Busso

A.A. 2006/2007

iii

…To My Family

4

AcknowledgAcknowledgAcknowledgAcknowledgeeeementsmentsmentsments

I would like to express my gratitude to my supervisor, Prof. Luca Gammaitoni,

whose experience, understanding, vast knowledge, ideas and patience, helped me

considerably at every step of this journey. I’ve appreciated, in particular, his great

availability whenever I needed and his constants encouragements. I would like to

thank Dr. Helios Vocca for the assistance, useful discussions he supplied at all levels

of the research project and most of all for his unavoidable irony (specially about the

religious issues). A very special thanks goes out to Dr. Paolo Amico for his skill and

the fruitful conversations and to Dr. Flavio Travasso for his real sympathy, lifestyle

and the many hours spent together in the “cold” laboratory! After the beast, I would

like to thank the beauty of our team who is Dr. Anna Dari for his exigent questions

about all human knowledge! She has been a constant source of inspiration and

incitement for me. Thanks to Chiara Molinelli for his continuous encouragements

and friendship. I would like to thank Dr. Ludovico Carbone for being a great advisor.

Appreciation also goes out to Dr. Leone Bosi and Dr. Igor Neri for the kindness and

all of their computer and technical assistance throughout my doctorate work. I would

like to thank Dr. Michele Punturo who made me know the research team and the

VIRGO project. Again, I have been honored to receive some useful suggestions by

Prof. Fabio Marchesoni.

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My special thanks to my university friends: Alessio, Filippo, Marco, Ruggero for

the great time I had with them and to all my colleagues. I enjoyed their friendship

and their support. I wish to thanks all my nearest friends Carlo, Ciro, Matteo, Daniele

together with all others which have been near me for moral support.

Finally, I doubt that I would not ever have been able to make such adventure

without the fundamental support and affection of my family. I owe them my eternal

gratitude. I also wish to thanks my fiancée, Martina, for her love and encouragement

specially in the last days of this hard work!

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AbstractAbstractAbstractAbstract

Ambient energy harvesting has been in recent years the recurring object of a

number of research efforts aimed at providing an autonomous solution to the

powering of small scale electronic mobile devices. Among the different solutions,

vibration energy harvesting has played a major role due to the almost universal

presence of mechanical vibrations: from ground shaking to human movements, from

ambient sound down to thermal noise induced fluctuations. Standard approaches are

mainly based on resonant linear oscillators that are acted on by ambient vibrations. In

spite of continuous optimizations and improvements, such linear oscillator-based

transducers present severe limitations like narrow bandwidth, need for continuous

frequency tuning, high resonant frequency at MEMS dimensions and low efficiency,

also independently by transduction technique used (piezoelectric, electromagnetic or

electrostatic). Here we propose a new method based on the exploitation of the

dynamical features of stochastic nonlinear oscillators. In particular, this work has

concerned the theoretical study, numerical and finite element modeling,

implementation and experimental test of stochastic bistable piezoelectric oscillators

employed for scavenging energy from vibrational noise. Such a concept is shown to

outperform standard linear oscillators and to overcome some of the most of present

approaches. Experimental tests have been carried out on a simple physical model

based on a bistable stochastic driven piezoelectric beam under repulsive magnetic

7

field and are in excellent agreement with all numerical expectations. We demonstrate

that the power performances of piezoelectric transducer in nonlinear dynamical

regime are almost greater by an order of magnitude as compared to linear dynamical

behaviour. In effect, the bistable system shows the ability to adsorb vibrational

energy from a wide bandwidth, mostly at lower frequencies. Moreover, this method

is not only restricted to bistable systems but even better to other kind of nonlinear

systems that should be investigated. We prove that the method proposed here is quite

general in principle and could be applied to a wide class of nonlinear oscillators and

different energy conversion principles. Finally, there are also potentials for realizing

micro/nano-scale power generators that is the natural continuation of this research

work.

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Table of ContentsTable of ContentsTable of ContentsTable of Contents

CHAPTER 1 ITRODUCTIO .................................................................................................... 9

1.1 MOTIVATIONS: MOBILE MICROPOWERING ............................................................................ 9 1.2 POWER DEMAND OF MICRO AND NANODEVICES .................................................................. 15 1.3 POTENTIAL POWER SOURCES .............................................................................................. 19

1.3.1 Energy storage systems ................................................................................................. 20 1.3.2 Power distribution methods .......................................................................................... 29 1.3.3 Power harvesting methods ............................................................................................ 31

1.4 COMPARISON OF POWER SOURCES ...................................................................................... 40

CHAPTER 2 VIBRATIO DRIVE MICROGEERATORS ............................................... 45

2.1 EXISTENT VIBRATION TO ELECTRICITY CONVERSION METHODS.......................................... 45 2.1.1 Electrostatic generators ................................................................................................ 46 2.1.2 Electromagnetic generators .......................................................................................... 51 2.1.3 Piezoelectric generators ............................................................................................... 54 2.1.4 Energy density of transduction mechanisms ................................................................. 60

2.2 DYNAMICS OF LINEAR TRANSDUCER: LINEARITY AND TRANSFER FUNCTION ...................... 61 2.3 NONLINEAR ENERGY HARVESTING SYSTEMS IN A DUFFING-LIKE POTENTIAL.................... 66

2.3.1 The inverted piezoelectric pendulum ............................................................................ 69 2.4 BISTABLE PIEZOELECTRIC BEAM IN A REPULSIVE MAGNETIC FIELD .................................... 74

2.4.1 Energy balance ............................................................................................................. 78

CHAPTER 3 UMERICAL AALYSIS AD EXPERIMETAL RESULTS...................... 81

3.1 ANALYSIS OF BISTABLE STOCHASTIC OSCILLATORS .......................................................... 81 3.1.1 &umerical Approach ..................................................................................................... 81 3.1.2 Simple Duffing Oscillator ............................................................................................. 82 3.1.3 Theoretical Considerations ........................................................................................... 92

3.2 PIEZOELECTRIC DUFFING GENERATOR ............................................................................... 95 3.3 PIEZOELECTRIC INVERTED PENDULUM IN A MAGNETIC FIELD ......................................... 109

3.3.1 Experimental Setup and Characterization .................................................................. 110 3.3.2 Finite Element Analysis .............................................................................................. 114 3.3.3 &umerical and Experimental Results .......................................................................... 117

CHAPTER 4 COCLUSIOS .................................................................................................. 126

4.1 EFFECTIVENESS OF NONLINEAR APPROACH VERSUS LINEAR ............................................. 126 4.2 MINIATURIZATION PERSPECTIVES OF NONLINEAR POWER HARVESTING SYSTEMS ............ 130

APPEDICES ................................................................................................................................... 137

BIBLIOGRAPHY ............................................................................................................................. 147

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Chapter 1Chapter 1Chapter 1Chapter 1

IntroductionIntroductionIntroductionIntroduction

1.11.11.11.1 Motivations: mobile Motivations: mobile Motivations: mobile Motivations: mobile micromicromicromicropoweringpoweringpoweringpowering Feynman's vision seems to be finally realized thanks to ever more sophisticated

microelectronics. Last decade have seen an incredible development of miniaturized

devices and today micro and nano-scale machines are yet incorporated in all kinds of

electronic devices. The strong expansion of multibillion-dollar market of portable

electronics has led to large research efforts in size reduction as power consumption

as well. Nowadays the MEMS (Micro-Electro-Mechanical-Systems) and emerging

&EMS (Nano Electro-Mechanical Systems) technology have permitted the

development of so called smart devices: sub-millimeter wireless sensor and

actuators. These micro devices have wide applications that cover from military

applications to the industry of consumption. Initial research was mainly funded by

Chapter 1. Introduction

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DARPA in military research projects focused to the realization of self-organized

wireless networks of a large number of sensor nodes (e.g.,Smart Dust, NEST[1]). A

wireless sensor network could be defined as a wireless, ad hoc, multi-hop,

unpartitioned network of heterogeneous, tiny, mobile sensor nodes that could be

randomly distributed in the area of interest and they may make use of existing

communication infrastructures [2]. Each node can be an autonomous complete

complex system with sensors, actuators, memory, processor, radio or optical

communication interface and power supply. More recently those devices have been

employed in wide variety of civilian applications such as environmental monitoring,

biomedical sensors, RFID, interactive control, integrated biology, agriculture,

structural monitoring, sensing harmful chemical agents, location of person,

transports.

source

destination

Streaming Data to/from the Physical World

Figure 1.1

Multihop Wireless Sensor etwork


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Wireless autonomous sensors and actuators with sizes under a centimeter and

below already surround us in an almost invisible way and in a forthcoming future for

example intelligent clothing and body-area networks could monitor our health

parameters. The data that's collected by miniaturized pressure sensors built into

buildings, roads, bridges and railways will be used by construction engineers. It's

almost impossible to find areas of our civilization that will not be affected.

While modern electronics continue to reduce past boundaries of integration,

increasing density and shrinking the systems, however, arise the problem of scaling

in the same time the on-board power supply [3]. Research continues to develop ultra-

low power circuit [4] and higher energy-density batteries but the amount of energy

available is not infinite and limits the system's life. Extended life is critical in many

systems with limited accessibility, such as biomedical implants or micro-sensors

inserted in building structure.

a

b

c

d

Figure 1.2

a) Intel Mote b) Implantable sensor

c) Smart Dust d) Intel chip


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For a series of motives, these devices cannot be easily powered by ordinary batteries:

• almost impossible to built micro/nano scale batteries • unpractical to replace a large number of batteries in a micro-devices once

these have finished their charge • batteries of these devices dispersed in the environment in large quantities

will produce a significant pollution • prohibitive cost of wiring power for dense network of nodes

It’s need to develop alternative methods of power these wireless microdevices that

must be economical, efficient and ecological. There are different ways to address the

problem:

• improve energy density of storage systems • develop innovative methods to deliver power to nodes (e.g. wireless

power transmission[5]) • develop self-powering nodes that harvest and convert the energy directly

from the ambient [6]

In the last years many efforts have been done to scale down power energy

devices. Fuel cells technologies, micro heat engine or micro-nuclear batteries, for

example, promise energy densities several time higher than chemical batteries and

are capable of far higher maximum power output [7]. Electronic Radio Frequency

Identity tags, smart cards and other many passive electronic devices are yet powered

by a close energy transmitted to them to perform their operations [8]. However, this


13

method is not a good solution when considering dense networks of wireless nodes. In

fact, it is not suitable for distances beyond 5-10 meter where high power transmitter

is required with a consequent efficiency loss. In that case, this technology would

probably present a health risk and may exceeds local or international regulations of

maximum radio-frequency human exposition.

The best solution to avoid battery replacement is that each node must be

autonomous and self-powered, adsorbing for example a renewable source of energy

continuously from the ambient [9] (solar, vibrations, electromagnetic, thermal), but

this method is that less explored as fully as the powering by storage systems.

For such reasons a large research effort has been devoted in designing on-board

power generators that could supply the necessary amount of energy when and where

necessary. Among the different energy sources available in a generic environment,

kinetic energy available through random vibration is probably the most common

form. Random vibrations come in a vast variety of forms, amplitude, spectral shapes

and durations. It is quite difficult to imagine a single generator that is capable of

harvesting energy from sources as diverse as wind induced movements, seismic

ground shaking and car motion.

Present working solutions for vibration-to-electricity conversion are based on

oscillating mechanical elements that convert kinetic energy into electric energy via

capacitive, inductive or piezoelectric methods[10], to mention the most common

physical principles exploited. Linear oscillators are usually designed to be resonantly

tuned to the ambient dominant mechanical frequency. However, in the vast majority

of cases the ambient vibrations come with their energy distributed over a wide


14

spectrum of frequencies, with significant predominance of low frequency

components. This is the case for example of the omnipresent seismic vibrations. In

order to take advantage of such energy spectral distribution it is necessary to tune the

oscillator resonant frequency as small as possible. Due to the geometrical/dynamical

constraints that the dimensions of the device pose, the efficiency of such mechanical

resonant oscillator is sometimes seriously limited.

To overcome these difficulties we propose a different approach based on the

exploitation of the nonlinear properties of non-resonant oscillators. Specifically we

demonstrate that a bistable oscillator, under proper operating conditions, can provide

better performances compared to a linear oscillator in terms of the energy extracted

from a generic wide spectrum vibration. This dissertation is focused on exploitation

of stochastic nonlinear dynamics with a focus on bistable systems for improving

powering scavenging methods, useful for low-consumption devices at the micro and

sub-micron scale.

The starting point is power demands constraints of mobile computing electronics

with a special focus on wireless sensor node but in perspective open to all MEMS

and NEMS (Nano Electro-Mechanical Systems) world powering issues. In the next

paragraphs of this chapter will be shown a survey of state of the art techniques and

methodologies about mobile powering. Then, will be reviewed various potential

power sources with a focus on vibration noise. In chapter two will be discussed the

existents vibration-to-electricity conversion methods, the theory of linear oscillator


15

and will be showed the idea to take advantage from nonlinear dynamics of a bistable

piezoelectric oscillator in order to obtain an hyper efficient vibration energy

harvesting system. Numerical simulations, Finite Element Analysis and experimental

test of some bistable piezoelectric systems will be presented in chapter three. Finally

in chapter four will be exposed the conclusions: a comparison of performances of

nonlinear vs linear energy harvesting systems and perspectives of implementation of

these generators at micro and nano-scale.

1.21.21.21.2 PowerPowerPowerPower demanddemanddemanddemand ofofofof micromicromicromicro and nanoand nanoand nanoand nanodevicesdevicesdevicesdevices

The Moore’s law states that the transistors doubling every couple of years and the

Bell’s law that a new computing class born every ten years. Although electronics

became smaller and smaller, enabling today’s mobile technologies explosion, in

parallel the need for energy scale-down became a serious challenge. Semiconductor

miniaturization is followed by the decrease in the power demand of single transistors,

but this savings is being counteracted by a higher structural density of transistors and

higher power leakage caused by quantum effects (fig. 1.3). In order to decrease the

thermal dissipation and consequently the working temperature of processor, it has

been push down as much as possible the supply voltage. In the next figures it’s

shown the historical exponential increment of transistors, power consumption for a

processor and the decrement of the supply voltage[11].


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Figure 1.3 Upper: integrated circuit complexity

Lower left: processor power (Watt) active and leakage.

Lower right: processor supply voltage, (ITEL source)

For the mini scale down to nano level the development of ultra-low energy

consumption electronic devices constitutes a great challenge as for the macro scale as

well. A PDA (Personal Digital Assistant) or a cellular with a battery capacity of 1500

mAh is significantly more energy efficient than a PC using about 200-250 W and

likewise a Notebook (23 - 54 W) but the performance of portable power supply

systems is the major limitation on the amount of applications and computing

performance provided by portable electronics. The power demand constraints can be


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classified as a function of the overall average linear size but the functionalities and

computing performances do not necessarily follow the same trend.

Device class Linear size Power Requirements

Server, workstation 50cm-90cm above 100 W

Desktop PC 20-50cm 200-300 W

Notebook PC 20-35cm 20-50 W

Handled 1-10cm 80m W-10 W

Wireless Sensor Node 0.1-1cm 100 µW-100 mW

Nanodevices Nanorobots

0.01-1µm 0.1-100 µW

Table 1.1

Comparison of power demand of electronics devices

Since the main subject of this work regards the physics of energy harvesting

systems that is crucial for low-scale powering, we now shift our attention to the

micro and nano-world area. The main important features for sub-centimeter or sub-

millimeter devices, that constitutes a great challenge, is the availability of small,

lightweight, low-cost, energy efficient electronics while the computing performance

is not always critical. The main goals for the wireless sensor nodes are that they must

be smaller than one cubic centimeter, weigh less than 100 grams, and cost

substantially less than one dollar. Even more important, the electronic components

of the node must use ultra-low power to extend the battery life and to avoid frequent

replacement. However, The huge success in reducing the size of MEMS, in effects, is

limited by a lack of similar reductions in power supplies. Although new nano-


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materials are improving the battery technology, its energy density doesn’t follow the

exponential curve as the Moore’s law for miniaturization process and performance.

In order to explore the possibility of energy scavenging techniques for self-powering

device many WSN (Wireless Sensor Nodes) researchers have provided new

specifications. The most important specifications for the power supply system are the

total size and average power dissipation of an individual node (i.e. for a PicoNode

that communicates over a 10 meters range in PicoRadio network system[12]): the

size of a node must be overall less than 1cm3 and the target average power

dissipation of a completed node must be below 100µW. This power constrain is

particularly difficult, and it is likely that several technology efforts will be necessary

to achieve this goal but this is the upper limit that will address the energy harvesting

studies. Therefore, it’s a measure of acceptability for an energy scavenging system

design. This does not mean that power system solutions which don’t meet this

feature are not worthy of further exploration, but simply that this constrain will

constitute a desirable limit or standard for the most projects of wireless sensor nodes.

Going into more tiny dimensions, in the domain of EMS from 10nm to 3µµµµm the

there are even more problems about energy issue. Energetics represent a serious

limitation in nanotechnology robot design. Mechanical motions, pumping, electronic

process, computing, chemical transformations etc. require energy. Nanodevice could

in the next future metabolize in vivo local glucose and oxygen for energy, or power

could be externally supplied from a sound wave of radio-frequency sources. Heat

dissipation is also a major question in nanomachine design, particularly when large

numbers of nanomachines are deployed in vivo. It’s interesting to envisage a


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possible estimate order-of-magnitude of nanorobots power consumption. An energy

harvesting nanosystem could use organism metabolism as an energy reservoir.

Freitas [13] makes an estimation using as a first crude approximation the power law

P=(4.13)m3/4 , where ‘m’ if mass of organism to calculate the available power

density. Supposing P=100 Watts for an m= 70 kg human body mass and assuming

water density for nanorobots, then P = 23pW for a 1 µµµµm3 nanorobot, therefore a

power density of d~2x107 watts/m

3. For nanorobots chemically powered by an

oxyglucose engine it’s need to consider the fundamental limits on power density that

are imposed by diffusion limits on glucose molecules. For a spherical nanorobot of

radius rn~0.5 micron in arterial blood plasma it is possible to estimate a maximum

chemical power density of d~109 watts/m3 or 0.1nW/µµµµm 3.

1.31.31.31.3 Potential power sourcesPotential power sourcesPotential power sourcesPotential power sources

The current state of research on vastly different potential power sources for micro

and nano systems is not so simple to discuss. Furthermore there are many good

works about this topic[10, 14, 15] and it’s not the aim of this thesis to do an ulterior

deep inspection. Here we want only to make a survey and a comparison among

storage systems and renewable sources. Power sources are distinguished as energy

storage systems, power distribution methods or power harvesting methods,

which enable micro-devices to be completely self-sustaining. We will deem that a

power source acceptable when it’s capable of providing power density on the order


20

of 100µµµµW/cm3 for at least ten years of duration. For a sake of simplicity the

principal metric which we will use in this work for evaluating power sources is

power per volume, specifically µW/cm3 and J/cm3 for energy density.

1.3.11.3.11.3.11.3.1 EEEEnergy nergy nergy nergy storagestoragestoragestorage systemssystemssystemssystems

Today there are many forms of energy storage that may be used in micro-systems

such as wireless sensor node dependently by the type of energy: electrochemical,

electromagnetical, chemical, nuclear, kinetics. Each of these forms present

advantages and disadvantages. The more recent storage systems suitable to be

embedded on board of wireless node can be summarize here:

• Batteries and Micro-batteries

• Ultracapacitors • Micro-fuel cells • Micro-heat engines • Nuclear Radioactive power sources

From the first zinc-copper cell invented by A. Volta in 1779 to the last years the

electrochemical battery has seen constant technological improvements and today, as

never before, it is the most diffused mean of energy storage for medium scale devices

and household hardware. In effects, this device are probably the easiest and most

practical solution for electronic devices because its flexibility and availability.


21

Among its major advantages there is the stability of the output voltage that allow to

the systems to run directly without any power transformation, so reducing the

dissipation of extra energy. As we have already stressed, the energy density and

lifetime are the crucial features that discriminate the various storage systems.

Common batteries are grouped in the two main classes: primary and secondary

batteries. Primary even called disposable batteries reversibly transform chemical

potential energy to electrical energy, once the initial supply of chemical agents is

exhausted, energy cannot be further restored to the battery by electrical means.

Secondary batteries can be recharged, that is, have their chemical reactions reversed

by supplying electrical energy to the cell, restoring their original composition.

Among the disposal most commercial batteries we find Zinc-carbon, Zinc-air,

Alkaline, Mercury, and Lithium. These are most commonly used in portable devices

with light current drain, mostly in circuits where electric power is used intermittently

such as sensors, alarm, radio communication, small calculators but even in a fairly

high and constant consumption electronics like hearing aids and watches. Among

that we have listed, Lithium batteries are the most expensive but they possess high

energy density (2880 J/cm3), high voltage (3-4V per cell) and almost the best

duration. For example, a lithium battery with a capacity of 1000mAh can provide

energy to a wireless sensor node with an average consumption of 100µW for at least

one year. For these reasons the Li-ion battery is one of best ready-to-use solution for

powering current wireless sensor nodes.

Rechargeable batteries have less energy density (Lithium-ion 1080 J/cm3) than

non-rechargeable ones but for their intensive use are mostly employed in notebook


22

computer, cell phones, PDA’s, digital camera and so on. In the context of micro-

sensor devices another primary power source must be used to charge them. It’s clear

that periodically connect the nodes to a power grid is almost impossible. Indeed it

could be possible to recharge the on-board battery by solar cell as a possible solution.

However, we must taking into account of the extra dissipation due to the control

electronics for charging process. In any case, the more electrolyte and electrode

material there is in the cell, the greater the capacity of the cell. Thus the capacity of a

cell scale with its size a this is the principal reason for that it doesn’t follow the

miniaturization historical trend (Fig. 1.4).

