Optimization of Coil Parameters for a Nonlinear Two

Degree-of-Freedom (2DOF) Velocity-ampliﬁed Electromagnetic

Vibrational Energy Harvester

Eliabetta Boco

,Valeria Nico

, Declan O’Donoghue

, Ronan Frizzell

, Gerard Kelly

and Jeff Punch

CTVR, Stokes Institute, University of Limerick, Plassey, Limerick, Ireland

Efﬁcient Energy Transfer (ηET) Dept, Bell Labs, Alcatel-Lucent, Dublin, Ireland

Keywords:

Energy Harvesting, Nonlinearity, Multiple Degree of Freedom, Electromagnetic Optimization, Modelling.

Abstract:

A 2DOF velocity ampliﬁed electromagnetic vibrational energy harvester is analyzed. The system consists of

two masses, one larger than the other, oscillating relative to each other in response to external excitation. The

large mass is designed with a centrally located cavity into which a second smaller mass is placed. This con-

ﬁguration allows the larger mass to impart momentum to the smaller mass during impact, which signiﬁcantly

ampliﬁes the velocity of the smaller mass. By coupling high strength magnets (placed on the larger mass) and

a coil (embedded in the smaller mass), an electric current is induced in the coil through the relative motion of

the two masses. To intensify the magnetic ﬁeld, the magnets are arranged with alternating polarity within the

soft-iron body of the larger mass. Between the two masses, and between the larger mass and the support, four

springs are placed. The smaller mass is designed to disconnect from the larger mass, when input vibrations of

sufﬁcient magnitude are present, and this leads to signiﬁcant nonlinearity in the system response, which is well

described by its transfer function. The nonlinearity leads to an increased bandwidth over which the system

can harvest energy. As a further improvement, the energy harvester is optimized by changing the properties

of the coil. Four different coils are compared in terms of their voltage and power output. Finally, a theoretical

model is proposed in order to predict the optimal conﬁguration.

1 INTRODUCTION

From the invention of the ﬁrst transistor in 1947

made by John Bardeen, Walter Brattain, and William

Shockley, the number of transistors in Integrated Cir-

cuits (ICs) has followed Moore’s law, basically dou-

bling approximately every two years. An increas-

ing number of transistors, however, has not only led

to more powerful devices, but also increased power

consumption. Moreover, nowadays Information and

Communication Technologies (ICTs) are used in al-

most all ﬁelds of everyday life, so that the global

power demand is constantly increasing, mainly in-

volving a greater number of sensors and micro and

nano-scales device (i.e. wireless sensors networks for

temperature or pressure in buildings, industrial plants

or in the environment). Energy harvesting comes

from the necessity to address this increasing power

demand and aims to extract energy already present in

the environment in many forms, such as temperature

gradients, vibrations and electromagnetic waves, and

use this energy to power low-power consuming elec-

tronic devices.

Vibrations are one of the most appealing kinds of

ambient energy: they are always present, at any scale,

and their intensity can be very different depending on

the surrounding conditions. There are many possi-

ble ways to convert vibrational energy: piezoelectric,

electromagnetic or variable capacitors. In this work,

the electromagnetic conversion method is used, be-

cause it is applicable at many scales and it can pro-

duce quite high power densites provided that there

is high relative velocity between the magnet and coil

(Waters et al., 2008). Nonlinear systems are advanta-

geous compared to linear systems since although lin-

ear systems have a higher response at resonance, non-

linear systems are more ﬂexible because they do not

need to be tuned, i.e. they are naturally able to harvest

energy from broad band excitations. In this manner, a

single nonlinear device can be used in many different

applications and can be efﬁcient even if the excitation

frequency spectrum is not constant in time, which is a

feature of many real vibration proﬁles (Cottone et al.,

2009; Leadenham and Erturk, 2014).

119

Boco E., Nico V., O’Donoghue D., Frizzell R., Kelly G. and Punch J..

Optimization of Coil Parameters for a Nonlinear Two Degree-of-Freedom (2DOF) Velocity-ampliﬁed Electromagnetic Vibrational Energy Harvester.