Figure 1.4 - Historical ICT improvements

with battery energy density trend [16]


23

1.3.1.11.3.1.11.3.1.11.3.1.1 BBBBatteriesatteriesatteriesatteries

Going beyond this trend requires developing innovative storage technologies or

searching for a new energy source but recently, battery research seems to find a

rebirth helping by nanotechnology. A research team at Rensselaer Polytechnic

Institute has implemented a “paper battery” designed to function as both a lithium-

ion battery and a supercapacitor by infusing carbon nanotubes into a cellulose

substrate with a lithium hexafluorophosphate solution[17]. The nanotubes works as

one electrode and the lithium metal that cover the white side of the film is the other.

The sheets can be twisted, rolled, folded, or modeled in to numerous shapes with no

loss of efficiency or stacked, like printer paper to boost total output (every film can

produce 2.5volts of electrical potential). Their light weight and the inexpensive

material make them attractive for portable electronics, aircraft, automobiles and

medical devices. In addition, they are biodegradable a major drawback of chemical

cells. This discover can revolutionize the micro-batteries research field. In fact, the

possibility to stack many layers solves the problem of low surface area and low

current output typical of the small electrode of an cubic centimeter in-chip

battery[18]. Either bi-dimensional thin film or three-dimensional micro-battery with

electrode surface of 3-4cm2 could have maximum current throughput of 20mA at

4.2volts but there are problems of inherent non-uniformity of current[19], durability

and of containing aqueous electrolyte.


24

1.3.1.21.3.1.21.3.1.21.3.1.2 Micro Micro Micro Micro FFFFuel cellsuel cellsuel cellsuel cells

Another promising storage device is the micro fuel cell that new state of the art

fabrication technologies have permitted to realize. The main important feature is that

their energy density is higher than an order of magnitude then conventional batteries

(18 kJ/cm3 Vs 2 kJ/cm

3 of ordinary battery) . This technology can drive a cellular

phone on standby for 6 months as opposed to 2 weeks with lithium ion batteries, a

notebook computer for a week. At large scale fuel cells can produce sufficient power

for an electric car engine or an house backup energy system. These devices differ

from conventional electrochemical cells and batteries. Both technologies involve the

conversion of potential chemical energy into electricity. But while a conventional

cell or battery employs reactions among metals and electrolytes whose chemical

nature changes over time, the fuel cell actually converts the chemicals hydrogen and

oxygen into water or another fuel such as methanol from which extracting hydrogen

(DMFC Direct Methanol Fuel Cells), and in the process it produces electricity

leaving nothing but an empty reservoir or cartridge. A proton membrane separate the

proton from the hydrogen atom and with electrons recombine with oxygen atoms on

the other side. Micro fuel cells also offers an higher power density (100mW/cm2

for µDMFC up to 250mW/cm2 of UltraCell RMFC [20] has demonstrated in 2005)

than micro-battery thanks to high surface to volume ratio but seems to operate good

only at higher temperatures. So, even if at large scale these device have reached 45-

90% of efficiency, at micro scale they show of it only a 20% of maximum efficiency

for methanol type. Although these technologies presents many advantages and are


25

attractive for the micro-scale world, they are not yet mature. There are also

disadvantages like not so small size (the most commercial are of to order of

centimeter), high costs, membrane corrosion, bad tolerance to wide temperature

range.

1.3.1.31.3.1.31.3.1.31.3.1.3 UUUUltra capacitorsltra capacitorsltra capacitorsltra capacitors

A middle way between rechargeable batteries and common electric capacitors is

represented by ultra-capacitors. They are like capacitors that store electrostatic

energy via charge separation but using electrode-electrolyte interface instead of

classical dielectric layer. No chemical reactions are involved in their energy storage

mechanism so that they provide a very high efficiency. Because of their very long

lifespan (even one million recharge cycles), short charging, high performance to

release high power in a short time, they are attractive for many applications even

working in parallel with classical batteries. While they reach significantly higher

power density (10kW/kg) their energy density if only one order of magnitude lower

then common battery (∼∼∼∼100J/cm2 Maxwell Technologies, NEC). Many applications

that need high power peak performance can benefit from ultracapacitors but they

could be employed in wireless sensor nodes only working in conjunction with

ordinary batteries as a secondary power sources. This imply more energy dissipation

due to the power control electronics. Furthermore, they are already limited by high

costs.


26

1.3.1.41.3.1.41.3.1.41.3.1.4 Micro HMicro HMicro HMicro Heat eat eat eat EEEEnginesnginesnginesngines

The energy density of a liquid hydrocarbon fuel is of the order of 30kJ/cm3 so, if

a tiny engine can convert it into electrical power, such a technology would provide at

almost 10 times the energy density of a Li-ion battery. Nowadays, the micro heat

engine research field is one of the most funded all over the world. Many approaches

have been taken such as millimeter-scale gas turbine engine[21], MEMS-scale

Wankel rotary engine[22, 23], free and loaded piston internal combustion engine[24],

thermo-photovoltaic micro-generator[25] and recently the very innovative smallest

piezoelectric heat engine called P3[26].

a

B

Figure 1.5 - a) SiC coated MEMS Wankel engine components.

b) The P3 vapor cycle heat engine concept

The P3 is based on a thin film piezoelectric PZT transducer that convert fuel

energy into electrical. It is capable to run off a variety of sources, from diesel fuel to

solar energy or even waste heat from an hot surface or an exhaust pipe. A voltage 4V

and 1mW of power have been achieved with membrane generators 9mm2 in area and

2-3 µm thick[27]. The power performance predicted for micro heat generators ranges

from 0.1 to 20W within a linear dimension of some millimeter up to 5-10cm.

However, this so large power output does not always represents a benefit, specially

for wireless sensor micro node where the power request is small whereas their


27

autonomy in time, that depends on energy storage density, is more important. So, the

engine would intermittently charge up a secondary battery or capacitor. If the energy

efficiency doesn’t overcome 20% the energy density of about 7 kJ/cm3 gains only a

factor 2-3 relative to a Li-Ion battery. Furthermore, devices that burn fuel potentially

involve issues with heat, exhaust pollution, noise, thrust, or safety.

1.3.1.51.3.1.51.3.1.51.3.1.5 Nuclear Nuclear Nuclear Nuclear micromicromicromicro----batteriesbatteriesbatteriesbatteries

More exotic emerging power technologies like nuclear micro/nano-batteries have

great potentialities as an on-board MEMS power supply. For example, an alpha or

low-energy beta emitter can provide energy to MEMS for decades dependently only

by the half-life of radioisotope (63Ni source has an half-life of 100.2 years). In

effects, nuclear energy density is 3 to 5 orders of magnitude greater than chemical

energy density (∼∼∼∼105kJ/cm

3 Vs 7kJ/cm

3 of Lithium battery). The radioactive

material could be in both solid and liquid form: current best candidates as nuclear

sources of electric charges are, for example, 63i,

3H,

210Po. Evidently, no gamma

emitters are possible sources such as 238U for commercial application because of the

heavy shielding needed to avoid heath risks and electronics damages. There are two

main viable methods under investigation and at micron-scale some prototypes have

been realized. The first concept is that of a semiconductor junction-type battery[28]

that make use of betavoltaic effect. The second concept is based on self-

reciprocating cantilever[29, 30]. The betavoltaic effect is the generation of

electrical potential due to net positive charge flow of the β-particle induced electron


28

hole pairs (EHPs). As electric field of the depletion region sweeps the induced

charges across the junction a resulting current is created from n to p-type layer.

a b

c

Figure 1.6

a) Betavoltaic microbattery based on a pn-junction with the the inverted pyramid tank of

Liquid 63iCl/HCl solution. b) View of the bulk-icromachined inverted pyramid array.

c) Picture of a packaged sample of betavoltaic micro-battery based on a planar Si pn-diode with

electroplated 63i [28].

In the cantilever concept, the emitted charged α or β particles are collected by a

cantilever plate faced in front of radioisotope source. The increasing electric force

created deflects the beam until it contacts a ground-electrode. After this point the

beam initiate to oscillate while the plate restarts to collects other charges for another

cycle. In this mode, the cantilever became like an intermittent oscillator and its

kinetic energy can be converted into electrical by means of piezoelectric material or a

magnetic transducer (fig. 1.7a).


29

a

b

Figure 1.7 – a) Drawing of a 63i radioisotope piezoelectric cantilever.

b) Sensor microchip with on-board PZT nuclear generator [30].

Although these technologies are particular attractive for long-lasting applications

(i.e. space missions, pacemakers or any other medical implantable micro-sensors),

they have no high efficiency: only 0.5% it’s been demonstrated for betavoltaic

working prototypes up to 2% for a 2cm x 1cm x 0.5cm PZT nuclear resonator.

Moreover, they are not yet suitable for high power devices. The electric power of

them range from 10nW/cm3 for betavoltaic technology to about 10µµµµW/cm

3 of PZT

cantilever resonator type.

1.3.21.3.21.3.21.3.2 Power distributionPower distributionPower distributionPower distribution methodsmethodsmethodsmethods

Direct power distribution to sensor nodes or generic mobile devices is an other

viable method in addition to energy stock method. In this case, the energy density or

power density per unit volume is not a good metric to measure the performances

because the power received primarily depends by the efficiency of the power

transferred to them. The effectiveness of this method is on the effective quantity of

the power adsorbed over that transmitted.


30

1.3.2.11.3.2.11.3.2.11.3.2.1 Electromagnetic Radio Frequency DistributionElectromagnetic Radio Frequency DistributionElectromagnetic Radio Frequency DistributionElectromagnetic Radio Frequency Distribution

Wireless Power Transmission through RF electromagnetic field is not a new idea.

Wireless technologies were being investigated and implemented by many physicists

during the early 1900s. Nikola Tesla designed his own transmitter with power-

processing capability some five orders-of-magnitude greater than those of its

predecessors. Then, in a demonstration performed by Bill Brown between 1969 and

1975 a microwave ray of 30Kw was beamed over a distance of 1 Mile at 84%

efficiency. After that, far field wireless power transfer systems based on traveling

microwaves have had no great success because of the health and safety risks due to

strong interaction of focalized microwave beam with biological tissues. However

with the use of resonant coupling, wavelengths produced are far lower making it no

more dangerous than being exposed to radio waves. For instance, WiTricity

(Wireless-Electricity) technology developed recently by a MIT research group[5] is

based on near field inductive coupling through magnetic fields like RF ID tags. They

were able to transfer 60 watts with ~40% efficiency over distances at about 2 meters.

Nevertheless, this technology is not suitable for mid-range and long-range (5-10

meters and beyond) power distribution. The power transmitted to a node is expressed

by P(r) = Po λ2/(4πr2) where ‘Po‘ is the transmitted power, ‘λ’ is the wavelength of

the signal and ‘r’ is the distance between transmitter and receiver. Assuming a

maximum distance of 10 meters in the frequency band of 2.4-2.485 GHz and that a

single node consumes at max 100µW, the power transmitter needs to emit 10-14

watts of radio-frequency radiation. Consequently, the safety limitation would need to


31

be exceeded to power a dense wireless sensor network. Besides, as the power

transmitted fall more realistic as 1/r4 indoor the efficiency will follow the same trend

as well.

1.3.2.21.3.2.21.3.2.21.3.2.2 Wires, acousticWires, acousticWires, acousticWires, acoustic, lasers, lasers, lasers, lasers

The advantages of distribute power to sensor nodes by means of wires are very

limited. For example, it would be convenient method only in new architectures or

devices where the power grid for sensors should be foreseen in the design. But for

dense sensor network this way is not practicable due to high costs, prohibitive

maintenance and reduced flexibility.

Extracting energy from acoustic wave could be feasible power source only for

ultra-low power devices or nano-devices. In effects, a sound wave of 100 dB has a

power density less then 1µW/cm2 that is far from the 100µW/cm3 target discussed

before. Finally, if at first glance deliver energy to sensors with a laser directly

focused toward them is possible, this method does not present so many benefits and

it is very complex and expensive for sensing applications.

1.3.31.3.31.3.31.3.3 PowerPowerPowerPower harvesting methodsharvesting methodsharvesting methodsharvesting methods

Doubtless, the most attractive way to provide “perpetual” power to sensor

avoiding the refueling and making it completely self-supporting seems to be the

“energy harvesting” method. In fact, a power scavenger would be limited only by

failure of its own electro-mechanical components. On the other hand, these methods


32

are not yet deeply explored because of the complex variety of environments, each

one with its forms of renewable energy source. So that, there is not a unique solution

suitable for all environments and applications. Unlike energy reservoirs, power

scavenging sources provide the energy for the time during which the source is in

operation. Therefore, they are primarily characterized by the power density, rather

than energy density.

1.3.3.11.3.3.11.3.3.11.3.3.1 Solar sourceSolar sourceSolar sourceSolar source

Solar energy of an outdoor incident light at midday holds an energy density of

roughly 100mW per square centimeter and up to 0.15mW/cm2 on cloudy days.

Instead, the lighting power density in indoor environments ranges from about

0.45mW/cm2 provided by a 60W desk lump down to 0.010mW at the surface of an

office desk[12]. Commercially off-the-shelf single crystal solar cells offer

efficiencies of about 15% and up to 20-40% for the state of the art expensive

research cells. However, these type of solar cells are not suitable for indoor

environments because they are affected by severe degradation of the open circuit

voltage[31]. Thin-film polycrystalline cells are not expensive but show efficiencies

of only 10 – 13%. A cadmium telluride (CdTe) thin-film cell type has a very wide

spectral response. So, it has good performance in both indoor light conditions and

outdoor environment with an efficiency that ranges from 8 to 13%.


33

a

b

Figure 1.8

a) Integrated beacon circuit with an on-board 3x2cm2 Panasonic BP-213318 CdTe solar cell [32]

b) 16 mm3 mock-up with an integrated millimeter solar panel (Smart Dust [33])

For these reasons, it is selected which best candidate as a power generator that can

operate directly or in conjunction with rechargeable battery for wireless sensor

applications (fig. 1.8a). Even though it needs a proper power electronics to transform

the current for the battery and to optimize its lifespan. More cheaper plastic organic

photovoltaic devices have been recently fabricated[34] and exhibit an efficiency of

2.5%. This is too low for our scopes but researchers promise to reach values like that

of the inorganic cells at an half cost.

1.3.3.21.3.3.21.3.3.21.3.3.2 AAAAir flowir flowir flowir flow

The wind flow power goes as cubic power of its velocity “v” and it is direct

proportional to air density “ρ” and cross sectional area “A” by the relation

P=(1/2)ρAv3. Assuming an air density of 1.22 Kg/m3 at standard atmospheric at a

velocity between 2m/s and 6m/s the power density ranges from about 20µW/cm2 to

10mW/cm2 for a conversion efficiency of 20%. But we must take into account that at


34

low velocity the efficiency normally does not overcome 5%. Unlike large-scale

windmills have reached efficiency of 40% thanks to even more sophisticated material

technology and shapes, at small scale the research is currently quite poor because the

applications are bonded to airy environments.

1.3.3.31.3.3.31.3.3.31.3.3.3 Temperature gradTemperature gradTemperature gradTemperature gradientsientsientsients

Several approaches to convert thermal gradients into electricity are currently

under investigation (through Seebek effect, thermo-couples, piezo-thermal effect).

All of those have the efficiency related to the Carnot law expressed by equation

η=(Tmax-Tmin)/Tmax. So that, for a temperature difference of 10°C the efficiency is

about 3.3%. Considering a silicon device with thermal conductivity of 140W/mK,

the heat power that flow through conduction along a 1cm length for a ∆T=5°C is

7W/cm2. Hence, the electric power obtained at Carnot efficiency will be

117mW/cm2. At first sight this could be seems an excellent result but the real devices

have efficiencies well below the simple Carnot rule. Exploiting the Seebeck effect

some research groups[35, 36] have implemented a silicon micro-thermoelectric

generator µTEG capable of generating from 10 up to 40µW/cm2 at 10˚C temperature

differential. Others[37] groups have recently demonstrated a thermoelectric

efficiency factor of 0.83 µW/K2cm2. A more efficient approach that has already been

catalogued as “micro heat engine” is that of external combustion engine, in which

thermal power is convened to mechanical power by means of a thermodynamic cycle

that approaches the ideal vapor Carnot cycle. Mechanical power is converted into


35

electrical power using a thin-film piezoelectric membrane generator. This called P3

thermo engine[26, 27] is theoretically capable of ~ 1 mW/mm2 and over.

1.3.3.41.3.3.41.3.3.41.3.3.4 Human powerHuman powerHuman powerHuman power

The energy burnt by an average human body every day is about 10.5MJ that

corresponds to an average power dissipation of 121 Watts within an interval of

80 1600 Watts . Many research groups and industries are currently working on the

most efficient technologies to tap the energy worn by human body. For example,

piezoelectric insert embedded into a shoe can capture energy "parasitically" from

footfalls (theoretically available from 5-8 Watts up to max 68W) while walking[38].

However, the efficiency of this technology does not overcome on the average 17%,

excluding some advanced prototype, so that the effective mean power harvested is

about 1.5 Watt. Wristwatches powered by both the kinetic energy of a moving arm

and the heat flow from the surface of the skin are yet available. These make use of

the so called Inertial Power Generators based on 2 gram “proof” mass mounted off-

center on a spindle. As the user moves during the day, the mass rotates on the spindle

and winds the mechanism. Some models such as ETA Autoquartz Self-Winding

Electric Watch or Seiko AGS system Seiko creates 5µW on average when the watch

is worn and 1mW when the watch is forcibly shaken. But generally, also scaled up

these kind of scavenging energy systems do not produce more that 10mW. While it is

possible to obtain a power density of 300µµµµW/cm2 (mostly from walking energy

scavenging), the problem of a no loss transmission of electrical energy to wearable


36

sensors still remain. So, these ways could be both impractical and not cost efficient

when applied to dense network of sensor nodes.

1.3.3.51.3.3.51.3.3.51.3.3.5 Pressure variationsPressure variationsPressure variationsPressure variations

Another renewable power source could comes from atmospheric pressure and/or

thermal variation. The possible energy available E for a fixed volume V and a

pressure variation of ∆P is merely given by the equation E=∆P*V . If the pressure

varies of 677 Pa once per day the available power density would be 7.8nW/cm3.

While, considering that a pressure changing due to temperature variation for a fixed

volume of ideal gas follow the state equation ∆P=mR∆T/V where m is the mass of

the gas (i.e. Helium) and R a gas constant, for a ∆T=10°C thermal variation per day

the corresponding energy change would be 1.4 Joules, which is about 17 µW/cm3.

So, we are tens time below desirable power density for sensor nodes. Although there

are yet some devices that incorporate power supply system that make use of phase

changing of a fluid like “Atoms clock”, however, there are no recent advances in

implementing large-scale systems.

1.3.3.61.3.3.61.3.3.61.3.3.6 VibrationVibrationVibrationVibrationssss

The first important virtue of random mechanical vibrations as a potential power

source is that it is present almost everywhere. Mechanical vibrations occur in many

environments such as building, transports, terrains, humans activities, industrial

environments, military devices and so on. Their characteristics are various: spectral


37

shape from low to high frequency, amplitude and time duration are manifolds

dependently by the surroundings. Theory and experiments of many research work

shows that the power density that can be converted from vibrations is about

300µµµµ/cm3. Another strong point is that as solar the vibration source is a renewable

source as well, so, it has no lifetime and in addition it is not limited to the sunlight

areas. In order to establish how much power comes from vibrational excitation

Roundy[10] from Berkley and a other MIT groups[16] have performed a

characterization of most common environments like typical office building,

manufacturing plant, machines, human activities and household appliances.

Figure 1.9 - List of vibration sources [10] .

In figure 1.9 are shown a list of vibration sources measured with a standard

accelerometer and ordered from greatest amplitude of acceleration to least.

Frequently, the most of vibrational energy is located at fairly low frequencies (below

500Hz). As an example, vibration spectra of a microwave casing, illustrated in figure

1.10, shows sharp peaks in magnitude around 120Hz and 250Hz. Even for a milling


38

machines the peaks of fundamental mode fall around 70Hz with few higher

harmonics close to 150 and 200Hz. Likewise for a wooden deck the first vibration

modes appear at 350Hz and at 240Hz for a refrigerator. The sharp peaks at low

frequency indicate the fairly sinusoidal shape of displacement and acceleration signal

in time domain. While their narrowness is proportional to quality factor Q of the

oscillating system.

Figure 1.10 Displacement and acceleration spectra

for a turned on microwave oven and milling machine [10].

Another important characteristic that is common to most vibration source is that

the power spectrum tends to fall off as ωωωω2. Other source such as water jet assisted

drilling gives rise, as the most of macroscopic systems, to large mechanical

vibrations in bandwith below 1KHz. Vibration spectrum of this source measured

with an accelerometer by a group[39] have two main resonant peaks to consider and

those peaks at about 400 Hz and 1400Hz. Many meso and micro-scaled energy


39

scavenging generators have been developed in the last five years by an increasing

number of research groups[4, 9, 10, 16, 38, 40-42]. Three are the principal concepts

to convert mechanical vibration power into electrical one: piezoelectric, electrostatic,

electro-magnetic. These models will be more deeply analyzed in the next chapter.

The predicted power density that a such kind of micro-scaled generators can extract

from vibrational source ranges from 4µW/cm3 (human motion—Hz) up to on

800µW/cm3 (machines—kHz). But, we take into account that excitation is highly

dependent by the environment. All of these works are addressed to realize

scavenging generators that must be tuned to the fundamental vibration frequencies

of the source, where the most of energy of the vibration spectrum is sited. This

constitute a great limit. In facts, although larger structures can achieve relatively

higher power densities (for instance, a simple shake-driven flashlight can delivers 2

mW/cm3 at 3 Hz), at small scale the narrow-tuned oscillator posses a natural

resonance frequency ωn of some kilohertz (roughly ωωωωn=k/m where k is the effective

elastic constant and m the inertial mass). Far from fundamental frequency of the

environment, that’s for hypothesis within a band below 300-400Hz, the resonator is

mismatched and the efficiency falls down.

a

b

Figure 1.11 a) SEM photo of the fabricated cantilever prototype[43].


40

b) cross section of piezoelectric cantilever.

For example, a demonstrated prototype showed in figure 1.11 of a 5mm cantilever

piezoelectric resonator with a inertial Nickel mass 0.02 grams has a natural

frequency of about 608Hz and its power output is 2.16 µW. In order to obviate to

frequency detuning problem we have exploited the properties of nonlinear dynamic

oscillator that will be discussed in a comprehensive manner in chapter 2 and 3.

1.41.41.41.4 CCCComparison of power sourcesomparison of power sourcesomparison of power sourcesomparison of power sources

In order to make a direct comparison between fixed-energy source, such as

batteries and fuel cells, and renewable energy sources, it is difficult to use the same

metric. Because some sources like batteries are benchmarked by energy density

while other are characterized by power density such as solar cells. Some devices do

not need of the third dimension so, they can be characterized by power per square

centimeter rather then cubic centimeter.


41

Table 1.2 – Comparison of various potential power sources.

Green highlighted are renewable energy sources while the red ones

are the fixed energy alternatives. S. Roundy [15]

Roundy et al. have analyzed the characteristic of various power sources so far

discussed (tab. 1.2, fig.1.12). This is a starting point for the choice of optimal way for

sensors. How it has been already stated before, it’s improbably that any single

solution will satisfy all applications, because each method has its own constraints.

Solar cells require sunlight, thermal gradients need sufficient temperature variation,

and vibration-based systems need sufficient vibration sources. Conversely vibration

sources are generally omnipresent and can be readily found in inaccessible locations

such as building walls or inside of machines.


42

Figure 1.12 – Average power available Vs time from batteries

and scavenged energy sources. S. Roundy [15]

It can be noted that solar and vibration power density can be range within an

interval of 10-1000µW because of environmental conditions (outdoor or in indoor

office light condition for solar cells, low level vibrating environment) but they are no

function of lifetime. While both energy drain and leakage determine a variation in

time for chemical batteries with inflection point for rechargeable types. So that,

within one year the batteries can support efficiently a wireless sensor nodes

(assuming a 100µW of consumption) but going beyond 2 year arise the refilling

problem for those rechargeable and over 5 year the primary batteries cannot provide

the same power level of solar cells or vibration-based generators. Others interesting

indicators for a comparison among the various methods (especially for batteries) are

the specific power defined as power over weight ratio and specific cost per watt.


43

Figure 1.13 - (a) the specific power range and

(b) the power density for different methods.

Figure 1.14 - Specific cost of all energy storage systems. Flipsen[44].

Besides the combustion engines that have the highest specific power (fig. 1.13),

the renewable power sources converted by means piezoelectric and photovoltaic

transducers present performances comparable with fuel cells. In any case, these


44

methods do not have the problem of high noise output, toxic exhaust fumes and

instability of electrical power from the system typical of combustion engines. On the

front of cost-effective the piezoelectric seems to be the worst solution (fig 1.14) but

this research was been conducted for power demands in 100mW–30W, then for

meso-scaled devices. Instead, for micro devices both specific power and cost-

effective are not so critical, while, size scalability and lifespan are the most important

parameters.

In the next chapter we deal of existent methods for vibration to electricity

conversion with a focus on theoretic mathematical models for linear, nonlinear

bistable oscillators and vibration noise source as a basis from which simulations and

the experiments have been implemented.

45


VibrationVibrationVibrationVibration drivendrivendrivendriven

microgeneratorsmicrogeneratorsmicrogeneratorsmicrogenerators

2.12.12.12.1 Existent vibration to electricity conversion Existent vibration to electricity conversion Existent vibration to electricity conversion Existent vibration to electricity conversion

methodsmethodsmethodsmethods There are three possible devices that can transform ambient vibrations into

electrical energy:

• variable capacitor (electrostatic fields)

• electromagnetic inductor (electro-magnetic fields)

• piezoelectric transducer (straining a piezoelectric material)

Chapter 2. Vibration driven microgenerators

46

These three methods are commonly used for inertial sensors (i.e. accelerometers)

as well as for actuators. The best transducers systems should be those that can

maximize the coupling between the kinetic energy of the source and the conversion

mechanism dependently entirely upon the characteristics of the environmental

vibrations. Vibration kinetic energy is best suited to generators with the mechanical

component attached to an inertial casing which acts as the fixed frame. The case

transmits the vibrations to a suspended inertial mass producing a relative

displacement between them.

A brief analysis of strength and weakness points of the existent transducer models

will be outlined in this paragraph. Further details on dynamic and equivalent circuit

models will be explained in the next paragraph 2.2. with particular focus on

piezoelectric cantilever model.

2.1.12.1.12.1.12.1.1 Electrostatic generatorsElectrostatic generatorsElectrostatic generatorsElectrostatic generators

This conversion methods is based on use of a variable capacitor. It simply consists

in two plates which are electrically isolated from each other by a dielectric (typically

air, vacuum or an insulator). Unlike the simple fixed capacitor the metallic plates of

variable capacitor can be in motion in order to vary its capacitance. As the separation

between the plates (typically nanometer or microns for a MEMS) varies the energy

stored in the charged capacitor changes due to the work done by an external vibrating

force. The capacitance for a parallel plates capacitor in term of the insulator

dielectric constant k=ε/ε0 is given by:


47

0

AC k

dε=

(2.1)

where A is the plate surface, d the relative distance, ε and ε0 are the permittivity of

the dielectric material and vacuum, while, the voltage across the plates is expressed

by definition from

/V Q C= (2.2)

hence,

0

QdV

Aε= (2.3)

and the electrostatic energy stored within capacitor that is given by

2 21 1 1

2 2 2E QV CV Q C= = = (2.4)

At constant voltage, in order to vary the energy it’s needed to counteract the

electrostatic force between the mobile plates that is

2

2

1

2e

AVF

dε= (2.5)

then, the mechanical work against this electric force done by an external force like

vibrating excitation is transformed into electrostatic potential energy when varying

the capacitance. A current flow through a load shunted to plates in order to balance

the fixed voltage. A similar method like fixed voltage is that of charge constrained,

with the difference that if a constant charge is held into the plates (i.e. by means of a

battery or another capacitor), the electrostatic force is given by

1 2

2e

dF Q

Aε= (2.6)


48

but in general, the voltage constrained offers more energy than the charge

constrained approach. A manner to increase the output electrical energy for the

charge constrained method is add a capacitor in parallel with the variable harvesting

capacitor. This parallel storage capacitor effectively constrains the voltage on the

energy harvesting capacitor. A base circuit was designed by Roundy (fig. 2.1) where

Cv is the variable capacitor, Cpar the parasitic capacitance associated with the variable

capacitance and interconnections, finally, the switches which transfer the electric

current toward the storage capacitor and regulate the charging that can be substitute

by diodes.

Figure 2.1 - Simple circuit sketch for an electrostatic converter.[10]

The maximum potential energy per cycle that can be harvested by this configuration

is expressed as first approximation by the following formulas

max2

min

1

2par

in

par

C CE V C

C C

+= ∆ +

(2.7)

max

1

2 inE V V C= ∆ (2.8)

with ∆C=Cmax-Cmin and Vmax which represents the maximum allowable voltage

across a switch.


49

Up to now, there are three kinds of electrostatic generators (fig.2.1) that are based

on both constrained charge and voltage[44, 45]: In-plane overlap varying, In-plane

gap closing, Out-of-plane gap closing.

a

b

c

Figure 2.2 – a) in-plane overlap varying b)in-plane gap closing

c) Out-of-plane gap closing

For the in-plane overlap topology (a) the capacitance changes by changing overlap

area of interdigitated fingers that implements the multi-plates capacitor. While for

the other two types the capacitance changes by changing gap between fingers (b) or

large plates (c). In Table 2.1 it is shown the electrostatic force variation for the three

configurations in function of the displacement x of the inertial mass.

Table 2.1 - Electrostatic force variation for the three configurations.[46]

For the Out-of-plane gap closing type there are several problems. The gap x must

become very small in order to obtain a large capacitance change but, as the fluid

damping force is proportional to 1/x3, the loss becomes very large as the plates move


50

close together. A possible solution may be to set the MEMS device under very low

pressure. Furthermore, this design concept exhibits the problem of short-circuit

contact as the plates get close together. In-plane gap closing converter solves this

problem. For this type of configuration the motion of plates is in the plane of the

substrate, therefore, the minimum dielectric gap, and thus the maximum capacitance

can be precisely fixed by incorporated mechanical stops. As it has been investigated

by Roundy[47] (fig. 2.2) the In-plane gap closing type offers the highest power

output with an optimized design producing 100 µW/cm3; out-of-plane gap closing is

the next highest and the last in performances is in-plane overlap varying. It can be

noted that the maximum power occurs at very small dielectric gaps.

a

b

Figure 2.3 –Power output vs. dielectric gap for

a) in-plane overlap varying and b) in-plane gap closing converter

for different device thicknesses.

The planar design of an electrostatic converter has the potential to be tightly

integrated with silicon based microelectronics that are readily available. Therefore,

the scalability of its size through MEMS technology is the first reason why

electrostatic converter is attractive. On the other hand, one of the principal negative

side is the high working frequency (∼5-10KHz) of these generators. Nevertheless,


51

recently some groups [48] reports on MEMS electrostatic converter with high

electrical damping capable to operate over a wide low frequency range (<100 Hz): a

silicon microstructure of volume 81mm2×0.4mm with a 2×10−3kg inertial mass

driven by a vibration amplitude of 95µm at 50Hz is capable to produce a scavenged

power of 70 µW.

Figure 2.4 –Interdigitated fingers of MEMS prototype variable capacitor.[47]

Many other group are focusing in realization of low frequency operating converters

like Tashiro [49] which has developed an honeycomb structured electrostatic

generator that harnesses ventricular motion operating at heart beat frequency 1-2Hz

with the aim of driving a cardiac pacemaker permanently. Anyway the power output

for a square centimeter variable capacitors that have been developed so far range

from 10 to 100µW.

2.1.22.1.22.1.22.1.2 Electromagnetic Electromagnetic Electromagnetic Electromagnetic generatorsgeneratorsgeneratorsgenerators

Amirtharajah et al.[9] have previously proposed and developed electromagnetic

generators that exploits the relative motion of an electrical conductor in a magnetic


52

field produced by a permanent magnet. The device simply consists of a mass m

connected to a spring with elastic constant k that is attached to a rigid case (fig. 2.5).

The ambient vibration excites the housing which transmits the mechanical

displacement to the inertial mass. The consequent variation of magnetic flux through

the coil generates an inducted current in accordance with Faraday’s law. In this way

the part of kinetic energy stored in the movement of mass-spring system is converted

into inducted current (fig. 2.5a).

a

b

Figure 2.5 – a) Drawing of inductor generator, Amirtharajah [9]

b) Cross-section of the wafer-scale electromagnetic generator proposed by Williams[50]

There are many other preferable configurations: for example with the magnets

attached to a cantilever beam acting as inertial mass[51] or that proposed by

Williams et al. in fig. 2.5b.

Faraday’s Law states that the induced electromagnetic field produced by a

changing magnetic flux ΦB is given by

Bd

dtε

Φ= − (2.9)

hence, for a coil moving through a perpendicular constant magnetic field, the

maximum open circuit voltage across the coil is


53

oc

dxV &Bl

dt= (2.10)

where & is the number of turns in the coil, B is the strength of the magnetic field, l is

the length of a winding and x is the relative vertical distance between the coil and

magnet. Making a few assumptions: baseline vibrations of 2.25 m/s2 at 120 Hz,

maximum device size is 1cm3, and about the magnetic field intensity and coil design,

it can easily be shown that output voltage does not overcome 100mV. Far more

realistic estimates of present technology range within 50mV, otherwise always less

then 1 Volt. For a typical 5mm x 5mm x 1mm device, the predicted power generation

was 1µW for an excitation frequency of 70Hz, and 100µW at 330Hz. Though the low

voltage represents an bad limit, because it requires a rectifier and transformation

electronics to be raised, electromagnetic transduction has some strength sides. First,

high output current levels are achievable. Second, unlike electrostatic conversion, no

separate voltage source is needed to get the process started. Moreover, the almost

total absence of mechanical contact between any parts improves reliability and

reduces mechanical damping.

There is a wide variety of magnetic spring-mass concepts implemented with

various types of material[50-53] that are well suited and proven in cyclically stressed

applications. Some of those with size of 5mm x 5mm x 1.5mm achieves 35mW of

maximum power at 12.6KHz of resonant frequency. But, at sub-micron scale, many

problem arise relatively to the implementation of planar permanent magnets,

minimum line and space for coils fabrication and most of all limited amplitude of

vibrations (∼10µm).


54

2.1.32.1.32.1.32.1.3 Piezoelectric Piezoelectric Piezoelectric Piezoelectric generatorsgeneratorsgeneratorsgenerators

Piezoelectric ceramics have been used in many applications for many years to

convert mechanical energy into electrical energy. The direct piezoelectric effect was

early demonstrated by Jacques and Pierre Curie in 1880. They found that when

certain ceramic crystals were subjected to mechanical strain, they became electrically

polarized and the degree of polarization was proportional to the applied strain.

Conversely, these materials deform when exposed to an electric field. Materials

which show piezoelectricity are widely available in many natural and man-made

forms: single crystal quartz, cane sugar, Rochelle salt, piezoceramic materials (e.g.

Lead Zirconate Titanate, PbTiO3, BaTiO3 composites [54]), thin film (e.g. sputtered

zinc oxide), screen printable thick-films based upon piezoceramic powders [55],

polymeric materials such as polyvinylidenefluoride (PVDF) [56] and nanostructured

material[57].

Figure 2.6 - Piezoelectric elementary cell; (1) before poling (2) after poling.


55

The origin of the piezoelectric phenomenon is due to the asymmetry in the cell unit

of the material. When it is subjected to mechanical distortion along one direction,

aligned electric dipoles are formed due to spontaneous separation of electronic

clouds from their individual atomic center and this lead to a macroscopic net

polarization of the crystal lattice (fig. 2.6). The compressive and tensile stresses

along one single direction will generate a parallel electric field and a consequent

force that opposes to the length variation. It is also reciprocal, the same crystal

exposed to an electric potential will experience an elastic strain causing its length to

decrease or increase according to the field polarity. Each of these effects result

almost linear within small length variation relative to the crystal size.

Figure 2.7 - Behaviour of piezoceramic material. a) on-polarized state, b) polarized state,

c) electric applied after poling.[58]

Groups of dipoles with parallel orientation form the so called Weiss domains

(fig.2.7). the raw piezoelectric material has these domains randomly oriented.

Applying an electric field (> 2KV/mm), the material expands along the axis of the

field and contracts perpendicular to that axis. After poling action the material


56

presents a remanent polarization (which can be degraded by exceeding the

mechanical, thermal and electrical limits of the material) and it is grown in the

dimensions aligned with the field and it’s contracted along the axes normal to the

electric field. When an electric voltage is applied to a poled piezoelectric material,

the Weiss domains increase their alignment proportional to the voltage causing the

expansion/contraction of the piezoelectric material. In this way the piezoelectric solid

is ready to work as a sensor of actuator transducer.

The coupling between the electrical and mechanical behaviour of the material has

been approximated by static linear relations between electrical and mechanical

variables:

E

T

s d

d ε

= +

= +

S T E

D T E (2.11)

where: S is a strain tensor, T is a stress tensor, E is an electric field vector, D is an

electric displacement vector, sE is an elastic compliance matrix when subjected to a

constant electric field E, d is a matrix of piezoelectric constants, εT is a permittivity

measured at a constant stress.


57

Figure 2.8 – otation of axes

The Piezo ceramics structure is anisotropic, thus the piezoelectric effects is

dependent on direction. To identify directions the axes, termed 1, 2, and 3, are

introduced (analogous to X, Y, Z of the classical right hand orthogonal axial set).

The axes 4, 5 and 6 identify rotations (shear). The direction of polarization (3 axis) is

established during the poling process by a strong electrical field applied between two

electrodes. When a compressive strain is applied perpendicular to the electrodes that

extract the voltage the d33 coefficient determines the electro-mechanical coupling

whilst if a transverse strain is applied parallel to the electrodes piezoelectric

generator exploits the d31 coupling-coefficient. Though compressive strain can

produce much more high voltage then that transverse, however, is not a practical

coupling mechanism for vibration energy harvesting in the majority of applications.

In general, the elements of piezoelectric beams or films are coupled in the transverse

direction because such a configuration is more practical and it multiplies the applied

mechanical stress.

3

(y)

(z)

2

1 (x)

4

5

6

dir

ecti

on

of

pola

riz

ati

on


58

Table 2.2 - Coefficients of common piezoelectric materials [46].

Table 2.2 shows the fundamental constants that characterize piezoelectric material

such as k, g, εεεε and s. The matrix element kij is defined as the ratio

e

iij m

j

Ek

E= (2.12)

between the electrical energy Eie stored along the i-axes and the mechanical input

energy Eim along the j-axes. This describes the efficiency of energy conversion of

the material between electrical and mechanical form in a given direction. The matrix

gij is defined as the electric field generated per unit of mechanical stress, or the strain

developed for an applied charge density. Finally, εεεε is the electrical permittivity of

the material which is defined as the dielectric displacement per unit electric field and

s which is the compliance matrix namely the strain produced per unit of stress. In

figure 2.9 an equivalent circuit of a piezoelectric element that works as a voltage

generator (on right) and a conceptual design of a common cantilever implementation

of transducer (on left) are represented.


59

Figure 2.9 – Conceptual design and equivalent circuit of piezoelectric generator.

The voltage source represents the voltage that develops due to the excess surface

charge on the crystal. The series capacitor Cp represent the capacitance of the

piezoelectric layer which is proportional to the film permittivity and area and

inversely proportional to the film thickness. Rp represents the internal piezo-element

resistance and RL a purely resistive load. Assuming that the mechanics take place

along a single axis then each variable or constant is treated as a single scalar quantity

rather than a tensor. The open circuit voltage that results from an external mechanical

stress σ on piezoelectric beam with thickness t is defined by the expression

out

d tV σ

ε⋅

= − (2.13)

while the average power dissipated by a simple resistive load will be PL=VL2/2RL. In

reality, we have to consider an input impedance of power electronics attached to the

generator rather than a simple purely resistance.

Among the main advantages of piezoelectric transducers we find the possibility of

direct generation of suitable voltages and currents [44], small mechanical damping

material and unlike the variable capacitors they do not necessitate of separate voltage

source. Moreover, the recent manufacturing process has permitted to implement

Vp

Cp Rp

RL

Piezoelectric generator


60

piezoelectric MEMS with thin film under one millimeter[59] but some problems of a

decrement of piezoelectric coupling and an high resonant frequency still remain.

2.1.42.1.42.1.42.1.4 Energy density of transduction mechanismsEnergy density of transduction mechanismsEnergy density of transduction mechanismsEnergy density of transduction mechanisms

A fundamental starting point to benchmark the three methods so far showed can

be made by considering the theoretical inherent energy density and summarizing the

strength and weakness side of each one. The following table realized by Roundy et

al. [10] shows the practical and theoretical maxima of energy density for each

transduction techniques.

Table 2.3 - Summary of maximum energy density of three types of transducers

From the table 2.3 it jumps to the eye at once the gap between practical power

density of piezoelectric mechanism that is 17.7mJ/cm3 and the other techniques that

produce only 4mJ/cm3. Finally, the advantages and disadvantages points of each

mechanical-to-electrical energy conversion method which have been discussed are

summarized in the following table 2.4.


61

Table 2.4 - Summary of the comparison of the three transduction mechanisms. [15]

From this comparison it is clear that the most desirable conversion method results

that piezoelectric one which presents the major number of advantages. So, it is for

these reasons that this is currently the best choice to realize the micro vibration-

driven generator for energy harvesting to power sensor nodes.

On this basic choice a generic theory of linear and non-linear resonator with

piezoelectric coupling will be discussed in the next paragraphs. Anyhow, the nucleus

of this thesis, that is the idea to exploit the non-linear dynamics to enhance the

efficiency of energy conversion, can be also applied to electromagnetic and

capacitive techniques.

2.22.22.22.2 Dynamics of linear Dynamics of linear Dynamics of linear Dynamics of linear transducertransducertransducertransducer: : : : linearity and linearity and linearity and linearity and

transfertransfertransfertransfer functionfunctionfunctionfunction


62

Making use of the linear system theory it is possible to express a simple generic

model for the conversion of kinetic energy of a mass that undergoes to a vibrating

excitation into electrical energy. Vibration converters can be seen, in effects, as

second-order spring-mass oscillators. Figure 2.10 shows a generic model proposed

early by Williams and Yates[50] of such a system based on a seismic mass, m, on a

spring of stiffness, k.

Figure 2.10 – Drawing of a generic vibration-to-electricity transducer.