DOI: 10.5220/0005411901190128

In Proceedings of the 4th International Conference on Smart Cities and Green ICT Systems (SMARTGREENS-2015), pages 119-128

ISBN: 978-989-758-105-2

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

The ﬁrst part of the paper presents the experi-

mental characterization of a two degree-of-freedom

(2DOF) velocity-ampliﬁed electromagnetic energy

harvester. To exploit nonlinearities, and to provide

velocity ampliﬁcation, an uncoupled 2DOF system

is used: a 2DOF harvester can enlarge the energy

conversion bandwidth (Wu et al., 2012b; Wu et al.,

2012a; Jang et al., 2011), but it does not improve the

harvesting capability for random excitations as long

as the masses are coupled (Jang et al., 2012). More-

over, for electromagnetic conversion the most impor-

tant feature is the high relative velocity between coil

and magnets: in order to have velocity ampliﬁcation

through the momentum transfer between two impact-

ing masses, the masses need to beuncoupled. A trans-

fer function analysis is used to describe the observed

nonlinearity in detail. Following this, an optimization

process for designing the coil of the electromagnetic

generator is proposed. Finally, a numerical model is

presented, which can be used to predict the optimal

coil settings for a given volume.

2 TRANSFER FUNCTION

CHARACTERISATION

The main purpose of this section is to investigate the

nonlinear response of a 2DOF vibration energy har-

vester. In order to do so, the system was tested experi-

mentally under different levels of acceleration and the

transfer function is used to highlight nonlinearities in

the system response.

2.1 Experimental Setup and Procedure

The system consists of a large mass, where four mag-

nets are orientated in the conﬁgurations shown in

Fig.1. The large mass can move between two springs

which are attached to the outer housing that serve to

transfer energy from the vibrating base into the larger

mass itself. A smaller mass, enclosed within the cav-

ity of the larger mass is designed to separate from the

larger mass when sufﬁcient excitation is present. This

conﬁguration allows the larger mass to impart mo-

mentum to the smaller mass during impact, which sig-

niﬁcantly ampliﬁes the velocity of the smaller mass.

(Cottone et al., 2014; Nico et al., 2014; O’Donoghue

et al., 2014). All impacting surfaces have high Q

springs attached to mediate the impacts and efﬁciently

transfer loads. The volume of the outside mass is

1.29 · 10

-4

. The closed loop system used to con-

trol the LDS V406 permanent magnet shaker from

Bruel&Kjaer is shown schematically in Fig.2. Out-

put signals from the Dactron Comet shaker control

(a) Conﬁguration (1): outer mass is steady

(b) Conﬁguration (2): the movement of the outer mass is

constrained by the cap. The springs are always connected

to the large mass, so that it always experiences the elastic

force.

Figure 1: Schematic of the 2DOF energy harvester. The

coil is embedded in a smaller mass which oscillates between

four magnets that provide a strong magnetic ﬁeld. The big

mass oscillates outside in conﬁguration (2) and (3) while it

is ﬁxed in conﬁguration (1). The collisions between the two

masses provide the velocity ampliﬁcation.

system were ampliﬁed using the LDS TPO 25 Power

Oscillator and used to drive the shaker and the voltage

response of the harvester for each coil was measured

using LabView through an appropriate data acquisi-

tion card. For each coil, data havebeen acquired using

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the load resistance which maximized the power out-

put. A high sensitivity (1.96mV/g) PCB Piezotronic

accelerometer was employed to provide feedback to

the controller in order to ensure that the correct accel-

eration levels were applied.

To fully characterize the system, two conﬁgura-

tions were tested (see Fig.1 (a) and (b)): (1) mo-

tion of the larger mass was prevented and so only

the smaller mass was free to oscillate and (2) mo-

tion of the smaller mass was free but motion of the

larger mass was restricted by the top of the housing

which was lowered to ensure the larger mass could

not detach from its supporting springs (this meant that

spring forces were always active on the larger mass).