The energy losses by friction are related to the internal mechanical damping which is

expressed by dm , while, de defines the electrical induced damping coefficient due to

the electro-mechanical conversion. For simplicity, it can be assumed that the inertial

mass m of generator is much smaller then the vibrating mass of ambient (wall, floor,

machine) and that the vibration source is an infinite energy reservoir. In this mode

the vibrating source is unaffected by the movement of the generator. An external

vibration moves out of phase with the inertial mass when the generator housing is

vibrated at resonance resulting in a net displacement that we call for simplicity x(t),

between the mass and the frame. If we consider the displacement y(t) of the vibrating

y(t)

m

k

dm+de

x


63

housing rather than the external force, the differential equation of motion for the

mass m is the next

( ) ( ) ( ) ( ) ( )m emx t d d x t kx t my t+ + + = −ɺɺ ɺ ɺɺ (2.14)

the force on the mass is equal to the force on the mass-spring-damper, that, for a

sinusoidal vibrating excitation is

0( ) sin( )f t my Y tω= − =ɺɺ (2.15)

The simple steady-state solution for the mass displacement of the equation (2.14) is

2

022

2

( ) sin( )( )e m

x t Y t

d dk

m m

ωω φ

ωω

= −+ − +

(2.16)

setting dT =dm+de the total damping coefficient, the phase angle φ is given by

1

2tan Td

k m

ωφ

ω− = −

(2.17)

Maximum energy can be extracted when the excitation frequency is tuned to the

natural frequency of the system that is given by

/n k mω = (2.18)

The instantaneous kinetic power p(t) transferred to the mass is the product of the

force on the mass and its velocity.

( ) ( )[ ( ) ( )]p t my t y t x t= − +ɺɺ ɺ ɺ (2.19)

Taking the Laplace transform of equation 2.14 and 2.19, the transfer function is

2

2 2

( )( )

( ) 2 ( )xf

e m n n

XH

Y i

ω ωω

ω ω ω ζ ζ ω ω= =

− + + + (2.20)


64

and the power dissipated by total electro-mechanical damping ratio, namely

ζT=(ζe+ζm)=dT/2mωn, is expressed by

2 22( )

diss T n T n xfP m X m f Hω ζ ω ζ ω ω= = ⋅ɺ (2.21)

that is

3

2 30

2 22

1 2

T

n

diss

T

n n

m Y

P

ωζ ω

ω

ω ωζ

ω ω

=

− +

(2.22)

At natural resonance frequency, that is ω=ωn , the maximum power is given by

2 3

0

4n

diss

T

mYP

ωζ

= (2.23)

or as a function of excitation acceleration amplitude A0=ωn2Y0.

20

4diss

n T

mAP

ω ζ= (2.24)

For steady-state solutions like these, power remains limited and does not tend to

infinite as the damping ratio tends to zero. Separating parasitic damping ζm and

transducer damping ζe for a particular transduction mechanism forced at natural

frequency ωn, the power can be maximized from the equation

2

24 ( )e

el

n m e

m AP

ζω ζ ζ

=+

(2.25)

for fixed acceleration amplitude A, when the condition ζζζζe=ζζζζm is verified.

Mechanical dissipation cannot be avoided in a real system and as a matter of fact

it can be regulated to improve the conversion mechanism. As we can see from the


65

graph 2.11, for a sufficient acceleration, increasing the total damping coefficient will

results in a broader bandwidth response or the oscillator which loss the power

transferred. So, the damping factor control the selectivity of the device.

Figure 2.11 – Frequency spectrum of power generation

around the resonance frequency of the generator for different damping factor

It is clear from 2.22 that the inertial mass of generator should be maximized within

the geometrical constrains in order to obtain the maximum electrical output. For a

given acceleration level, power output is inversely proportional to the frequency.

Furthermore, it is necessary to know the spectral shape of vibration noise source well

to properly design the transfer function of a linear resonator. So, it is critical that the

natural frequency of a linear generator match the fundamental frequency of the

driving vibrations. Unlikely, it is not always simple to find a source that concentrate

the vibrational energy around a single frequency constantly. Often, the most part of

kinetic energy (like that of a seismic vibration) is present at low frequency or in a

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

6

ω/ωn

Power

ζ=0.05

ζ=0.1

ζ=0.2

ζ=0.3


66

wide bandwidth that changes in time. For instance, inside a car tyre or on aircraft the

spectral power density could vary a lot in time domain. Up to now, there are no so

many solutions to face up this limitation. Systems with active tuning or multimodal

resonator are under investigation [60, 61] but they present some problems of size

scaling and energy losses due to the control electronics and mechanical configuration

for auto-tuning process.

This linear model proposed by Williams and Yates is only a first approximation

and neglects the details of transduction mechanism. It is a fairly good model for

electro-magnetic transducers, but we must consider that the electro-mechanical

coupling term is not always linear (most of all for piezoelectric system) and

necessarily proportional the velocity as the mechanical damping as well.

Nevertheless, the relationships 2.22-2.24 are useful to characterize linear generators

and this model can be a starting point to compare the efficiency between linear and

nonlinear harvesting devices. In facts, it’s quite simple to calculate the electro-

mechanical induced damping term ζe for each type of conversion mechanism in first

approximation.

More detailed conversion models for a piezoelectric system in linear and

nonlinear regime will be discussed in the next paragraphs.

2.32.32.32.3 NNNNonlinear onlinear onlinear onlinear EEEEnergy nergy nergy nergy Harvesting SHarvesting SHarvesting SHarvesting Systemsystemsystemsystems in a in a in a in a

DuffingDuffingDuffingDuffing----like potentiallike potentiallike potentiallike potential


67

For the purpose to explain the main idea on which is based our research, we show

the basic dynamic model of a bistable damped oscillator which is forced by random

vibration noise. For the sake of simplicity we restrict ourselves to one-dimensional

case. The equation of motion of a mass-spring damped oscillator which is moving in

a constant Duffing-like[62] potential is

( )

( ) ( ) ( )dU x

mx t x t tdx

η σξ= − − +ɺɺ ɺ (2.26)

where m is the inertial mass that we can normalize to 1, η is the damping coefficient,

σξ is the stochastic external excitation and the conservative Duffing potential is the

quartic well

2 4

( )2 4

x xU x a b= − + (2.27)

from such a potential the 2.26 differential equation becomes the Langevin equation

3( ) ( ) ( )x t x t ax bx tδ σξ= − + − +ɺɺ ɺ (2.28)

Accounting the driven random force, for instance, a Gaussian distributed white noise

with zero-mean-value and variance σ 2 that is ( ) ( ') 2 ( ')t t t tξ ξ δ= − , the stochastic

differential equation 2.28 cannot be calculated analytically but only numerically. For

now, here we only give the basic equations and qualitatively description. As it can be

seen from figure 2.12, for a fixed b>0 and a<<0 the potential resemble to quadratic

harmonic well and in fact the oscillator behaves likewise a linear resonator. With a

close to zero the potential still has only one equilibrium position at x=0, but the basis

shape of the well becomes even more flat until the critical condition a=0 is reached.


68

Figure 2.12 – Biquadratic potential well for fixed b>0 and various values of a.

When a>0 bistability arises with the formation of two relative energy minima at

/x a b± = separated by a potential barrier with height 2 / 4U a b∆ = . In this way,

maintaining b fixed at a positive value and varying a it is possible to pass from a

linear dynamical regime to more complex soft/strong nonlinear dynamical behavior

of the oscillator. Dependently by the statistical characteristics of noise such as

standard deviation σσσσ and its autocorrelation time, a certain dynamical regime can

outperforms the kinetic energy transfer from the source to the system.

a<<0

a=0

UHxL

x

-4 -2 2 4

50

100

150

200

250

300

350

barrier height

a>0

a>>0

UHxL

x-4 -2 2 4

-20

20

40

60


69

Inserting inside the oscillator an electro-mechanical transduction mechanism we

obtain a scavenging energy generator whose dynamics is tunable by few parameters

and it will able to adsorb vibrating energy from a narrow or wide frequency

bandwidth. Bistable systems have been extensively studied in the presence of noise

both in the classical[63] and in the quantum domain[64], but some aspects related to

the energy spectrum and dynamics requires a deeper insight into the stochastic

dynamics of the oscillator because analytical descriptions do not exists. These will be

explained with the help of simulations and experiments discussed in the next chapter.

We are going to expose in the next paragraphs the basic models of bistable vibration

based transducers which were used in the simulations and experimental tests.

2.3.12.3.12.3.12.3.1 The iThe iThe iThe inverted nverted nverted nverted piezoelectric piezoelectric piezoelectric piezoelectric pendulumpendulumpendulumpendulum

The inverted pendulum is one of the simplest mechanical bistable device to

realizing at macroscopic scale. So, it has been chosen to test the main concept of the

present work. If part of its bar is made by a flexible piezoelectric beam it realizes a

nonlinear vibration-based energy harvesting system. A basic model of an inverted

pendulum with a restoring spring is sketched in fig. 2.12. An effective mass is

attached at the upper tip of a rigid bar with a global moment of inertia I. The

clamping base is forced to vibrate by a random shaking force f(t). Looking the

system from to the reference frame of the clamping base, for the sake of simplicity,

the inertial mass m sees as a stochastic force ξ(t) applied to itself. Furthermore, the

mass is subjected to a restoring elastic force relates to the effective stiffness Keff and

it is damped by a viscous force relates to the coefficient η. Using a piezoelectric


70

laminated beam as a support, it works also as a transducer element. Then, it will be

included in the model as electro-mechanical coupling term which is characterized by

effective piezoelectric coefficient Kv and its inherent capacitance Cp.

Figure 2.13 – a) Basic models of a vibrating inverted pendulum

and b) its equivalent with piezoelectric element

In the mechanical inverted pendulum represented in figure 2.13a, the motion

equation in the unidimensional angular variable θ is

sin ( )effI K mgl tθ θ ηθ θ ξ= − − + +ɺɺ ɺ (2.29)

Approximating the periodic term with the Taylor series expansion around θ=0

3sin (1/ 6)θ θ θ≈ − to the second order, we obtain the stochastic Duffing equation

3 1( )

6effmgl K mgl

tI I I I

ηθ θ θ θ ξ

− = − − +

ɺɺ ɺ (2.30)

which can be written

3 '( )a b tθ θ θ δθ ξ= − − +ɺɺ ɺ (2.31)

setting the following parameters:

X

a)

x

b)

k η

m

ξ(t)

Vp

I f(t)

piezoelectric element

Y ξ(t)

k -mg

θ

l

x

f(t)

m


71

1

'

; 6

;

mgl Keffa

I

I I

mglb

I

ηδ ξ ξ

−=

= =

= (2.32)

From the equation 2.31 we have two cases. One equilibrium position exists for

mgl<Keff (a<0) that corresponds to linear behavior with m exactly above the rotation

axis for θ=0. Two equilibrium positions: one at right and one left when mgl>Keff

(a>0) that may be approximately calculated equating the opposing torques and are

equal to 1,2 6(1 / )effK mglθ ≈ ± − or in terms of reduced parameter 1,2 /a bθ ≈ ± .

Adding an effective piezoelectric coupling term KθV(t) that can be derived by

structural geometry (relations 2.35) of the piezoelectric bender [6, 65, 66], the

governing equations system are so expressed:

3( ) ( ) ( ) ( ) ( ) '( )

( ) ( ) ( )

p

p p p

t a t b t t K V t t

C V t K t I t

θ

θ

θ θ θ δθ ξ

θ

= − − − +

= −

ɺɺ ɺ

ɺɺ (2.33)

Where Ip is the current flowing in the equivalent circuit that could be assumed a

purely resistive parallel load impedance (figure 2.9b) so that Ip(t)=Vp/RL. Otherwise,

it could be used a more complex AC-DC harvesting electronic circuit (i.e. rectifier

sketched in fig. 2.14) and with a more efficient current regulator system such as

SSDS Synchronous Switch Damping on Short[6].

Figure 2.14 – A typical AC-DC harvesting circuit

Rout Ce Vc

Piezoelectric transducer

Vp

Ip


72

As it has been previously mentioned, the piezoelectric pendulum can be easily

implemented using a bimorph piezoelectric beam as shaft with a rigid steel or

tungsten mass attached to its tip. This is the configuration that we have chosen for

experimental setup which is outlined in figure 2.15.

Figure 2.15 – Basic design of the Inverted Piezoelectric Pendulum.

Choosing as more practical observable the structural deflection ‘x’ in the

configuration of figure 2.15 rather than ‘θ’, the dynamical coupled equations (2.33)

accordingly become the (2.34)

3 1( ) ( ) 2 / * ( ) ( ) ( ) ( )

( )( ) ( )

eff va eff b p

p

p c

L b

K Kx t F x t K m x t F x t V t t

m m m

V tV t K x t

R C

δ σ ξ

= − − + − + ⋅ ⋅ = −

ɺɺ ɺ

ɺ ɺ

(2.34)

where it has been considered that the ratio x/lb<<1, hence, from the Taylor series

expansion 1tan ( / ) /b bx l x lθ −≈ ≈ truncated at first order. The effective coefficients

related to material constants and particular structural geometry selected can be

derived making use of modal analysis, Euler beam equation piezoelectric linear

m

bimorph

piezo-bender

s

strain

tp

tsh

lm

lb

m

deflection

x


73

theory. Some of them can be calculated or measured directly by an experimental

setup. Using the equations of piezoelectric linear theory 2.11 applied to piezo-

cantilever modeling[61, 67] the electromechanical coupling constants are given by:

31

1

31 1

1

2

a) 2

b)

3( / 2 / 2)c)

( )

4d)

( / 2 / 2)(4 3 )

eff

v

p

E

p p

c

p

p h

b c b

p h b m

K d aK

t k

t d Y kK

a

t tk

l l l

Ik

t t l l

ε

=

=

+=

+

=+ +

(2.35)

that are defined as

Kv the first coupling term of the piezo-electrostatic restoring force KvVp,

Kc the second coupling term relating the voltage-displacement ratio,

k1 the average strain to vertical displacement S/x,

k2 the input force to average induced stress ξ/σin .

The parameters Fa and Fb of 2.34 are used to tuning the shape of elastic-gravity

bistable potential and they depend in this case by the gravity acceleration g and

stiffness Keff. Moreover, d31, εp, YE

p and I are respectively dielectric displacement

coefficient, absolute dielectric constant, piezoelectric elastic modulus and composite

moment of inertia of the beam. While, the other are geometrical parameters showed

in fig. 2.15.

The choice of a bistable system as nonlinear system somehow simplifies the

implementation of experimental test, but it is not restrictive at all. As we will see

later, it could be possible to imagine oscillating systems which work in other kinds of


74

anharmonic potentials for our scope. For example, oscillators moving on a periodic

or complex multistable potential well are alternative systems that shall be

investigated.

2.42.42.42.4 Bistable Bistable Bistable Bistable PPPPieieieiezoelectric zoelectric zoelectric zoelectric beambeambeambeam in a in a in a in a repulsive repulsive repulsive repulsive

magnetic fieldmagnetic fieldmagnetic fieldmagnetic field

Even though we chose a commercial material with an high mass density (i.e.

tungsten 74110W density 19 gr/cm3), below millimetric dimensions the gravity force

becomes negligible respect to the elastic restoring force of a piezo-cantilever.

Moreover, it is unpractical to constrain the orientation of a wireless sensor with its

energy harvesting generator along only the axis of gravity force. An alternative

bistable system feasible on both the macro and micro scale is an oscillator in a

constant magnetic field that creates the bistable potential. A magnetic force acting to

inertial mass on the beam end could be generated by a proper combination of

permanent magnets and/or coils. This force can be repulsive or attractive in order to

counteract or reinforce the restoring elastic force of the beam. In this way, we can

adjust the shape of the potential well passing from mono to bistable dynamics. In

figure 2.16 is represented a modified version of design in fig.2.15 that implements a

piezoelectric beam with permanents magnets.


75

Figure 2.16 –Piezoelectric inverted pendulum with permanent magnets

In the hypothesis of a relative distance ∆>>hm larger then the size of each magnet,

we can assume the interaction between two permanents magnets likewise of two

“point” magnetic dipoles. The force between two dipoles having the same magnetic

moments M and their axes aligned equals[68]

2 403 / 2m rF a M rµ π= ± (2.36)

where µ0 is the magnetic permeability, r is the dipole-dipole relative distance and ar

is the unit radial coordinate vector. This supposition is not ever valid since it depends

by the specific configuration. For short distance, in facts, the magnetic force goes as

square of the distance. Nevertheless, the experimental test was resulted in better

agreement with 2.36 rather than the inverse square of distance law. Making another

permanents

magnets

lm

lb

m

∆

Y

X

ξ(t) excitation

Piezo bender

‘xd’ deflection

θ

M2

M1

a) b)

r

hm


76

strong assumption that the deflection xd is small relative to the beam length lb, we can

consider the two dipole always aligned. In this case the tangential magnetic force can

be neglected, then, the radial component of the magnetic force sees by tip magnet M1

projected along its cartesian coordinates results

0 1 22 2 5/ 2

3

2 ( )m

M M xF

x

µπ

=+ ∆

(2.37)

where the magnetic moment are considered in antiparallel configuration. So the

conservative energy of the system is expressed

( )2 2 0 1 22 2 3/ 2

1( , )

2 2 ( )eff

M MU x K x

x

µπ

∆ = + ∆ ++ ∆

(2.38)

Accordingly , the mass normalized governing equations of the system 2.39.

0 1 22 2 5/ 2

a) ( ) ( / ) ( ) 2 / * ( )

3 ( ) (1/ ) ( / ) ( ) (1/ ) ( )

2 ( ( ) )

( )b) ( ) ( )

eff eff

v

c

L b

x t K m x t K m x t

M M x tm K m V t m t

x t

V tV t K x t

R C

δ

µσ ξ

π

= − − + ≈

≈ + ⋅ − + ⋅ ⋅+ ∆

= −

ɺɺ ɺ

ɺ ɺ

(2.39)

The dynamics and stability points of the system are now controlled by the relative

magnets distance ∆ that now plays the same role of the parameter a in Duffing

potential 2.27 as it can be seen from the plot 2.17. Adjusting this parameter the

system passes from quasi-linear monostable to bistable behavior. For large ∆ the

system oscillates around the minimum located at zero displacement. When ∆ reaches

a critical value the potential well becomes flat, thus, the system remains monostable

but shows an anharmonic dynamics. After ∆ has overcomes this critical value, the

potential shows two minima separated by a rising barrier with the decreasing of ∆.


77

Figure 2.17 – Effective potential at different values of parameter ∆∆∆∆.

The position of two minima can be computed by differentiating the energy

expression (2.38) respect to x.

5

2 2 2( , )

3 ( )eff m

U xK x xK x

x

−∂ ∆= − + ∆

∂ (2.40)

5 72

2 2 2 2 22 22

( , )3 ( ) 15 ( )eff m m

U xK K x K x x

x

− −∂ ∆= − + ∆ + + ∆

∂ (2.41)

with 20 / 2mK Mµ π= considering magnets with equals magnetic moments.

Expanding 2.38 in Taylor series around the x=0

2

0 0

2 3 5

( , ) (0, ) ' (1/ 2) '' ...

1.. 3 ...

2

x x

eff m eff m

U x U U x U x

K K K K

= =

− −

∆ = ∆ + + + =

= ∆ + ∆ + − ∆ + (2.42)

The condition whereby the point x=0 becomes maximum is

-10 -8 -6 -4 -2 0 2 4 6 8 100.02

0.04

0.06

0.08

0.1

0.12

0.14

x (a. u.)

U(x)

∆ = 5

∆ = 7

∆ = 8∆ = 10

∆ = 15

∆ = 20


78

1

55 3

3 0 meff m

eff

KK K

K

−

− ∆ < ⇒ ∆ <

(2.43)

Setting 1/ 5(3 / )m effK Kα = , when α∆ > there is an absolute minimum and the

excited system must oscillates around it with frequency

( )2 50 0

(1/ ) '' (1/ ) 3eff mxm U m K Kω −

== = − ∆ (2.44)

In the limit case for ∆ → ∞ the angular frequency becomes 20 /effK mω = namely the

classical resonant frequency for a linear harmonic oscillator.

For α∆ < the potential has three different zeroes derived by the equation U’(x)=0,

and apart x=0, the others two are

( )2 /5 23 /m m effx K K= ± − ∆ (2.45)

The barrier height is derived from the difference ∆U=U(0)-U(xm) thus

3

52 31 5

2 2 3eff

eff m m

m

KU K K K

K

− ∆ = ∆ + ∆ −

(2.46)

Where for a decreasing ∆ the cubic term dominates the concurrent square term.

2.4.12.4.12.4.12.4.1 Energy balanceEnergy balanceEnergy balanceEnergy balance

Consider now the energy balance. Let 2.39a be multiplied by xɺ and 2.39b be

multiplied by pV . Integration of the addition of these two equations from time ti to tf

gives the equation of the energy balance


79

2 2 2

2 2 5/ 2 2

1 1 1

2 2 2

( )

f

f f f

i i i

i

f f f

i i i

tt t t

eff p pt t tt

t t t

m p p

t t t

xdt m x K x C V

K x x xdt x dt V I dt

σξ

η−

= + + + ≈

≈ − + ∆ + +

∫

∫ ∫ ∫

ɺ ɺ

ɺ ɺ

(2.47)

where, considering a purely resistive load the electrical converted energy term is

2

2 21 1

2 2

f f f

f f

i i

i i i

t t tt t p

v p p p p p p pt tLt t t

VK V xdt C V V I dt C V dt

R= + = +∫ ∫ ∫ɺ (2.48)

and the last integral represents the electrical energy dissipated on pure resistive load

RL. The physical meaning of all terms in energy balance equation (2.47) is described

in the following table 2.5.

Expression Physical meaning

f

i

t

t

xdtσξ∫ ɺ Input energy

21

2

f

i

t

tm xɺ Kinetic energy

21

2

f

i

t

eff tK x Elastic energy

2 2 5/ 2( )f

i

t

m

t

K x x xdt−− + ∆∫ ɺ Magnetic energy

2f

i

t

t

x dtη∫ ɺ Mechanical losses

21

2

f f

f

i

i i

t tt

v p p p p ptt t

k V xdt C V V I dt= +∫ ∫ɺ Converted electrical energy


80

Table 2.5 – Energetic terms definitions

The relationships discussed here allow us to carry out an analytic interpretation of

important characteristics of the nonlinear oscillator: the frequency of oscillation, the

rate of the intra-well jump e the distribution P(x) of the position as a function of

noise strength. On the other hand, being the excitation a random force, the motion

equations become stochastic nonlinear differential equations that can be solved only

through numerical methods whose results will be exposed in the next section. The

energy balance and efficiency will be computed through numerical evaluation of the

time averaged integral of mechanical x and electrical V variables multiplied by the

velocity xɺ . As it be easily seen from the expression 2.46 and numerically computing

the discrete average value of the power terms in table 2.5, in order to evaluate the

enhancement of the electrical and mechanical energy, the important observables such

as transferred mechanical energy and electrical converted energy are essentially

related to the root-mean-squared value of xrms and Vrms. These were numerically

simulated and in the next section will be shown and compared with experimental

results.

81


Numerical Analysis And Numerical Analysis And Numerical Analysis And Numerical Analysis And

Experimental ResultsExperimental ResultsExperimental ResultsExperimental Results

3.13.13.13.1 Analysis of Analysis of Analysis of Analysis of BBBBistableistableistableistable Stochastic Stochastic Stochastic Stochastic OscillatorsOscillatorsOscillatorsOscillators

3.1.13.1.13.1.13.1.1 Numerical ApproachNumerical ApproachNumerical ApproachNumerical Approach

The nonlinear model of the piezoelectric generator in a Duffing-like potential,

presented in paragraph 2.3, is described by nonlinear stochastic Langevin equations

(2.33), since the oscillator was assumed to be driven by random excitation. In order

to solve such an equation system, it is necessary to recur to numerical integration

methods. In particular, we used the Euler−Maruyama method for discrete numerical

integration of stochastic differential equations [69]. Without giving a deeper insight

Chapter 3. &umerical Analysis And Experimental Results

82

of this method, we want to describe now the statistical properties of the vibration

noise source, the equations used and the structural parameters of the prototype

model. A concise explanation of the numerical integration method is shown in

appendix A.1. Unlike the linear analysis of precedent works in which a vibration

source tuned to the natural frequency of the harvesting system was postulated, here

we have considered/examined a stochastic driving force like an exponentially

correlated Gaussian noise ECG&. It can be defined as a stochastic process which

satisfies the following differential equation [70]

( ) 1

( )ee w

c

d tg t

dt

ξξ σ

τ= − + (3.1)

where Gw(t) is a Gaussian white noise with zero mean and δ-autocorrelation

( ) 0

( ) ( ') ( ')

w

w w

g t

g t g t t tδ

=

= − (3.2)

and correlation time τc. While the autocorrelation function of process ξe is given by

2( ) ( ') exp ' /e e e ct t t tξ ξ ξ τ= − − (3.3)

with 2 2 / 2e cξ σ τ= .

3.1.23.1.23.1.23.1.2 Simple Duffing OscillatorSimple Duffing OscillatorSimple Duffing OscillatorSimple Duffing Oscillator

First of all, we want to study the phenomenology of the simple non-coupled

inverted pendulum; therefore, we can start from the non-coupled prototype Duffing

equation (2.31) in the one-dimensional variable x, where we assumed m=1

3( ) ( ) ( ) 2 ( )x t ax t bx t x tγ σξ= − − +ɺɺ ɺ (3.4)


83

In order to perform a numerical integration, we must discretize in time this nonlinear

stochastic differential equation SDE. Let the time interval [ ]0,T be divided by &

points tj for j=1,..,& with & positive integer and ( / )jt j T & j t= = ∆ , and, let the

above second order SDE be turned into a system of two first order differential

equations

3

2 1 1 2

1 2

2X aX bX X

X X

γ σξ = − − +

=

ɺ

ɺ (3.5)

where 1 2,X x X x= = ɺ . As integral forms this equations system becomes

0 0

0

32 2 0 1 1 2

1 1 0 2

( ) ( ) ( ) ( ) 2 ( ) ( ) ( )

( ) ( ) ( )

t t

t t

t

t

X t X t aX s bX s X s ds s dW s

X t X t X s ds

γ σξ

= + − − + = +

∫ ∫

∫ (3.6)

Adopting the Euler-Maruyama method (A.11), we are now able to rewrite it in terms

of discrete equations for two contiguous time steps ti, ti+1

3

2 1 2 1 1 2

1 1 1 2

( ) ( ) [ ( ) ( ) 2 ( )] ( )

( ) ( ) ( )i i i i i i i

i i i

X t X t aX t bX t X t t t W

X t X t X t t

γ σξ+

+

= + − − ∆ + ∆

= + ∆ (3.7)

This equation system can be easily numerically computed by means of a simple

code. A proper time step ∆t has been chosen sufficiently small to prevent the

summation from diverging at the end point t=T since from (A.12) the error results

1/ 2: ( )strong

t &e X X T C t∆ = Ε − ≤ ∆ . Furthermore, we accounted for a sampling error that

decreases as 1/ & , taking a sufficiently large number of samples &. Instead, we

neglected the inherent errors in the random number generator and the floating point

roundoff errors.


84

Digital simulation programs shown in A.2 were realized in MATHLAB code.

This choice for code language was made for various reasons: the ease of use, the

possibility to run the simulation on parallel architecture and the abundant

mathematics tools for signal analysis, random generators function and flexible

plotting procedures.

(The) listing 1 in the file wise_duffing.m (A.2) is structured as a function with the

following sections:

1. Declaration of constants

2. Generation of exponentially correlated Gaussian noise

3. Initialization and integration of non-linear SDE system

4. Plotting

5. Saving

The input parameters of the function: T, damp, Fa, Fb, sigma, tau represent

respectively the time interval, the damping coefficient, quartic potential parameters

(a, b), noise intensity and autocorrelation time of the exponentially correlated

Gaussian noise. Keeping fixed parameter b of the quartic well and varying a, we

examined the dynamics of the Duffing oscillator for different potential shapes and

noise strength σ. The parameters used for the first test are listed in table 3.1.


85

Parameter Value

σ [0.025,1] with step 0.025

γ 0.05

m 1 [Kg]

T 100-500 [s]

∆t 10-4 [s]

τ 0.1 [s]

a [-1,1] with step 0.025

b 1

Table 3.1 –Parameters for Duffing model.

Some realizations of simulated displacement x versus time are shown for a

negative coefficient ‘a’ in fig. 3.1 and for its positive values in fig. 3.2. It is obvious

that the quartic well is similar to the parabolic one (fig. 2.12) while a<0, therefore,

the oscillator behaves like a linear spring-mass damped system and most of the noise

power is located near its resonance frequency.


86

Figure 3.1 – Displacement [m] Vs. Time at noise magnitude σσσσ=0.1, γγγγ=0.05 and a<0.

The well base becomes even more flattened as the ‘nonlinear’ parameter ‘a’

approaches to zero. According to the relation (2.44), The first vibration mode

presents a pulsation ω0 proportional to a , that is 20 0

(1/ ) '' (1/ )x

m U m aω=

= = − . It

is worth stressing that passing from quasi-harmonic to non-harmonic potential, the

oscillation amplitude enhances, even keeping constant the intensity of excitation

force. In the case of simple inverted physical pendulum as a Duffing oscillator,

parameter a is derived from the moment of inertia and the effective elastic constant

(2.32); thus, if we want to vary only this parameter taking b fixed, it is sufficient to

change the effective elastic constant of the oscillator.

0 50 100 150 200 250 300 350 400-1

0

1

time [sec]

displacement

0 50 100 150 200 250 300 350 400-1

0

1

time [sec]

displacement

0 50 100 150 200 250 300 350 400-1

0

1

time [sec]

displacement

0 50 100 150 200 250 300 350 400-1

0

1

time [sec]

displacement

a=-0.05

a = -1

a =-0.5

a=-0.25


87

Figure 3.2 - Displacement Vs. Time at noise magnitude σσσσ=0.1, γγγγ=0.05 and a>0.

Displacement traces for positive values of a shown in fig. 3.2 give reason of the

origin of two equilibrium positions at /x a b± = separated by a barrier with height

equal to 2 / 4a b . Therefore, the mono-stable oscillation around zero is substituted by

intra-well jumps between the two minima. These jumps occur with even more low

frequency as the barrier height increases. Indeed, when the potential barrier grows,

the jump probability decreases exponentially according to Kramer’s escape rate [71]

given by

2min max(1/ 2 ) ''( ) ''( ) exp( / )jump cP U x U x Uπ σ τ±= −∆ (3.8)

0 50 100 150 200 250 300 350 400-1

0

1

time [sec]

displacement

0 50 100 150 200 250 300 350 400-2

0

2

time [sec]

displacement

0 50 100 150 200 250 300 350 400-2

0

2

time [sec]

displacement

0 50 100 150 200 250 300 350 400-2

0

2

time [sec]

displacement

a = 1

a = 0.05

a = 0.25

a = 0.5


88

where σ and τc are respectively the intensity and the autocorrelation time of the

excitation noise. After that, a critical value of barrier height has been crossed, the

system is constrained to oscillate mostly around one of the local minima with very

rare jumps. The variance of the random variable x accounts for the average

magnitude of vibrations around its mean value along a time series and it is given by

22 2 2 2

x rmsTTx x x xσ = − = − (3.9)

This is an important quantity for analyzing the dynamics and energy balance, as we

will see later. The experimentally measured value of the variance (or rms if we have

zero mean) is thus affected by the measurement time. If the measurement time is

long enough, compared to the inverse of Kramer’s rate, then the measured rms

coincides with the rms of the equilibrium process. On the contrary, if the

measurement time is significantly shorter than the inverse of Kramer’s rate, the

system is confined around one local minima and thus a lower rms is measured.

Figure 3.3 below contains the measured standard deviation STD(x)=σx versus a at

different values of noise intensity. Each value is calculated over a realization of x

series of 4 6/ 500 /10 5 10& T t −= ∆ = = ⋅ samples for 100 values of a. We must point

out that the time step ∆t affects in a crucial way the numerical convergence of the

solution of the SDE. Taking a sufficient small integration step size ∆t is one of the

key things for ensuring precision of the Euler-Maruyama approximation to equation

(3.4), but at the same time it is necessary to find a trade off with the computing time.

Indeed, this one can grow as a polynomial function of matrix size.


89

Figure 3.3 - Standard deviation in arbitrary units of displacement as a function of ‘a’ ,

b=1 fixed and noise strength σσσσ = 0.025, 0.05, 0.075, 0.1 .

This plot shows clearly the dynamical behavior discussed above. Three distinct

regimes can be identified:

1) a << 0. The potential is monostable and the dynamics is characterized by

quasi-linear oscillations around the minimum located at x=0. There is a slight

growing of the vibration amplitude when passing from the quasi-linear to the

flattened potential for a d 0.

2) a > 0. A raising peak in the nonlinear region around zero where the dynamics

is characterized by frequent jumps between the forming two wells.

3) a >> 0. The potential is bistable with a very pronounced barrier between the

two wells. There is an abrupt drop due to an insurmountable potential barrier

∆U over a critical value of a. The dynamics is mainly trapped inside one

minimum. As we have stated before, this depends on the length of the finite

-1 -0.5 0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

a

STD(x)

σ = 0.025

σ = 0.05σ = 0.075

σ = 0.1

quasi-linear region nonlinear region


90

time series on which we measure the rms. In fact, for an infinite time interval,

the jumps between the two minima would not terminate and the variance

would grow indefinitely. Such a behavior could be seen as a high pass

filtering effect, indeed, the low frequencies are primarily generated by intra-

well jumps in the bistability regime.

Figures 3.4 and 3.5 show the x standard deviation Vs parameter a and

excitation noise variance σ2.

Figure 3.4 – STD(x) versus ‘a’ and square of noise strength σσσσ=[0.05,0.2]

b =1 and autocorrelation time ττττc=0.1[s].


91

Figure 3.5 – Top view. The solid black line traces the fitting curve 23a σ= from

theoretical prediction of the maximum shift.

The top view of the 3D plot 3.4 clearly shows the evolution of the maximum σx. The

solid line is a theoretical prediction obtained with the following argument. The root

of the variance of x in regime 3 can be roughly modelled as governed by two main

contributions: i) the raising, mainly due to the growth of the separation between the

two minima; ii) the drop, mainly due to the decrease in the jump probability

measured by the crossing probability defined in (3.13), caused by the increase of the

potential barrier height ∆U. The solid black line remarks the dependence of the

maximum from parameters a and σ whose form is explained in the next theoretical

analysis. Since in a purely open circuit case the output voltage Vp of a Duffing

piezoelectric oscillator is directly proportional to displacement x, as shown by the

equations (2.33) and (2.34), the larger the standard deviation ( )x STD xσ = is, the

larger average extracted voltage Vp is. Indeed, for the open circuit case we get


92

2 22 2 2 2

p

p p

V p p x

p p

K KV V x x

C Cσ σ = − = − = (3.10)

The remarkable result shown by fig. 3.2-3.3-3.4 is that the vibration magnitude in

nonlinear region is six times larger than that in a linear region for some value of

noise amplitude. The increasing of the average mechanical and converted electrical

power output will follow the same trend since they are proportional to the root mean

square of the displacement and electric potential. These results will be discussed in

details in the next paragraphs.

3.1.33.1.33.1.33.1.3 Theoretical Considerations Theoretical Considerations Theoretical Considerations Theoretical Considerations

An attempt of theoretical interpretation of the curve in fig. 3.2 and 3.3 could be

made starting from qualitative considerations and statistical properties of the bistable

system under observation. First of all, since it is simulated and measured in a finite

time interval, we cannot consider a stationary state persisting for an infinite time.

Therefore, the rate of intra-well jumps in a bistable regime depends on the rate

between Kramer’s time and the total interval of time ∆T over which the dynamics is

generated. For a<<0 we identify quasi-linear behavior or RMS of x that can be easily

characterized; for a>>0 we are in an entrapment region of x around one of the two

minima and even in this case the dynamics is quite similar to that of a linear

oscillator oscillating close to its stable equilibrium position. Instead, for the intra-

well jumping region at positive a not far from zero, the interpretation appears more


93

complex. So we can try to approximate the variance σx using the root mean square of

the state variable x for a stationary bistable system[71, 72] driven by a additive

random noise, multiplied by Kramer’s jump rate (3.8). The approach is the

following.

Let the variance be

2 22 2 2

;x rms ststx x x xσ = − = − (3.11)

The rms value of variable x in stationary case corresponds to

2

2 2;

( )

( )

st

rms st st

st

x P x dx

x x

P x dx

∞

−∞∞

−∞

= =∫

∫ (3.12)

and the stationary probability distribution is given by

2 2 40 0 2

1( ) exp( ( ) / ) exp

2 4st c

c

a bP x & U x & x xξ

ξ

σ τσ τ

= − = −

(3.13)

with &0 as a normalization constant. Assuming positive values for a,b, 2cξσ τ and

computing the integral (3.12), we get a function of a,b, 2cξσ τ as a combination of

modified Bessel functions In(r) of the first kind:

3 1 1 3

2 4 4 4 4

1 1

4 4

( ) ( )

( ) ( ) ( ) ( )

2st

I I I I

xI r I r

r r r ra

b

− −

−

+ + +

=+

(3.14)

with r expressed by

2

2 2

1 1

2 2 2c c

a Ur

bξ ξσ τ σ τ∆

= = (3.15)

which represents the half ratio between barrier height and noise strength.


94

The relation (3.14) is a positive monotonic function in variable a proportional to

the square position of the minimum 2 /x a b± = , keeping the other parameters fixed.

To make qualitative considerations we equate 2 2rmsx x±≈ as a first approximation.

Moreover, by virtue of the arguments discussed above, we can consider the squared

mean of x as the product between the complementary Kramer’s jump probability

Pjump and 2x± , because, as the intra-well jumps decrease, the oscillations will happen

even more around one of the two local minima. Otherwise, this mean will be null for

the symmetry of the potential.

22

(1 )jumpT

x P x± ≈ − (3.16)

In this way, the equation (3.11) becomes equal to

22 2 2 2 2

; (1 ) ( )x rms st jump jumpx x & x P x & P xσ ± ± ± = − ≈ − − = ⋅ (3.17)

and, computing the jump probability times 2x± for the Duffing potential, it results

2

2 2

2

1exp

4x jump

c

a ax & P &

b bξ

σσ τ±

≈ ⋅ ⋅ = ⋅

(3.18)

where & is a constant that can be properly chosen in order to fit the numerically

simulated curve 2 ( ; , , )x cf a b ξσ σ τ= .

For the sake of identifying the dependence of the maximum position, we compute

the partial derivative of σx with respect to a,

2 2

2

0,

1 10

22c c

x

U U

c

a

a ae e

b bab

ξ ξσ τ σ τ

ξ

σ

σ τ

∆ ∆− −

∂=

∂

+ − =

(3.19)


95

obtaining that σx reaches a maximum when

2c

a b ξσ τ= (3.20)

for a given noise intensity.

As it can be seen from the solid line in figure 3.5, the theoretical prediction

obtained by this formula is in good agreement with the numerical calculus.

3.23.23.23.2 Piezoelectric Piezoelectric Piezoelectric Piezoelectric Duffing Duffing Duffing Duffing GGGGeneratoreneratoreneratorenerator

So far, we have discussed some important features of the nonlinear dynamics of a

bistable system our idea is based on. This model, anyway, does not yet incorporate

mechanical-into-electrical conversion terms. Right now, we want to examine a

generic Piezoelectric Duffing Generator (PDG) whose equivalent electrical circuit is

the same designed in fig. 2.9; so, we have to simulate the two coupled equations

(2.34) with coefficient relationships (2.35) in the case of a pure resistive load. The

model geometry is the same exposed in figure 2.15. The governing equations are

really those of a simple inverted pendulum shaken by a vibrating force at clamping

base containing the Duffing term 3/ ( )a bk m F x F x− responsible to control the

bistability.

Note that in the case of the inverted pendulum this term arises from the gravity

force applied on the mass, therefore, the coefficients Fa and Fb are function of mass

and moment of inertia, hence, they are not uncoupled. It is not possible to vary these

parameters independently, and most of all a sub-centimeter inverted pendulum is


96

unpractical to use. Nevertheless, apart from the gravity, an other conservative force

directed toward the clamping base can reproduce the repulsive effect that

counterbalances the elastic restoring force. This mechanism can be realized by using,

for example, elastic spring, electrical or magnetic field. Thus, for now, in order to

study the dynamical and electrical behavior more generically, we have opportunely

chosen these parameters of a simulated mathematical model to change the bistability

of the Duffing potential and observe displacement and output voltage.

Using a dimensionless form of such an equations system makes the numerical

simulation much easier. After a little algebra, we can express the new form with

reduced coefficients:

23

2

( ) ( )( ) ( ) ( ) ( )

( ) ( )( )

d x t dx tax t bx t V t t

dtdt

dV t dx tV t

dt dt

ζ σξ

= − − − Θ + = Θ − Ω

ɶ ɶɶ ɶɶ ɶɶɶ ɶ ɶ ɶɶ ɶɶɶ

ɶ ɶ ɶɶɶ ɶ

ɶ ɶ

(3.21)

where all variables , ,x V tɶ ɶɶ and parameters are dimensionless.

To achieve this form we made the choice of the following unknowns

( )( )

( )( )

( )( )

/

t tt

x t tx t

x

V t tV t

V

t t t

ξξ

ξ⋅

=

⋅=

⋅=

=

ɶɶ ɶ

ɶɶɶ

ɶɶ ɶ

ɶ

(3.22)

With following reduced parameters


97

2

0

0

/

/

1

6 con

2

/

v

c

a

b

v c

C

L p L P C

t m K

K m x

Kx

V K K

a F

Fb b

b x

K K

K

t m K

R C R C

ξ

ζ ζ

τ ωτ ω

=

= ⋅

=

= −

= =

=

Θ =

Ω = = = =

ɶ

(3.23)

where Ω represents the ratio between the piezoelectric cutoff frequency cω and

the natural frequency at short circuit 0 /K mω = in the limit of linear oscillator that

is for a=-1 and b=0. This influences the electrical power dissipated on load. So, Ω is

the dimensionless cutoff frequency of the equivalent piezoelectric circuit. Actually,

the piezoelectric element works as an high pass filter and, as it will be shown next,

the bistability pushes the power absorption from noise towards low frequencies.

Likewise, for dimensional equation, if the piezoelectric cutoff frequency cω is

relatively high, for a high bistability it brings a loss of electrical power transferred to

the resistive load, because most of it is filtered lying at a bandwidth under cω .

Moreover, the open-circuit situation corresponds to the limit case of 0Ω = , that is

the situation of infinite load resistance for real model or infinite capacitance and the

electric voltage Vɶ is directly proportional to displacement xɶ . The other important

piezoelectric coefficient Θ regulates the mechanical-to-electrical conversion

strength and derives from the electrical and geometrical piezo-bender characteristics.


98

In order to evaluate the power balance, starting from dimensional governing

equations (2.34), where the first is multiplied by mass and velocity and the second

one by the electrical potential, then, using the set (3.23) we obtain the following

dimensionless average power formulas listed in the table 3.2 below.

Power expression Description

1vin

c p

K dxP

K C dtσξ= ⋅

Ω

ɶɶɶ

ɶ

Average Input Power

2elP V=ɶ ɶ

Average Electrical power dissipated on resistive load

21v

loss

c p

K dxP

K C dtζ = Ω

ɶɶɶ

ɶ

Average power dissipated by mechanical friction

m in loss elP P P P= − −ɶ ɶ ɶ ɶ

Average total mechanical power (potential plus kinetic)

Table 3.2 – ormalized power expressions.

The resulting mean power terms and efficiency are simulated for a set of plausible

parameters listed in table 3.3. Simulations were made simultaneously running many

time series for equal and/or different parameter space on a 20-nodes computer

cluster. Then, averaging the statistical observables of the same constants and

parameters series.