These conﬁgurations allowed different modes of op-

eration of the 2DOF system to be investigated to bet-

ter understand the response of the system.

Figure 2: Experimental setup.

In conﬁguration (1), the response of the smaller

mass could be investigated in isolation, while the

coupled response was examined using conﬁguration

(2). The response of each of these conﬁgurations

varied considerably depending on the magnitude of

the input acceleration, as this controlled whether the

masses decouple from each other or not. For exam-

ple, when the input acceleration was sufﬁciently low

(from a=0.05g to a=0.2g) the larger mass did not os-

cillate in any conﬁguration and the spring on the un-

derside of the smaller mass remained in contact with

the larger mass. Under such excitation levels, conﬁg-

uration (1) behaved linearly, while system (2) could

be considered to be a a pair of coupled harmonic os-

cillators. In addition there where further nonlineari-

ties due to the electromagnetic conversion.

Under higher levels of excitation (a=0.5g), the

smaller mass in conﬁguration (1) received sufﬁcient

energy to break contact with the larger mass, lead-

ing to nonlinearity in the system response. This was

caused by the fact that while the smaller mass is de-

tached from larger mass, there is no elastic spring

force acting on it. This means that the effective elas-

tic constant experienced by the system depends on the

time that each mass is moving freely in the detached

conﬁguration. Higher accelerations also allowed such

changes in the system’s effective elastic constants to

affect conﬁguration (2), resulting in the system acting

as coupled 2DOF oscillators with a softening nonlin-

earity.

To verify these empirical observations, the differ-

ent conﬁgurations were tested under increasing and

decreasing sine wave frequency sweeps (from 5 to

100 Hz in 260 seconds) with different amplitudes of

accelerations, as in Fig.3 and Fig.4, and the voltage

output was recorded. Finally, the transfer function

was calculated for each conﬁguration as described in

the following section.

2.2 Transfer Function Analysis

Let y(t) be the output of our system, and let x(t) be the

input. Let X(f) and Y(f) be the Laplace transforms of

the input and the output respectively.

X(s) =

+∞

−∞

x(t)e

−st

dt (1)

Y(s) =

+∞

−∞

y(t)e

−st

dt (2)

Where

s = jω (3)

The Laplace transform is equal to the Fourier trans-

form, and represents the frequency behaviour of the

system.

So, for a linear system, the transfer function is de-

ﬁned as:

H(ω) =

Y(ω)

X(ω)

(4)

When the system is linear, H(ω) is a well-deﬁned

mathematical function, because it has a unique value

for each value of ω, whereas for a nonlinear system it

depends also on the shape and amplitude of the input

signal. This means that the transfer function is not

mathematically well deﬁned for nonlinear systems.

Nonetheless, transfer function analysis is a very

useful method to know if a system is linear or not for

different input signals, (Muller and Massarani, 2001),

as deviations from linearity result in distinct varia-

tions in the transfer function, as will be discussed in

the next section.

2.3 Transfer Function Results

Comparing the resulting spectrum from the transfer

function analysis in Fig.3 to a simple linear response

is the ﬁrst method to detect nonlinearity in a system.

A linear response is characterized by a sharp res-

onant peak that is symmetric about that peak. Fig.3

shows that nonlinearity due to the change in the ef-

fective stiffness of the springs is evident as the ac-

celeration increases. This is caused by the smaller

OptimizationofCoilParametersforaNonlinearTwoDegree-of-Freedom(2DOF)Velocity-amplifiedElectromagnetic

VibrationalEnergyHarvester

121

mass spending more and more time detached from the

springs with increasing acceleration which led to a re-

duction of the resonant frequency of the system due

to a decrease in the effective stiffness, an effect that is

more pronounced for higher acceleration levels. The

overall shape of the power spectrum varies for the

different cases indicating deviation from linear be-

haviour. For example at very low acceleration ampli-

tude (a=0.05g), in conﬁguration (1) , the behaviour is

linear and the resonant peak is sharp and symmetric,

as shown in Fig.3 (b). By increasing the acceleration,

the resonant frequency decreases to lower frequencies

becoming more and more asymmetric but displaying

a broader band response (Fig.3 (a)). Very similar re-

sponses are seen for conﬁguration (2) in Fig.4 (a) and

(b)

The second ﬁngerprint of nonlinearity in the trans-

fer function characterization is the hysteresis phe-

nomenon. In a linear system, a sine sweep that is

increasing or decreasing in frequency as the excita-

tion signal results in the same system response. In a

nonlinear system, however, the behaviour for increas-

ing or decreasing sweeps are different: as already

stated, the transfer function is not a mathematically

well-deﬁned function for nonlinear systems as it can

havetwopossible values for the same input frequency,

one for the increasing sweep and one for the decreas-

ing sweep (Leadenham and Erturk, 2014). The trans-

fer function analysis in Fig.3 shows this hysteresis for

conﬁguration (1).

The same analysis has also been conducted using

conﬁguration (2), in order to investigate the effect of

the impacts (Fig.4). The frequency shifting is quite

similar when the outer mass started to fully compress

the springs Fig.4 (c), (d) and (e). Then the effect of

the velocity ampliﬁcation became dominant, and the

frequency response shifted to very low frequency, due

to the larger mass, and the output increased by about

an order of magnitude Fig.4 (b). The noisy behaviour

outside resonance is due to the fact that using the

sine sweep method to measure the transfer function

is not the most robust to noise (Muller and Massarani,

2001).

An alternative more precise method, is an analysis

through single harmonics excitation. This method is

slower than a sine sweep, however, and since the non-

linear effects are very well evident despite the noise,

it was not considered necessary.

Improvements to the results could be achieved by

conducting a statistically relevant number of tests and

averaging them. The main problem with linear os-

cillators as energy harvesters, is that they are only

able to harvest energy effectively from a narrow fre-

quency range around their natural resonance. This

(a) Transfer function with the outer mass steady

(b) Hysteresis in the almost linear case. The output has

been acquired at a=0.05g so that the smaller mass cannot

detach from the springs

in input is a=0.2g, so that the smaller mass is starting to

detach from the springs

(d) Hysteresis with a=0.5g acceleration. The acceleration

a=0.5g is enough to let the smaller mass detach from the

springs and spend more time decoupled from the larger

mass leading to a decrease in the resonant frequency.

Figure 3: Measured transfer functions and hysteresis for

conﬁguration (1).

leads to a serious difﬁculty when reducing linear os-

cillator in size as, the smaller the system the higher

the resonance frequency: a mm scale harvester would

have a natural frequency in the kHz range, whereas

real-world vibrations are usually under few thousands

of Hz (Mizuno and Chetwynd, 2003; Rebeiz et al.,

1987).

High natural frequencies are not compatible with

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(a) Transfer Function with the outer mass starting to move

(b) Transfer Function with the outer mass fully compress-

ing the springs

(d) Hysteresis with a=0.3g acceleration

(e) Hysteresis with a=0.5g acceleration

Figure 4: Measured Transfer Functions and Hysteresis for

conﬁguration (2).

many real-world vibrations, which are often charac-

terised by low frequency, broad spectrum and are usu-

ally not constant in time. The analysis conducted here

reveals that the response of a nonlinear energy har-

vester is more suitable for a broad band excitation,

and that the scale is not the only factor to determine

the frequency response, since the softening effect can

also shift the resonance. There is the possibility also

to use the nonlinear response to control the resonance

frequency, however how much control is possible will

need to be determined on a system-by-system basis.

3 OPTIMIZATION

3.1 Theoretical Model

An electromagnetic energy harvester can be described

by a system of two coupled equations: the ﬁrst is the

equation of motion of the device, the second is the

voltage output. The value of the magnetic ﬂux into the

coil acts as the coupling coefﬁcient. (Bouendeu et al.,

2011; Poulin et al., 2004). In this paper, a numerical

model is proposed to predict the optimum coil param-

eters for a given magnetic ﬁeld. Of particular interest

are the wire diameter and number of turns that opti-

mize the voltage and power output of the harvester.