Dimensionless Value Description


99

parameters

σ 0.01– 0.2 step 0.0025 noise strength

tɶ 0 – 2000 step 0.001 normalized time

ζɶ 0.02 – 0.4 damping

a (-1,1) step 0.01 potential parameter

b 1 potential parameter

Ω 0.011–1 dimensionless cutoff frequency

Θ 0.336 coupling coefficient

Physical parameters Value Description

τξ 0.1 [s] noise autocorr. time

Keff 1.16 – 26.6 [N/m] effective elastic constant

tp, tsh 0.278e-3 , 0.2e-3 [m] geometrical thicknesses

lb, le, lc,

lm,Wb

200e-3, 60e-3, 10e-3, 140e-3, 4e-3 [m]

geometrical dimensions

m 0.0182 [Kg] effective inertial mass

f0 1.27 – 6.67 [Hz] natural frequency

Cp 1.10x10-7 [F] piezoel. capacitance

RL 1-100 [Mohm] load resistance

τc=RLCp 11 [s] discharge time

d31 190x10-12 [ m/V]

longitudinal piezoelectric strain

coefficient

(material PSI-5A4E Lead

Zirconate Titanate)

εp 1800 elative dielectric permittivity

Kv 1.85 [N/V] coupling coefficient

Kc 1.859x103 [V/m] coupling coefficient

Table 3.3 – Model parameters

The dimensionless electro-mechanical coupling coefficients Ω and Θ were

derived from a realistic set of physical parameters (table 3.3) of a physical

macroscopic model similar to our prototype test model (exposed in the next

paragraph), but these can be varied in order to investigate many different geometry

and material properties. Let us give a glance of some displacement and voltage

simulated outlines that show in time domain the dependency between them as a

function of bistability parameter ‘a’ and, in particular by Ω .


100

Figure 3.6 – Displacement and voltage traces of PDG model with Ω =0.014

for linear case (upper) a= -1, and strong nonlinear (lower) a=0.8, ζɶ =0.1, σ =0.2.

Note that at equal noise intensity the bistable regime shows an amplified

oscillation due to the mix of intra-well jump and vibration close to each local

minimum. Unless a little discharge effect, the voltage trace follows with almost

direct proportionality the displacement of the bender. This result is quite obvious

considering the second linear equation (3.21), because, a relative cutoff frequency

0 50 100 150 200 250 300 350 400 450 500-1

-0.5

0

0.5

1

normalized time

displacement

0 50 100 150 200 250 300 350 400 450 500-0.4

-0.2

0

0.2

0.4

normalized time

voltage

a = -1

a = -1

0 50 100 150 200 250 300 350 400 450 500-2

-1

0

1

2

normalized time

displacement

0 50 100 150 200 250 300 350 400 450 500-1

-0.5

0

0.5

1

normalized time

voltage

a = 0.8

a = 0.8


101

value of Ω ∼1 means that the converted electrical energy residing at lower

frequencies is swept out by the piezoelectric high pass filter.

Figure 3.7 - Displacement and voltage traces

with Ω =1.13, bistable parameter a=0.4, ζɶ =0.1 and σ =0.1.

This high pass filtering effect is straightway visible in figure 3.7, where a

frequency 0/ 1cω ωΩ = > was chosen and, jumps apart, the voltage bistability

completely disappears. Even more clearly, the following plots 3.8 and 3.9 remark a

strong dependence of the harvested electrical power on the cutoff frequency. As it

can be seen from these figures, while the position variance does not experience an

important variation as a function of Ω , the root-mean-square normalized voltage

strongly depends by it. In fact, for Ω <<1 the power peak in the bistability region

(a>0) is maximized like that of variance, whereas, for even more Ω ~1 close to one,

it drops and there is no difference in harvesting efficiency between the linear and

nonlinear dynamics cause to power filtering.

0 50 100 150 200 250 300 350 400 450 500-2

0

2

normalized time

norm

. displacement

0 50 100 150 200 250 300 350 400 450 500-0.2

0

0.2

normalized time

norm

. voltage

a = 0.4


102

Figure 3.8 - Displacement variance for different

parameter Ω and with ζɶ =0.1 and σ =0.1.

Figure 3.9 –Mean electrical power computed as RMS normalized voltage (see tab. 3.2)

at different ratios Ω and for ζɶ =0.1 and σ =0.1.

-1 -0.5 0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

a

<x2>-<x>2

Ω = 0.01

Ω = 0.1

Ω = 0.5

Ω = 1

-1 -0.5 0 0.5 1 1.5 20

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Norm

alized Power

Ω = 0.01

Ω = 0.1

Ω = 0.5

Ω = 1

a


103

Figure 3.10 – ormalized conversion efficiency

at different ratios Ω and for ζɶ =0.1 and σ =0.1.

So, in order to utilize this voltage amplification effect in real applications, it is

highly desirable to project the piezoelectric device and harvesting circuit with an

1Ω << as low as possible. Also the noise spectrum and, therefore, its autocorrelation

time is an important parameter to know. Indeed, the lowest ξτ is, the more noise

energy lies at high frequencies, probably far away from the cutoff frequency for a

certain design. This second optimization condition can be defined for dimensional

equation as 1/c cξ ξω τ ω ω<< ⇒ << . But relating the dynamical response of such a

bistable system to this parameter together with noise strength ξσ is quite complex

and needs to be further investigated. Actually, the probability of diffusion over the

intra-well barrier depends on the product 2ξ ξσ τ and we must bear in mind that the

system is not linear, so that its behavior changes with the noise strength. As it was

-1 -0.5 0 0.5 1 1.5 210

-3

10-2

10-1

100

a

Efficiency

Ω = 0.01

Ω = 0.1

Ω = 0.5

Ω =1


104

seen in the previous paragraph 3.1, the position variance 22x x− of a bistable

system increases as much as the intra-minima distance 2 / 4U a b∆ = while the noise

strength and autocorrelation time make possible the diffusion over the potential

barrier. This constrain corresponds to having 2 / 1Uξσ τ ∆ > . The broadening of RMS

displacement x for a bistable system is a well known phenomenon, but looking at the

next plots 3.10-3.11 it is remarkable how much the harvested electrical power gets

advantage from it.

Figure 3.11 – Variance of normalized displacement Vs a,

with Ω =0.014 and ζɶ =0.1.


105

Figure 3.12 – ormalized electrical power

dissipated by resistive load.

Figure 3.13 – ormalized conversion efficiency

given by electrical power over noise input power.


106

It is noticeable that, for a fixed noise intensity 2σ , bistable sector (a>0) shows up

an enhancement around a factor five-six of the harvested power with respect to the

monostable zone (a<<0). In terms of conversion efficiency (fig. 3.12) defined as the

ratio between electrical power dissipated on load and input power supplied by the

mechanical noise (as described in table 3.2), we have the same amplification in

nonlinear zone. In this plot it is better visible that the peak shifts as the square root of

2σ on the plane ( 2σ ;a) just like the relationship (3.20) predicts.

Figure 3.14 – Efficiency (top view of plot 3.12).

A spectral analysis helps us to confirm what we have so far stated and permits to

retrieve much information about generated power as a function of normalized

frequency for different potential shape. At a fixed RMS of the Gaussian noise σ =0.1

and b=1, varying the potential parameter -1<a<1, we performed the spectra of

normalized position and voltage by means of an FFT algorithm, respectively plotted

in figure 3.15 and 3.16. When a=-1, the conservative potential is parabolic-like and


107

for small vibration close to equilibrium the oscillator behaves like a linear harmonic

oscillator. As the potential well becomes even more flattened as a∼0 approaches to

zero, the peak of resonance moves toward low frequencies and widens.

Figure 3.15 - Single-Sided Amplitude Spectrum of ( )x tɶɶ Vs a and

normalized frequency 1/ tω = ɶɶ (3D and top view).

Figure 3.16 - Single-Sided Amplitude Spectrum of ( )V tɶ ɶ Vs a and

normalized frequency 1/ tω = ɶɶ (3D and top view).

In the range of bistability (a>0), until the barrier height is relatively small, the

intra-well jumps are frequent and we get a broadening of the power peak down to

very low frequency in both position amplitude and voltage. As a consequence within


108

a frequency band ω∆ ɶ the mean converted electrical power increases being equal to

the integral of square voltage amplitude 2

( ) ( )elP V dω

ω ω ω∆

∆ = ∫ɶ

ɶɶ ɶ ɶ . This is directly

reconfirmed by the previous graphs 3.12-14 where the average power was computed

in time domain. The entrapment of the system within one of the two potential

minima roughly happens for a≥0.5 and depends on the choice of factor 2ξσ τ as we

have previously mentioned. After that, increasing a the two wells become even more

narrow, the system restarts to become harmonic-like and the amplitude peaks of

displacement and voltage shift toward higher frequency. Thus, the harvested power

quickly goes down as the subtended area becomes even more narrow.

Up to now, we have discussed many aspects of a Duffing model used in

conjunction with a piezoelectric transducer to show the effectiveness of the key idea

on improving the energy harvesting from vibrations. But, by virtue of theoretical and

practical considerations made at the beginning of this chapter, a piezoelectric bender

in an elasto-magnetic tunable potential is a better way to implement a bistable

system. Let us discuss the experimental observations of such a model compared with

the simulations in the next paragraph.


109

3.33.33.33.3 Piezoelectric Inverted Pendulum Piezoelectric Inverted Pendulum Piezoelectric Inverted Pendulum Piezoelectric Inverted Pendulum in a in a in a in a

Magnetic FieldMagnetic FieldMagnetic FieldMagnetic Field

As previously mentioned, one of the simplest technique to implement bistable

piezoelectric device for energy harvesting is using a piezoelectric beam with a

magnetic tip coupled with magnetic coil or permanent magnet placed at a certain

distance. Setting properly the magnetic dipole alignment and the distance or field

intensity, we can reproduce monostable or bistable dynamics through a repulsive

force. The physical model has been exposed in paragraph 2.4 and the geometry is the

same drawn in picture 2.16. The choice of a vertical pendulum configuration was

made to reduce at the minimum the influence of the gravity force on inertial mass

avoiding asymmetry. As a matter of fact, the bistability is yielded by magnetic

interaction rather than gravity which can be neglected in this case (the inertial mass

attached to piezo-bender is relatively small). Moreover, we have the great advantage

as regards to Duffing oscillator to be able to control only one parameter (for example

the relative distance between magnets) to tune the potential and regulate the

nonlinear dynamics. In addition, this is an easy-to-realize design that can work at

small scale yet. Indeed, at sub-millimetric size and small masses, the gravity force is

negligible and the device orientation does not constitute a problem.

Now, we’re going to describe the real macroscopic model used for the

experimental tests, the experimental setup and the comparison of numerical and

experimental results.


110

3.3.13.3.13.3.13.3.1 Experimental Setup Experimental Setup Experimental Setup Experimental Setup and and and and CharacterizationCharacterizationCharacterizationCharacterization

The following picture shows a schematic draw of the apparatus employed in the

experiment. Here are described the piezoelectric bender, the excitation system and

read out electronics.

Figure 3.17 – Experimental apparatus.

The inverted pendulum is realized with a four-layer piezoelectric beam (mod.

T434-A4-302 4-Layer Bender by Piezo system inc.) made of Lead Zirconate Titanate

xy micrometric stage

NI DAQ 16bit LabView sw

clamping

reading displ. xr CCD Laser

displacement sensor

excitation coils

Vout - Piezoelectric

4 layers piezoelectric

micrometric xyz base

Power supply

excitation magnets Ch1 displ. “x”

Ch2 Vout across RL

Ch3 noise

Low noise filter

RL Load variable resistance

tip magnets ∆

beam

defl. x

xm tip displ.


111

(PSI-5A4E) 40 mm long, clamped at one end. The pendulum base position can be

adjusted via a micrometric xyz displacement system. The pendulum mass is a 60mm-

long steel beam with three magnets (dipole magnetic moment M=0.051±0.002 Am2)

attached. The tip magnet is faced by a similar magnet with inverse polarities placed

at a distance ∆ and held in place by a massive structure. The distance ∆ can be

adjusted via a micrometric displacement control system.

a)

b)

Figure 3.18 – Photos of experimental setup.

a) global view. b) excitation coils.

The displacement is measured via an optical read-out with a CCD-Laser

displacement sensor (KEYENCE model LK-501/503) with sensitivity of

1mV/10micron. The signal from the displacement sensor is sampled by a digital

signal processing board (DSPB) controlled by a personal computer with sampling

frequency 1Mhz (National Instruments interface DAQ). The output voltage signal

from the piezo-bender is measured through terminals of a variable load resistor RL

placed in parallel with output contact of beam and sampled by the DSPB. The

digitalized signals are stored in the computer memory for post-processing


112

elaboration. The DSPB is also employed to drive a current generator that produces,

through a couple of coils, the magnetic excitation that mimics the ground vibration.

The signal generated by the DSPB is filtered and conditioned in order to reproduce

the desired statistical properties. The design parameters of the test model are

displayed in table 3.4.

&oise parameters Value Description Source

σ 0.3, 0.6, 1.2 [mN] RMS noise effective force Calculated by fitting

τξ 0.1 [s] noise autocorr. Time set

fs 1 [MHz] sampling frequency set

ζ 0.016 ±0.002 Loss factor measured

Model parameters Value Description

Keff 26.6±0.5 [N/m] effective elastic constant measured

f0 6.6±0.1 [Hz] (6.58±0.05 FEA)

first mode frequency measured and FEA

m 0.0155 [Kg] effective inertial mass modal analysis

tp, tsh 0.278e-3 , 0.2e-3 [m]

geometrical thicknesses specific properties

lb, le, lc, 72.4e-3, 60e-3, 12.9e-3 [m]

geometrical dimensions specific properties

lm,Wb 143e-3, 4e-3 [m] geometrical dimensions specific properties

S1=lxr/lb 2.32 geometrical scale factor calculated

S2=(lm+lb)/lb 2.97 geometrical scale factor calculated

M 0.051±0.002 [Am2]

magnetic moment measured

Cp 1.10x10-7 [F] piezoel. Capacitance specific properties

RL 1-100 [Mohm] load resistance set

τc=RLCp 1.4-11 [s] discharge time set

d31 190x10-12 [ m/V]

longitudinal piezoelectric

strain coefficient

(material PSI-5A4E Lead

Zirconate Titanate)

specific properties

εp 1800 elative dielectric

permittivity

specific properties

k31 0.33 coupling coefficient specific properties

Kv 1.85 [N/V] coupling coefficient Calculated

Kc 1.859x103 [V/m] coupling coefficient Calculated

Table 3.4 – Test model parameters


113

Some of them are derived from the relations (2.35) and modal analysis, others

from the technical data of the piezoelectric material and by measurements. The

damping factor ζ has been extracted from averaging a series of ring down

measurements carried out at open circuit. Figure 3.16 shows a sample of fitted ring

down of RMS voltage referred to displacement channel.

0 2 4 6 8 10 12

0,0

0,2

0,4

0,6

0,8

1,0

Data: QFACTOR_B

Model: ExpDec1

Chi^2/DoF = 0.00055

R^2 = 0.99155

y0 0 ±0

A1 0.8543 ±0.02275

t1 1.46707 ±0.07524rms (volt)

Time (s)

Ring down

Fitζ= 1/ω0τ = 0,016

f0= 6,67 Hz

25 30 35 40 45 50 55

200

300

400

500

600

700

800

900

1000

Fit ω2=k(1/m) k=27,09 (err 0,8)

ω2 (s

-1)

1/m (Kg-1)

Figure 3.19 – (Upper) Ring down sample of RMS voltage of

displacement channel at open circuit. (Lower) Fitted elastic constant.


114

The effective elastic constant and mass was retrieved by modal analysis, that is

varying the inertial mass attached to the piezo-bender and measuring the variation of

first mode natural frequency.

3.3.23.3.23.3.23.3.2 Finite Element AnalysisFinite Element AnalysisFinite Element AnalysisFinite Element Analysis

The dynamical behavior of the real model and powering speculations are

supported by finite element analysis carried out both on magnetic force interaction

and global piezo-mechanics features of the prototype using COMSOL multiphysics.

Since it is quite complex to derive an exact analytical expression to describe all

the magnetic forces between two real permanent magnets in relative motion at

different configuration and angles, we first computed the axial repulsive force with

2D finite element analysis of the magnetostatic field and then searched a suitable

fitting function.

0,000 0,005 0,010 0,015 0,020 0,025

-10

0

10

20

30

40

50

60

70

80

90

100

Data: Br33dic2_B

Model: magnetic_force

Equation: 6e-7*(P1^2)/abs((P2+x))^P3

Weighting:

y No weighting

Chi^2/DoF = 13.94691

R^2 = 0.91776

P1 0.05 ±0

P2 0.00562 ±0.00038

P3 4.83888 ±0.0531

force (N)

∆ (m)

Figure 3.20 - (left) FEM of magnetic flux density.

(right) Repulsive axial force between magnets computed by FEA and

fitted by the relation (2.37).


115

Figure 3.20 on the left shows the magnetic field flux streamline and intensity in

air domain computed via finite element modelling for one of the many

configurations, whereas on the right it shows the fitting of repulsive force data along

the vertical axis for magnets with aligned dipoles using the relation (2.37). We must

specify that this analytical relation is valid for magnetic force between two aligned

dipoles at relative long distance (r>>d where d is the linear dimension of a magnet),

whereas for short distance the proper law approximately goes as the inverse square of

the distance [68]. Anyway, for small oscillations and at medium relative distance ∆,

the fourth order polynomial function seems to fit better the FEA and experimental

data than the inverse square law. Actually, we should also consider the term which

accounts for the angle between the two magnetic dipoles, but in first instance it is

possible to neglect it. The resulting dynamical behaviour and modal analysis of our

model has been tested even with mechanical FEM of the piezo-pendulum.

Figure 3.21 – FEM and tip displacement Vs time from monostable to bistable dynamics.

f0=6.58 Hz first vibrational mode 20000 degree of freedom


116

In figure 3.21 there are plots of dynamical response of the pendulum subjected to

vibrating noise at different bistable potential barrier height or rather from no

magnetic interaction (∆>20mm) to relative strong repulsion force (relative close

distance ∆∼10mm) before the total trapping around one stable position. The

decreasing of the jumps frequency with an increasing potential barrier at even

smaller distance is clearly visible. The mechanical power density spectrum (fig. 3.22)

of the two extreme situations: quasi-linear behavior for negligible repulsive force

(w=0.001) and strong nonlinearity with (w=0.1) respectively correspond to having a

narrow peak and a broader power density distribution on frequency domain.

Figure 3.22 – Mechanical power spectrum for ∆∆∆∆>20mm (norm. barrier height w=0.001)

and ∆∆∆∆∼∼∼∼10mm (w=0.1).

Evidently, the subtended area of the bistable power spectrum is much more than

that of linear case at low as well as at high frequencies, hence, the finite element

model further confirms

for bistable dynamics with respect to linear

3.3.33.3.33.3.33.3.3 Numerical Numerical Numerical Numerical and and and and

As for the acquisition of experimental

analysis has been carried out on two signals: displacement signal

piezoelectric output voltage

hoc LabView program.

controlled NI-DAQ (channel 3) and

order to obtain the desired exponentially correlated Gaussian distribution with

various autocorrelation time

figure 3.23 below. We have conjectured as a first step

distributed force in frequency domain for two

on how much energy could be extracted form a

generally dwells at low frequency rather than at high ones.

Figure 3.23


117

the efficiency enhancement on power transfer to the system

with respect to linear dynamics.

and and and and Experimental ResultsExperimental ResultsExperimental ResultsExperimental Results

As for the acquisition of experimental data, a real-time statistic and spectral

nalysis has been carried out on two signals: displacement signal x

piezoelectric output voltage Vout across the resistive load RL (channel 2)

program. Whereas the noisy input force is generated via software

DAQ (channel 3) and then the signal conditioned by pass


various autocorrelation time ττττc=0.001-0.1s. An example of its spectrum is sho

. We have conjectured as a first step this kind of uniformly

distributed force in frequency domain for two principal reasons: we want to

on how much energy could be extracted form a wide spectrum and the most part of it

generally dwells at low frequency rather than at high ones.

23 – Power Spectral density of input noise with ττττc=0.1s.

&umerical Analysis And Experimental Results

efficiency enhancement on power transfer to the system

time statistic and spectral

(channel 1) and

(channel 2) made by ad

Whereas the noisy input force is generated via software

the signal conditioned by pass-band filter in


spectrum is shown in

kind of uniformly

reasons: we want to speculate

wide spectrum and the most part of it

=0.1s.


118

The displacement variance and RMS voltage were computed on time interval ∆T of

1000 seconds with a sampling frequency fs of 1MHz. The agreement between

experimental data and numerical simulation performed through MATLAB program

piezo_magnetic.m listed in (A.2) seems to be very good. Indeed, as we can see from

fig. 3.25, the experimental distribution of moving position x of the mass is very close

to the one expected from the numerical simulation: say within the relative

experimental error: about 7-8% in both axis directions (error bars are not plotted to

avoid confusion due to the large number of experimental points). This plot reports

the distribution of moving mass for effective driving noise rms σ=1.2 mN at three

decreasing magnet distance ∆=24, 10, 7 mm which corresponds to an increasing

repulsive force and respectively to regimes of monostable quasi-harmonic, bistable

and high intra-well barrier height potential when the pendulum is bounded around

one minimum. The relative maxima of displacement distribution (fig. 3.24) perfectly

follow the stable equilibrium positions corresponding to the potential energy (fig.

3.24b). This is derived from the expression (2.38) slightly adjusted considering some

geometrical scaling factors which connect the various position points (x, xr, xm) of the

system as shown in fig. 3.24a which are 1rx s x= , 2mx s x= , with 1 /rs lx lb= and

2 ( ) /s lm lb lb= + .


119

a)

b)

Figure 3.24 – a) System outline. b) Effective magneto-elastic potential of the oscillator

with respect to scanned beam position xr and relative magnets distance ∆∆∆∆.