The procedure used in this analysis was to select

different wire diameters for a ﬁxed internal radius and

coil height, and then determine the resulting coil pa-

rameters; namely the coil length (l), coil resistance

) and coil inductance (L). The parameters for the

different wire diameters were then used to solve the

system equations for a single-dof (1DOF) system as

in conﬁguration (1) in orderto determine the optimum

wire diameter.

The system to solve is:

(

M¨z = −k

ef f

z− γ˙z−

BlV

− Asin(ωt)

V =

(Bl˙z−

V −V)

(5)

where z is the relative displacement between the mag-

net and coil, M is the inertial mass, γ is the mechanical

damping, B is the magnetic ﬁeld, l is the length of the

wire, R

is the load resistance, L is the inductance, R

is the coil resistance and V is the induced voltage.

To be able to predict the optimum coil wire diam-

eter, the coil internal radius (r

), the number of turns

(Nturns) and the wire diameter itself (d

) have been

ﬁxed in each simulation. From these parameters, it is

possible to obtain:

Nturns

t f

(6)

where Nturns

is number of wire turns along the

height of the coil, t is the thickness of the coil, f is

the ﬁll factor and d is the diameter of the wire.

OptimizationofCoilParametersforaNonlinearTwoDegree-of-Freedom(2DOF)Velocity-amplifiedElectromagnetic

VibrationalEnergyHarvester

123

Nturns

sur f

Nturnsf

Nturns

(7)

where Nturns

sur f

is the number of wire turns along

the radial direction.

The outer radius of the coil is dependent on the

number of turns and on the wire radius:

ro = ri+ dNturns

sur f

(8)

where ro and ri are respectivelythe outer and the inner

radius of the coil.

The coil length is given by:

l = Nturns(ro+ ri)π; (9)

The coil resistance can be determined from the wire

length and resistivity:

Rc =

ρl

(π(D/2)

)

(10)

where R

is the coil resistance, ρ is the resistivity of

the wire and D is the diameter of the coil.

The coil inductance (given in µH) can then be

determined from Wheelers formula (Wheeler, 1942;

Wheeler, 1928):

L = 0.02

[(ro + ri)/2]

Nturns

6(ro+ ri)/2+ 9t+ 10(ro − ri)

(11)

where ro, ri, t are in mm.

The mass of the coil is given by:

= t fπ(

ro + ri

)

(12)

where ρ

is the copper mass density and m

is the

coil mass.

is an important quantity since the coil is em-

bedded in the mass in this analysis and so contributes

to the total inertial mass (M) given by:

M = m+ m

(13)

where m is the mass of the coil housing.

The system of equations has been solved with the

Heun method, to avoid numerical divergence (Gras-

selli and Pelinovsky, 2008). The device is assumed

to be in conﬁguration (1) from Fig.1, with the smaller

mass always attached to the springs.

The Power output has been calculated using R

= R

and an effective spring constant has been cal-

culated in order to match the experimental resonance

frequency using the experimental value of the little

mass (around 0.02 kg, slightly vaying with the mass

of each coil).

Fig.5 shows simulation results for three different

wire diameters for the 1DOF system with an input ac-

celeration of a=0.4g. The results show that although

(a) Simulated voltage output with RL=RC for the three coils

(b) Simulated power output with RL=RC for the three coils

Figure 5: Simulation results in linear approximation for the

transducer only.

the smallest wire diameter gives the highest voltage,

the power is optimized using the intermediate wire di-

ameter of 170 µm. The ability to represent this type of

effect is useful in order to ensure optimum coil param-

eters are found. The approach can also be extended to

the analysis of multi-mass systems.

In the following section experimental results are

used to verify that the optimum coil wire diameter

found through simulation is correct for the actual sys-

tems of interest to this paper.