-15 -10 -5 0 5 10 15

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

distribution

displacement (mm)

∆ = 7 mm

∆ = 10 mm

∆ = 24 mm

Figure 3.25 – (Dotted) experimental distribution of xr displacement

with RMS noise σσσσ=1.2 m (σσσσ/m=0.08 ms2) at ∆∆∆∆=7,10,24 mm.

(Solid line) simulated distribution.

Y

∆

lm

lb

lxr

xr

xm

x

X

CCD


120

Finally, the most important confirmation of what we have claimed so far, that is the

efficiency enhancement of the harvesting power, is shown in the following plots. In

fig. 3.26 we plot the standard deviation xσ computed on values of the stochastic

displacement x(t) as obtained by the numerical solution of the Langevin equation of

motion (solid line) and by the experimental data (scatter) for three different rms of

effective noisy force (σ =0.3,0.6,1.2 mN) applied to effective mass of the oscillator.

The agreement between the experimental data and the model is very significant.

5 10 15 20 250

1

2

3

4

5

6

7

8

9

σ = 1,2 mN

σ = 0,6 mN

σ = 0,3 mN

σx (m

m)

∆ (mm)

Figure 3.26 – (Scatter) standard deviation xσ of experimental displacement

Vs magnets vertical distance ∆∆∆∆. (solid) numerical prediction.


121

In parallel, as we would have expected, the correspondent values of root-mean-

squared voltage Vrms (fig. 3.27) across the resistive load RL (100 MOhm for this

test) show the same trend with the same maximum peak position as well as .

5 10 15 20 250

1

2

3

4

5

6

σ = 1,2 mN

σ = 0,6 mN

σ = 0,3 mN

Vrm

s (Volt)

∆ (mm)

Figure 3.27 - – (Scatter) experimental data of voltage RMS across the load

Vs magnets vertical distance ∆∆∆∆. (solid) numerical prediction.

As we previously analyzed in paragraph 2.4, looking at the plots (3.25-26) we can

easily identify the three different regimes:

1) when ∆ is very large the dynamics is characterized by quasi-linear

oscillations around the minimum located at zero displacement, at the vertical

position of the pendulum. This condition accounts for the usual performances

of a linear piezoelectric generator.

xσ


122

2) when ∆ is very small, the potential energy is bistable with a very pronounced

barrier between the two wells. In this condition and for a fixed amount of

noise, the pendulum swing is almost exclusively confined within one well and

the dynamics is characterized once again by quasi-linear oscillations around

the minimum of the confining well.

3) in between these two regimes there is a range of distances ∆ where the xσ

(and the Vrms as well) reaches a maximum value. Such a regime is

characterized by a bistable potential with two clearly separated wells. In this

condition, the pendulum dynamics is highly nonlinear and the swing reaches

its largest amplitude with noise assisted jumps between the two wells.

It is more evident, even by looking at the plot of effective harvested power

(fig.3.28) (that dissipated across the load 2 /e rms LP V R= ), that the maximum values

of the rms exceed by a factor that ranges between 2 and 6 the value obtainable

when the magnet is far away and thus the pendulum operates in linear condition.

This behavior amounts at a potential gain for energy harvesting between 200%

and 600% compared to the standard linear oscillators, depending on the noise

intensity and on the other physical features of the pendulum. Moreover we would

like to briefly comment about two other important features: a) the maximum

position shifts toward larger distances ∆ when the noise intensity increases. b) In


123

the low ∆ regime, the rms values seem to reach a plateau that is smaller than the

similar plateau reached by the rms values in the large ∆ .

5 10 15 20 25

0

5

10

15

20

25

30

35

40

σ = 1,2 mN

σ = 0,6 mN

σ = 0,3 mN

Electrical Power (10-7watt)

∆ (mm)

Figure 3.28 – Electrical harvested power (dissipated on resistive load 2 /e rms LP V R= )

Vs magnets vertical distance ∆∆∆∆.

These features further support all the considerations presented in chapter 2 about

the stochastic dynamics and potential energy. In particular, when ∆ is very large the

potential ( , )U x ∆ shows a single minimum. The librational frequency around this

minimum is derived from (2.44) that is 0 0 0/ 2 1/ 2 '' /

xf U mω π π

== = =6.49 Hz (for

∆=30mm). Both the measured value 6.6±0.1 Hz and that simulated by FEM

6.58±0.05 are in good agreement to the limit case ∆ → ∞ when 0 (1/ 2 ) /f K mπ≈ .

The value here can be estimated in the linear oscillator approximation as

proportional to 0ωσ [73]. On decreasing ∆, ω0 decreases and then the value

xσ

xσ


124

increases. The potential becomes bistable when ∆ is lower than the critical value

1/ 52(3 / )c m effK s K∆ = : two distinct minima arise at piezo-bender displacement

( )2/5 23 /m effx K K± = ± − ∆ separated by a maximum at x=0. The agreement between

the analytical and experimental values is almost perfect: for example, at ∆=0.007mm,

analytical minima results rx± =±0.0067 (where the modified formula with geometrical

ratios ( )2/5 21 2 2( / ) 3 /r m effx s s K s K± = ± − ∆ has been used in fig. 3.29 and

experimental ones are rx± =±(0.0067±0.0001). From this point, the oscillator jumps

now between the two minima and the increases proportional to rx± . As

demonstrated by the relation (2.46), by decreasing ∆ the potential barrier height

(0) ( )rU U U x±∆ = − grows proportional to ∆−3.

Figure 3.29 – Barrier height ∆∆∆∆U and minimum

position shift rx±

versus ∆∆∆∆.

xσ


125

The continuous growth of the xσ is interrupted when, due to the increase of the

barrier height, the noise induced jump dynamics between the two minima becomes

less and less effective. By further decreasing ∆ the barrier height is so large that the

jump probability becomes negligible and the pendulum swing is permanently

confined within one well. Such a trapping condition happens at smaller values of ∆

(larger barrier) when the noise is larger. This explains the observed shift of the

maximum position toward smaller ∆ as observed in a). Here the dynamics is almost

linear with small oscillations around the potential local minima (x+ or x-) with xσ

value proportional to σ ω± with ω±2 = ′ ′ U (x±) m . Being ω± > ω0 we have

, 0 ,x xσ σ∆→ ∆→∞< as observed in the experiment.

All these considerations about the nonlinear stochastic dynamics are inline with

those previously exposed for the Duffing piezoelectric oscillator. The sole difference

is the expression of the conservative potential. The role of the distance ∆∆∆∆ is played in

the Duffing case by the parameter a. Basically, the three dynamical regimes are the

same as well as the noticed efficiency enhancement. Also in this case, we notice a

maximum values of the 2rmsV a factor 3-6 grater than the value obtainable in quasi-

harmonic domain.

In the next chapter we will synthesize again the key results of the thesis subject

and take a glance to the possible solutions based on this concept. The implementation

designs and miniaturization issues are also outlined with some design example

(bistable piezoelectric membrane) for which we have made an early inspection

through finite element analysis.

126


ConclusionsConclusionsConclusionsConclusions

4.14.14.14.1 EEEEffectiveness offfectiveness offfectiveness offfectiveness of nonnonnonnonlinear linear linear linear approachapproachapproachapproach versus versus versus versus

linearlinearlinearlinear

In spite of the fact that a growing interest and many studies have been recently

focused on vibration energy harvesting, most of these are left restricted within the

optimization techniques of conversion efficiency, transduction mechanism and

material research. Up to now, the fundamental transduction mechanism of the

vibrating mechanical energy into electrical energy is ever modelled as a linear

oscillator coupled with piezoelectric, electrostatic or electromagnetic system. All of

these inspections have been based on the maximization of the kinetic energy transfer

around its first resonant mode and of electro-mechanical coupling constants. This

Chapter 4. Conclusions

127

common approach is clearly optimal for a well known shaking source whose energy

is primarily centered around single frequencies, however, most of the power of many

random vibration sources is spread over a large spectral bandwidth and most of all at

frequencies under hundred hertz [10, 31, 38, 61, 74, 75]. This is also the reason

whereby it is complex to obtain an efficient sub-micro scale transducer, inasmuch,

the resonant frequency of the system is inversely proportional to its geometrical

dimension with a consequent mismatch between the driven frequency and natural

frequency of the device. Some attempts to design energy scavengers with a wider

bandwidth under investigation concern the use of a chain of & different spring-mass-

dampers [61]. The various oscillators in this sort of chain have resonant frequencies

almost overlapped such that at least one element is in resonance over the desired

frequency range. Others groups are improving the signal processing and storing

techniques of the output voltage of the piezoelectric element (synchronized switch

damping[76]). By the way, both these approaches present severe limitations such as

the scale reduction, dispersive control electronics and poor efficiency in the case of

wide band multimodal excitation. Moreover, the linear model elaborated by Williams

et al. [50] can describe quite well the behaviour of electromagnetic transducer whose

efficiency depends basically by the velocity of inertial mass, whereas, it’s not

completely valid for piezoelectric beam conversion technique that needs to maximize

its displacement and to reduce its cutoff frequency as much as possible in order to

obtain the maximum harvested power.

The pivot idea we’ve presented here concerns the use of nonlinear oscillators

instead of linear ones. The exploitation of the their dynamical features permit to


128

strongly improve the absorption of the ambient vibrational energy from a wideband.

Our work has been focused in particular on the use of a Duffing-like bistable

oscillator coupled with piezoelectric element, but the key principle can be explored

even for other nonlinear systems and transduction mechanisms.

Summarizing here the most important points of this work:

We have proposed a new method of energy harvesting from ambient

vibrations based on the utilization of the dynamical features of nonlinear

stochastic oscillators.

We have demonstrated that the overall efficiency of this method, by

applying it to a piezoelectric beam in a Duffing-like potential, can be

grater than a factor six with respect to a linear oscillator concept. The

maximum peak of the generated power is obtained for a certain trade off

among noise variance, intra-wells distance and barrier height, while the

piezoelectric cutoff frequency must be as low as possible to avoiding

energy filtering.

The nonlinear oscillator doesn’t need to be tuned on a precise resonance

frequency, but it’s able to extract kinetic energy from a wide bandwidth

of noise, specially at low frequencies.


129

Early theoretical considerations have been done about the power peak

position as a function of potential and noise variance and they explain

quite well the phenomenology of the oscillator dynamical behaviour.

The experimental results of a piezoelctric bistable oscillator in a repulsive

magnetic field were compared successfully with numerical simulations.

We have proven that the method proposed here is quite general in

principle and could be applied to a wide class of nonlinear oscillators and

different energy conversion principles.

The outcomes here obtained need to be understood at the light of a more

comprehensive theoretical interpretation. On the other hand, stochastic bistable

systems cover a wide range of physical and biological phenomena so that they are

under crescent interest far beyond their application as energy harvesting methods.

Potentials for realizing micro/nano-scale power generators are currently under

investigations, even though these studies are mainly focused on materials research

and various kinds of microlithography techniques [57, 77]. Our concept can be easily

applied to every system even at small scale. Indeed, while the physics remains

regulated under the classical mechanics laws, our results could be easily transferred

to sub-micro world. Moreover, the nonlinear stochastic concept could be also

inspected even in quantum domain, but this lies outside the scope of this work.


130

4.24.24.24.2 Miniaturization pMiniaturization pMiniaturization pMiniaturization perspectives of nonlinear erspectives of nonlinear erspectives of nonlinear erspectives of nonlinear

power harvesting power harvesting power harvesting power harvesting systemssystemssystemssystems

Finally we would like to emphasize that the results obtained here can led to a

significant increase in energy harvesting performances also in the small scale

domain. Thanks to an even more advanced miniaturization technology (RIE dry

etching, wet chemical etching and UV-LIGA etc.) a crescent number of research

groups are making evident progress in the realization of micro-electro-mechanical-

systems for energy scavenging. For example, Jeon et al.,[40] developed a 170µm x

260µm PZT beam that is able to generate 1µW of continuous electrical power to a

5.2MΩ resistive load at 2.4V dc. Although, the corresponding energy density

(0.74mWh/cm2) is comparable to the lithium ion batteries, this power output is

obtained under a driving force precisely tuned on its first resonant frequency of

13.9kHz, that is not easily to find in common ambients.

a)

b)

Figure 4.1 – a) SEM of the fabricated PMPG device with bond pads (Jeon [40]).

b) Micrograph of a 500 µµµµm PMPG of TIMA labs, Ammar et al. [78]


131

Other submillimeter MEMS-based micro piezoelectric generator with cantilever

design has been developed by Fang et al.[79], which results in about 2.16µW (0.89V

AC peak-peak) of power output under a resonant operating frequency of 608Hz with

strength of 1g acceleration. But this relatively low voltage is not practicable to

present applications. Beeby et al.[46, 51] tested a micromachined electromagnetic

silicon generator with a 300µm wide paddle beam that gives a natural frequency of

6.4 kHz. Although its output voltage of 0.36V could be amplified, even in this case,

the actual power density is not still promising. Going down to more tiny scales,

various interesting energy harvesting systems are under investigations. A direct

current nanogenerator has been fabricated by Wang et al.[57] using an array of 50

nm radius x 600 nm length zinc oxide nanowires. It is capable to produce a

respectable voltage of 0.3 V enough to drive the metal-semiconductor Schottky diode

at the interface between atomic force microscope tip and the ZnO NW.

Nanopiezotronics is recently making giant steps and permits to implement

submicrometer devices capable to produces electric power from pulsating blood

vessels, acoustic waves, flowing blood or a beating heart. With some adjustments,

the nonlinear dynamical concept shown here can be quite easily applied to all these

present linear designs, even down to nanoscale, in order to make them working on

nonlinear regime. On the other hand, nanomechanical Duffing-like resonators have

been yet realized and studied for other purposes by some research groups [80, 81].

Inserting micro-magnets into the tip of cantilever piezoelectric beams as outlined in

fig. 2.16, it’s not difficult to tailor a suitable bistable potential to the bandwitdth of

background stochastic noise. And unified array structure of these parallel/serial


132

combined cantilevers could disclose promising performances. Moreover, there are

lots of ways and different designs to produce a bistable dynamics by means of

permanent magnets. Some possible configurations are suggested in fig. 4.2.

Figure 4.2 – Clusters of bistable piezoeletric beams.

A) perpendicular repulsive force configuration.

B) transversal attractive force configuration.

In the picture above shown there are two possible configuration of permanent

magnets: a) represents a perpendicular repulsive configuration mentioned above

while in b) the piezo-beams counteract each other their elastic restoring force via

transversal attractive magnetic force.

piezo beams

B – transversal magentic piezo-beams cluster

permanent magnets

A – perpendicular magentic piezo-beams cluster

permanet magnets

+ − − + − + − +

Vout

Iout

piezo beams

stochastic force

stochastic force


133

Unlike the cantilever geometry, we could think to an array of thin film

piezoelectric membranes. Each one with an appropriate inertial mass attached to the

centre. The bistable dynamics could be created, without accessory magnets, simply

contracting one of its edge on its parallel plane as outlined for example in fig.4.3.

Figure 4.3 – Bistable piezoelectric thin-film membrane.

For now, we restrict ourselves to describe an early study of dynamical bistable

behaviour through finite element modelling of such idea. A 20mm length x 0.1 mm

thickness PZT membrane model with a steel sheet inside of thickness 0.05mm is

plotted in fig. 4.4. It was compressed setting its vertical edge displacement by 0.01-

0.1mm with respect to relaxed position. Applying then a perpendicular force

(sinusoidal and/or stochastic) pointed to its center, for different compression levels of

the edges, we’ve computed the time series distribution of central point displacement.

Furthermore, introducing a differential pressure across the two sides, it is possible to

settle the symmetry of bistability. This design is similar to that of a pressure sensor

and in effects it can works both as generator/sensor and actuator, with the essential

difference that, in this case, its dynamic is bistable and not monostable (see fig. 4.5).

stochastic force

seismic mass piezoelectric membrane


134

Exploiting its residence time, it is possible to estimate the pressure difference

between the two sides without encounter the typical problems of calibration of

common pressure sensors [82].

Figure 4.4 – FEM of a piezoelectric membrane displacement.

The geometrical scale XY are not equal for a better viewing.


135

0 2 4 6 8 10

-1,0

-0,5

0,0

0,5

1,0

displacement (m

m)

Time (s)

∆P = 100mbar

∆P = 1000mbar

∆P = 2000mbar

Figure 4.5 – Membrane displacement under

a sinusoidal driving force at different differential pressures.

Actually, one of the major difficulty of such a kind of model thought as energy

harvester is to know exactly the elastic potential which is not perfectly conservative.

Moreover, piezoelectric material is subjected to aging, so its piezoelectric and elastic

properties present hysteresis and variation in time. Nevertheless, such a configuration

could be a promising and simpler solution than magnetic piezoelectric cantilever and

needs to be investigated even in clustered configurations.

137

AppendicesAppendicesAppendicesAppendices

A.1A.1A.1A.1 EulerEulerEulerEuler----Maruyama methodMaruyama methodMaruyama methodMaruyama method

Let a generic one-dimensional stochastic differential equation be

( , ) ( , )tt t t

dXF X t G X t

dtξ= + (A.1)

where Xt is a stochastic process, F is in generally a nonlinear function that describes

the deterministic evolution and the term ( , )t tG X t ξ is a stochastic Gaussian term with

( , )tG X t a generic function of Xt. Integrating this equation 3.1 we obtain

0

0 0

( ) ( )t t

t s s sX X F X ds G X dsξ= + +∫ ∫ (A.2)

Where F is a Riemann integrable function. Instead the last term represents a

stochastic noise multiplied by a function of process Xt which cannot be easily

integrated. It can be shown that a Gaussian process tξ with uniform spectrum

satisfies the expression (A.3) [83]

0

t

s tds Wξ =∫ (A.3)

in which Wt indicates a Wiener stochastic process defined in the interval [0,T] as

continue function with the following properties (A.4):

138

1) (0) 0 1;

2) 0

;

3) 0

t s

2

W with probability

for s t T the random variable given by increment

W W is normally distribuited with zero mean and variance

t s

for s t u v T the increments variable give

σ

=

≤ < ≤

−

= −

≤ < < < ≤

t s v u

n by increment

W W and W W are indipendents.− −

(A.4)

From this relations the stochastic term of the (A.2) can be expressed as

0

( )tW

s sG X dW∫ (A.5)

hence, it can be seen like a limit of the Riemann sum

( ) ( ) ( )1

1

( )n n ni i i

n

n

i

S G W Wτ τ τ −

=

= −∑ (A.6)

Where ( ) ( ) ( )1 ,n n n

i i it tτ − ∈ . But the summation limit depends by the choice of these

integrations extremes[71], so this algorithm sets the problem to use a proper

criterion. Among the various consistent methods of integration the most successful

results that of Itô and Stratonovich that set

( ) ( ) ( )1(1 )n n n

i i it tτ α α−= − − (A.7)

where the Itô integration corresponds to equal α=0, while the Stratonovich

integration corresponds to α=1/2.

The Euler−Maruyama method consists in the integration of the stochastic term of

the equation (A.2) on Brownian process ( )W t , thus, in the Wiener increment dW(t)

139

0

0 0

( ) ( ( ), ) ( ( ), ) ( )t t

s sX t X F X s s ds G X s s dW s= + +∫ ∫ (A.8)

these two integrals can be discretized as follows. Let a time interval [ ]0,T be divided

by & points tj for j=1,..,& with & positive integer and ( / )jt j T & j t= = ∆ .Then, as

limit for 0t∆ → of the Riemann summations

1 1

10 0

( ( ), )( ) ( ( ), )& &

j j j j j j

j j

F X t t t t F X t t t− −

+= =

− = ∆∑ ∑ (A.9)

and, for the stochastic part choosing the Itô integration

1 1

10 0

( ( ), )( ) ( ( ), )& &

j j j j j j j

j j

G X t t W W G X t t dW− −

+= =

− =∑ ∑ (A.10)

where we have placed ( )j jW W t= , hence 1j j jdW W W+= − is the j-th increment of the

discrete Wiener process ( )W t . Finally, the Euler-Maruyama approach is essentially

the discrete integration of the equation (A.8) that becomes

1 1

00 0

( ) ( ( ), ) ( ( ), ) .& &

j j j j j

j j

X T X F X t t t G X t t dW− −

= =

= + ∆ +∑ ∑ (A.11)

A method is said to have strong order of convergence equal to γ if there exists a

constant C such that the expected value:

( )nX X C tγτΕ − ≤ ∆ (A.12)

for any fixed [ ]0,n t Tτ = ∆ ∈ and t∆ sufficiently small. Keeping in mind that ( )X τ

is the possible analytic value and Xn is the simulated variable. It can be shown that

the E-M method has strong order of convergence γ=1/2.

140

A.2A.2A.2A.2 Simulation programs Simulation programs Simulation programs Simulation programs

Listing n.1

File: wise_duffing.m

function wise_duffing(serial,T,damp,Fa,Fb,sigma,tau,plt1,plt2,sav) try tic % dependent parameters % Declaration of the time steps integration h=0.0005; % time step dt [s] (integration step size) N=ceil(T/h); % number of steps t=(0:h:T); % t is the time vector %% generation of exponentially correlated noise epsilon=zeros(1,N); dW=sqrt(h)*randn(1,length(t)); % brownian increment randn('state', sum(100*clock)); % set inintial state for g = 1:N epsilon(g+1) = epsilon(g)*(1-h/tau) + randn*h; end epsilon = epsilon/std(epsilon); epsilon = epsilon - mean(epsilon); %% Inizialization and Integration of non-linear SDE system Std_x = zeros(length(Fa),length(sigma)); %standard deviation Y=zeros(size(t)); % place to store locations Y X=zeros(size(t)); % place to store locations X for k=1:length(sigma)

141

pos = 0.0; Y(1)= 0.0; % initial conditions for j=1:length(Fa) X(1)= pos; Y(1)= 0.0; for i=1:N Y(i+1)=Y(i)+Fa(j)*X(i)*h-Fb*X(i)^3*h-2*damp*Y(i)*h... +sigma(k)*epsilon(i)*dW(i); X(i+1)=X(i)+Y(i)*h; end pos=X(length(X)); if plt1 figure(1); clf plot(t,X,'r') hold on xlabel('time [sec]'); ylabel('displacement'); end Std_x(j,k)= std(X,1); %standard deviation of position

end end %% plotting if plt2 figure(1); clf plot(Fa,Std_x); xlabel('nonlinear parameter'); ylabel('Xrms'); figure(2) surf(sigma.^2,Fa,Std_x,'EdgeColor','none'); xlabel('sigma^2'); ylabel('nonlinear parameter'); zlabel('Xrms)'); end %% saving in Pavg.mat if sav suffix = ['n' num2str(serial) 'date' date '_numerical']; save(['num_' suffix '.mat'],'T','Fa','Fb','Std_z','damp','sigma','tau'); end

142

toc catch lasterror end

Listing n.2

File: piezo_magnetic.m

function []=piezo_magnetic(T,damp,mag,dy,scale,sigma,plt1,plt2,sav) try tic %% CONSTANTS DECLARATION lz = 168e-3; % reading position of deflection % constants for piezo PZT-5A4E T434-A4-302 piezo.com e_r = 1800; % relative dielectric constant (at 1Khz) e_0 = 8.85e-12; % F m^-1 vacuum dielectric permittivity e_p = e_0*e_r; % piezoelectric absolute permittivity d_31 = 320e-12; % m/v piezoelectric strain coefficient mu0=4*pi*1e-7; % model design parameters lm=143e-3; % length of the mass [m] hm=4e-3; % Height of the mass Wm=4e-3; % with of the mass density=7800; % Kg/m^3 material density of inertial mass lb=72.4e-3; % length of the cantilever beam lc=12.9e-3; % length of the beam under clamp Wb=5.1e-3; % with of cantilever, it as been assumed that le=lb le=lb-lc; % length of electrode

143

Tp=0.33e-3; % thickness of piezoelectric layer Tsh=0.2e-3; % Thickness of the center shim thk=Tp*2+Tsh; % thickness of the beam m_mag = 0.49e-3; %magnet mass m_mass = 15.1e-3; %mass attached [Kg] m_beam = 2.6e-3; %mass of beam m_clamp = thk*lc*Wb*7900; %clamped mass m = m_mass+m_mag+m_beam; % inertial mass Cp=5.2e10; % N/m^2 Elastic Young modulus

for piezoelectric material PZT-5H [Pa] Csh=2.0e11; % Young modulus of the center shim material [Pa] Rl=100e6; % load resistance [Ohm] % dependent parameters b=Tp/2+Tsh/2; % distance between piezoelectric layer and shim layer I=2*((Wb*Tp^3)/12+Wb*Tp*b^2)+((Csh/Cp)*Wb*Tsh^3)/12; % moment of inertia of composite beam a=4; % a=1 series config, a=4 double parallel configuration % average strain to input force k1=(2*I)/(b*(2*lb+lm-le)); % average stress to vertical displacement k2=3*b*(2*lb+lm-le)/(lb^2*(2*lb+3/2*lm))/2; Cb=112e-9; % measured effective piezoelectric capacitance K=26.6; % measured effective elastic constant % voltage coefficient for diff. equation Kv=K*d_31*a/(2*Tp*k2); % coupling term between voltage and displacement Kc=2*Tp/2*d_31*Cp*k2/(a*e_p); %piezoelectric coupling coefficient k_31=d_31^2*Cp/e_p % Declaration of the time steps integration h=0.0001; % time step of sampling N=ceil(T/h); % number of steps t=(0:h:T); % t is the time vector %% GENERATION OF EXPONENTIALLY CORRELATED GAUSSIAN NOISE epsilon=zeros(1,N); tau=0.1; % colored noise correlation time [sec] dW=sqrt(h)*randn(1,length(t)); % brownian increment randn('state', sum(100*clock)); % set inintial state

144

for g = 1:N epsilon(g+1) = epsilon(g)*(1-h/tau) + randn*h; end epsilon = epsilon/std(epsilon); epsilon = epsilon - mean(epsilon); %% Inizialization and Integration of non-linear SDE system Pavg = zeros(length(dy),length(sigma)); %electrical power Pavg_z = zeros(length(dy),length(sigma)); %mechanical power pos = 0.0; sf =lz/lb; %scale factors for delfection at reading position point sf2 = (lb+lm)/lb*scale; k_mag = 3*mu0/(2*pi)*mag^2*sf2; % initialization Y=zeros(size(t)); % place to store locations Y Z=zeros(size(t)); % place to store locations Z V=zeros(size(t)); % voltage function for k=1:length(sigma) pos = 0.0; Y(1)= 0.0; % initial velocity V(1)= 0.0; % initial voltage for j=1:length(dy) Z(1)= pos; Y(1)= 0.0; % initial velocity V(1)= 0.0; % initial voltage for i=1:N Y(i+1)=Y(i) -(K/m)*Z(i)*h+3*mu0/(2*pi*m)*mag^2*Z(i)*sf2/((sf2^2*Z(i)^2+(dy(j))^2)^2.5)*h-2*damp*sqrt(K/m)*Y(i)*h... -(Kv/m)*V(i)*h+sigma(k)*epsilon(i)*dW(i)/m; Z(i+1)=Z(i)+Y(i)*h; V(i+1)=V(i)+Kc*(Z(i+1)-Z(i))-1/(Rl*Cb)*V(i)*h; end pos=Z(length(Z)); Z = Z*sf; [dist,xout]=hist(Z,-15e-3:0.1e-3:15e-3); dist=dist/sum(dist)*20; if plt1 figure(1); clf subplot(2,1,1); plot(t,Z,'r') hold on xlabel('time [sec]');

145

ylabel('displacement'); subplot(2,1,2); plot(t,V,'k') hold on xlabel('time [sec]'); ylabel('voltage'); end %standard deviation of voltage Vrms and position variance Pavg(j,k) = std(V); Pavg_z(j,k)= std(Z); end end %% plotting if plt2 figure(1); clf subplot(2,1,1); plot(t,Z,'r') hold on xlabel('time (s)'); ylabel('displacement (m)'); subplot(2,1,2); plot(t,V,'k') hold on xlabel('time (s)'); ylabel('voltage (V)'); hold off figure(2); clf subplot(2,1,1); plot(dy,Pavg); xlabel('dy'); ylabel('STD(V)'); subplot(2,1,2); plot(dy,Pavg_z); xlabel('dy '); ylabel('STD(Z)'); figure(3); clf bar(xout,dist); xlabel('x (mm)'); ylabel('distribution'); hold on end xout=xout'; dist=dist'; %% saving in Pavg.mat if sav

146

suffix = ['date' date '_num']; save(['num_' suffix '.mat'],'T','dy','Pavg','Pavg_z','K','damp'... ,'m','mag','Kv','Kc','Cb','sf','sf2','f_n','Rl','k_31','d_31','sigma'); save('distribution.txt','dist','-ascii'); end toc catch lasterror end

147

BBBBibliographyibliographyibliographyibliography

1. Brett A. Warneke, K.S.J.P., An Ultra-Low Energy Microcontroller for Smart Dust

Wireless Sensor &etworks. MEMS AND SENSORS, 2004. 17.

2. Akyildiz, I.F., et al., Wireless sensor networks: a survey. Computer Networks, 2002. 38(4): p. 393-422.

3. TomsÏic, J., et al., Late events of translation initiation in bacteria: a kinetic analysis. The EMBO Journal, 2000. 19(9): p. 2127-2136.

4. Chandrakasan, A., et al., Trends in low power digital signal processing. Circuits and Systems, ISCAS'98. Proceedings of the 1998 IEEE International Symposium on, 1998. 4.

5. Kurs, A., et al., Wireless Power Transfer via Strongly Coupled Magnetic

Resonances. Science, 2007. 317(5834): p. 83.

6. E. Lefeuvre, A.b., C. Richard, L. Petit, D. Guyomar, A comparison between several

approaches of piezoelectric energy harvesting. J. Phys. IV France, 2005. 128: p. 177-186.

7. Kordesh K., S.G., Fuel cells and their applications. 2001, New York: VCH Publisher.

8. Friedman D., H.H., Duan D.W. A Low-Power CMOS Integrated Circuit for Field-

Powerede Radio Frequency Identification. in Proceedings of the 1997 IEEE Solid-

State Circuits Conference. 1997.

9. Amirtharajah, R., A.P. Chandrakasan, and C. Mit, Self-powered signal processing

using vibration-based powergeneration. Solid-State Circuits, IEEE Journal of, 1998. 33(5): p. 687-695.

10. Shad Roundy, P.K.W., Jan M. Rabay, Energy Scavenging For Wireless Sensor

&etworks with special focus on Vibrations. 2004: Kluwer Academic Publisher.

11. Moore, G.E., &o exponential is forever: but" Forever" can be

delayed![semiconductor industry]. Solid-State Circuits Conference, 2003. Digest of Technical Papers. ISSCC. 2003 IEEE International, 2003: p. 20-23.

148

12. Rabaey, J., et al., PicoRadio: Ad-hoc wireless networking of ubiquitous low-

energysensor/monitor nodes. VLSI, 2000. Proceedings. IEEE Computer Society Workshop on, 2000: p. 9-12.

13. Freitas Jr, R.A., &anomedicine, Basic Capabilities Vol. I. 1999, Georgetown, TX: Landes Bioscience.

14. E. Lefeuvre, A.B., C. Richard, L. Petit, D. Guyomar, A comparison between several

vibration-powered generators for standalone systems. Sensors and Actuators A: Physical, 2006. 126: p. 405-416.

15. Roundy, S., et al., Power Sources for Wireless Sensor &etworks. Energy (J/cm 3). 3780(2880): p. 1200.

16. Paradiso, J.A. and T. Starner, Energy Scavenging for Mobile and Wireless

Electronics. Pervasive Computing, IEEE, 2005. 4(1): p. 18-27.

17. Robert J. Linhardt, e.a. Flexible energy storage devices based on nanocomposite

paper. in Proceedings of the &ational Academy of Sciences of the United State of

America. 2007.

18. Neudecker, B.J., N.J. Dudney, and J.B. Bates, “Lithium-Free” Thin-Film Battery

with In Situ Plated Li Anode. Journal of The Electrochemical Society, 2000. 147: p. 517.

19. Hart, R.W., et al., 3-D Microbatteries. Electrochemistry Communications, 2003. 5(2): p. 120-123.

20. The UltraCell RMFC. [cited; Available from: http://www.ultracellpower.com/pwp/tech_detail.htm.

21. Epstein, A.H., Millimeter-Scale, Micro-Electro-Mechanical Systems Gas Turbine

Engines. Journal of Engineering for Gas Turbines and Power, 2004. 126: p. 205.

22. Lee, C.H., et al., Design and fabrication of a micro Wankel engine using MEMS

technology. Microelectronic Engineering, 2004. 73: p. 529-534.

23. Fu, K., et al., Design and Fabrication of a Silicon-Based MEMS Rotary Engine. Proc. 2001 International Mechanical Engineering Congress and Exposition (IMECE), 2001.

24. Toriyama, T., S. Sugiyama, and K. Hashimoto, Design of a resonant micro

reciprocating engine for power generation. TRANSDUCERS, Solid-State Sensors, Actuators and Microsystems, 12th International Conference on, 2003, 2003. 2.

25. Nielsen, O.M., et al., A thermophotovoltaic micro-generator for portable power

applications. TRANSDUCERS, Solid-State Sensors, Actuators and Microsystems, 12th International Conference on, 2003, 2003. 1.

26. Richards, C.D., et al., MEMS power- The p 3 system. IECEC- 36 th Intersociety Energy Conversion Engineering Conference, 2001: p. 803-814.

27. Whalen, S., et al., Design, Fabrication and Testing of the P3 Micro Heat Engine. Sensors and Actuators, 2003. 104(3): p. 200-208.

28. Guo, H. and A. Lal, &anopower betavoltaic microbatteries. TRANSDUCERS, Solid-State Sensors, Actuators and Microsystems, 12th International Conference on, 2003, 2003. 1.

149

29. Li, H., Self-reciprocating radioisotope-powered cantilever. Journal of Applied Physics, 2002. 92(2): p. 1122.

30. Lal, A.D. and R.H. Li, Pervasive Power: A Radioisotope-Powered Piezoelectric

Generator. Pervasive Computing, IEEE, 2005. 4(1): p. 53-61.

31. Randall, J.F., On ambient energy sources for powering indoor electronic devices. 2003, Ecole Polytechnique Federale de Lausanne.

32. Roundy, S., et al., A 1.9 GHz RF Transmit Beacon using Environmentally Scavenged

Energy. optimization. 4(2): p. 4.

33. Warneke, B.A., et al., An Autonomous 16mm 3 Solar-Powered &ode for Distributed

Wireless Sensor &etworks. Proceedings of Sensors’ 02, 2002.

34. Shaheen, S.E., 2. 5% efficient organic plastic solar cells. Applied Physics Letters, 2001. 78(6): p. 841.

35. Stordeur, M. and I. Stark, Low power thermoelectric generator-self-sufficient energy

supplyfor micro systems. Thermoelectrics, 1997. Proceedings ICT'97. XVI International Conference on, 1997: p. 575-577.

36. Strasser, M., et al., Miniaturized thermoelectric generators based on poly-Si and

poly-SiGe surface micromachining. Sensors & Actuators A, 2002. 3179: p. 1-8.

37. Glatz, W., S. Muntwyler, and C. Hierold, Optimization and fabrication of thick

flexible polymer based micro thermoelectric generator. Sensors and actuators. A, Physical, 2006. 132(1): p. 337-345.

38. Kymissis, J., et al., Parasitic power harvesting in shoes. Proc. of the Second IEEE International Conference on Wearable Computing (ISWC), IEEE Computer Society Press, pages, 1998: p. 132–139.

39. Singh, U.K., Piezoelectric Power Scavenging of Mechanical Vibration Energy, in Australian Mining Technology Conference. 2007.

40. Y.B. Jeon, R.S., J.-h. Jeong and S.-G. Kim, MEMS power generator with transverse

mode thin film PZT. Sensors and Actuators A: Physical, 2005. 122: p. 16-22.

41. Wang, Z.L. and J. Song, Piezoelectric &anogenerators Based on Zinc Oxide

&anowire Arrays. 2006, American Association for the Advancement of Science.

42. Wright, S.R.a.P.K., A piezoelectric vibration based generator for wireless

electronics. Smart Material and Structures, 2004. 13: p. 1131-1142.

43. Fang, H.B., et al., Fabrication and performance of MEMS-based piezoelectric power

generator for vibration energy harvesting. Microelectronics Journal, 2006. 37(11): p. 1280-1284.

44. Amirtharajah, R., et al., A micropower programmable DSP powered using a MEMS-

basedvibration-to-electric energy converter. Solid-State Circuits Conference, 2000. Digest of Technical Papers. ISSCC. 2000 IEEE International, 2000: p. 362-363.

45. Meninger, S., et al., Vibration-to-electric energy conversion. Very Large Scale Integration (VLSI) Systems, IEEE Transactions on, 2001. 9(1): p. 64-76.

150

46. Beeby, S.P., M.J. Tudor, and N.M. White, Energy harvesting vibration sources for

microsystems applications. Measurement Science and Technology, 2006. 17(12): p. R175-R195.

47. Roundy, S., P.K. Wright, and K.S.J. Pister, MICRO-ELECTROSTATIC

VIBRATIO&-TO-ELECTRICITY CO&VERTERS. Fuel Cells (methanol). 220: p. 22.

48. Basrour, S., et al., Fabrication and characterization of high damping electrostatic

micro devices for vibration energy scavenging.

49. Tashiro, R., et al., Development of an electrostatic generator for a cardiac

pacemaker that harnesses the ventricular wall motion. Journal of Artificial Organs, 2002. 5(4): p. 239-245.

50. Williams, C.B. and R.B. Yates, Analysis Of A Micro-electric Generator For

Microsystems. Solid-State Sensors and Actuators, 1995 and Eurosensors IX. Transducers' 95. The 8th International Conference on, 1995. 1.

51. Beeby, S.P., et al., Micromachined silicon Generator for Harvesting Power from

Vibrations. 2004.

52. Kulah, H. and K. Najafi, An electromagnetic micro power generator for low-

frequency environmental vibrations. Micro Electro Mechanical Systems, 2004. 17th IEEE International Conference on.(MEMS), 2004: p. 237-240.

53. Mizuno, M. and D.G. Chetwynd, Investigation of a resonance microgenerator. Journal of Micromechanics and Microengineering, 2003. 13(2): p. 209-216.

54. Sághi-Szabó, G., R.E. Cohen, and H. Krakauer, First-Principles Study of

Piezoelectricity in PbTiO_ 3. Physical Review Letters, 1998. 80(19): p. 4321-4324.

55. White, N.M. and J.D. Turner, Thick-film sensors: past, present and future. Meas. Sci. Technol, 1997. 8(1): p. 1-20.

56. Lovinger, A.J., Ferroelectric Polymers. Science, 1983. 220(4602): p. 1115.

57. Wang, X., et al., Direct-Current &anogenerator Driven by Ultrasonic Waves. Science, 2007. 316(5821): p. 102.

58. Ikeda, T., Fundamentals of Piezoelectricity. . 1996, Walton St, Oxford, UK: Oxford University Press 263.

59. Jeon, Y.B., et al., MEMS power generator with transverse mode thin film PZT. Sensors and Actuators A: Physical, 2005. 122(1): p. 16-22.

60. Map, S., P. Room, and S. Cart, Improving Power Output for Vibration-Based Energy

Scavengers. Pervasive Computing, IEEE, 2005. 4.

61. Roundy, S., et al., Improving Power Output for Vibration-Based Energy Scavengers. Pervasive Computing, IEEE, 2005. 4(1): p. 28-36.

62. Duffing, G., Erzwungene Schwingungen bei veränderlicher Eigenfrequenz. Vieweg, Braunschweig, 1918: p. 134.

63. Rahman, M., Stationary solution for the color-driven Duffing oscillator. Physical Review E, 1996. 53(6): p. 6547-6550.

151

64. Grossmann, F., et al., Coherent destruction of tunneling. Physical Review Letters, 1991. 67(4): p. 516-519.

65. Shu, Y.C. and I.C. Lien, Analysis of power output for piezoelectric energy

harvesting systems. Smart Mater. Struct, 2006. 15: p. 1499–512.

66. Sodano, H.A., G. Park, and D.J. Inman, Estimation of Electric Charge Output for

Piezoelectric Energy Harvesting. Strain, 2004. 40(2): p. 49-58.

67. Hagood, N.W., W.H. Chung, and A. Von Flotow, Modelling of Piezoelectric

Actuator Dynamics for Active Structural Control. Journal of Intelligent Material Systems and Structures, 1990. 1(3): p. 327.

68. Kraftmakher, Y., Magnetic field of a dipole and the dipole–dipole interaction. European Journal of Physics, 2007. 28(3): p. 409-414.

69. Higham, D.J., An Algorithmic Introduction to &umerical Simulation of Stochastic

Differential Equations. SIAM Review - Society for Industrial and Applied Mathematics 2001. 43(3): p. 525-546.

70. Fox, R.F., et al., Fast, accurate algorithm for numerical simulation of exponentially

correlated colored noise. Physical Review A, 1988. 38(11): p. 5938-5940.

71. Risken, H., The Fokker-Planck Equation: Methods of Solution and Applications. 1996: Springer.

72. Mei, D.C., Y.L. Xiang, and C.W. Xie, &umerical simulation study of the state

variable correlation function of a bistable system subject to cross-correlated noises. Physica Scripta, 2006. 74(1): p. 123-127.

73. Gitterman, M., Classical harmonic oscillator with multiplicative noise. Physica A: Statistical Mechanics and its Applications, 2005. 352(2-4): p. 309-334.

74. Ottman, G.K., et al., Adaptive piezoelectric energy harvesting circuit for wireless

remote power supply. Power Electronics, IEEE Transactions on, 2002. 17(5): p. 669-676.

75. Shenck, N.S. and J.A. Paradiso, Energy scavenging with shoe-mounted

piezoelectrics. Micro, IEEE, 2001. 21(3): p. 30-42.

76. Guyomar, D. and A. Badel, &onlinear semi-passive multimodal vibration damping:

An efficient probabilistic approach. Journal of Sound and Vibration, 2006. 294(1-2): p. 249-268.

77. Wang, Z.L. and J. Song, Piezoelectric &anogenerators Based on Zinc Oxide

&anowire Arrays. 2006, American Association for the Advancement of Science. p. 242-246.

78. Ammar, Y., et al., Wireless sensor network node with asynchronous architecture

and vibration harvesting micro power generator. Proceedings of the 2005 joint conference on Smart objects and ambient intelligence: innovative context-aware services: usages and technologies, 2005: p. 287-292.

79. Fang, H.B., et al., Fabrication and performance of MEMS-based piezoelectric power

generator for vibration energy harvesting. Microelectronics Journal, 2006. 37(11): p. 1280-1284.

152

80. Aldridge, J.S. and A.N. Cleland, &oise-Enabled Precision Measurements of a

Duffing &anomechanical Resonator. Physical Review Letters, 2005. 94(15): p. 156403.

81. Almog, R., et al., &oise Squeezing in a &anomechanical Duffing Resonator. Physical Review Letters, 2007. 98(7): p. 078103.

82. Gammaitoni, L., Private communication. 2006.

83. Werner, M.J. and P.D. Drummond, Robust algorithms for solving stochastic partial

differential equations. Journal of Computational Physics, 1997. 132(2): p. 312-326.

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