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(a) Voltage output at a=0.2g ( b) Power output at a=0.2g

(e)Voltage output at a=0.6g (f)Power output at a=0.6g

Figure 6: Voltage and power output comparison at different acceleration levels in conﬁguration (1).

OptimizationofCoilParametersforaNonlinearTwoDegree-of-Freedom(2DOF)Velocity-amplifiedElectromagnetic

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125

(a) VoltageOutput at a=0.4g

(b) VoltageOutput at a=0.6g

Figure 7: Voltage output for the three coils at different level

of accelerations for conﬁguration (3).

3.2 Experimental Validation of

Optimum Coil Wire Diameter

The same experimentalsetup shown in Fig.2 was used

to generate the results in this section. Conﬁgura-

tions (1) and (3) from Fig.1 were tested. Three dif-

ferent wire diameters were analyzed in order to de-

termine the most reasonable wire to use for winding

the coil in terms of voltage and power output. The

wire diameters selected were based on the modelling

of the previous section and the experimental results

serve to verify the model. Different levels of accel-

eration (a=0.2g, a=0.4g, a=0.6g) were provided as

input to the two conﬁgurations of interest in order to

verify that the optimal conﬁguration did not change

due to the frequency shifting observed in Section 2.3.

In addition, three levels where imposed to show that

the frequency shifting due to the system nonlinearity

did not inﬂuence the optimal value of wire diameter.

(a) Power Output at a=0.4g

(b) Power Output at a=0.6g

Figure 8: Power output for the three coils at different level

of accelerations for conﬁguration (3).

The three coils used had the same volume, but they

used different wire diameters of 100µm, 170µm and

280µm.

Each test was conducted by ﬁnding the optimal

load resistance for the differentcoils and then measur-

ing the voltage and power output. As a ﬁrst stage, the

system was optimized for conﬁguration (1): the re-

sults are shown in Fig.6. Although the voltage output

increases with decreasing wire diameter, the power

output shows a maximum for the 170 µm diameter,

before decreasing again for the lowest wire diameter.

This is due to the increasing resistance of the coil it-

self. The same effect is shown for increasing accel-

eration amplitude, as evident by comparing Fig.6 (d)

and (f).

At a=0.6g acceleration for conﬁguration (1), the

difference between the power output produced by the

different coils reduces. This could be due to the fact

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126

that the optimization theory is linear, which means

that the system is required to maintain a ﬁxed reso-

nance frequency. This is clearly not true for a non-

linear system: the shifting in the frequency behaviour

affects parameters such as the impedance of the coil,

which is taken into account in the optimization. The

same process was repeated for conﬁguration (3) in

Fig.1, in order to verify the same trends are observed

in a multi-mass system which experiences the veloc-

ity ampliﬁcation effect. In this case a wire diameter

of d=330µm was used instead of the 280µm diameter

wire. Fig.7 and Fig.8 show that the optimal wire di-

ameter is again 170 µm. The results also show that

at a=0.4g, it is possible to see the two resonant peaks

due to the two masses in motion. At a=0.6g, however,

the nonlinearity of the whole system dominates, lead-

ing to just one large peak and the spectrum becomes

wider.

The broadening of the resonance peak demon-

strates an enhanced energy harvesting capability such

that the system responds to a wider band of input fre-

quencies. It is clear from the results that the model

is capable of predicting the optimum wire diameter,

indicating it is a useful design tool. Further improve-

ments to the model would involve accounting for the

insulation thickness in the coil and using a set of non-

linear equations to model the performance. These

points will be addressed in future work.

4 CONCLUSIONS

In this paper, a 2DOF velocity ampliﬁed electromag-

netic energy harvester has been presented. The non-

linearity occurring in the system for different con-

ﬁgurations has been analyzed, which highlighted a

softening nonlinearity. This feature has the effect of

shifting the resonance frequencies of the harvester to

lower values and to enlarge the bandwidth of the re-

sponse. Thus it is possible to harvest energy from

a wider band of frequencies, making it unnecessary

to tune the harvester each time the driving frequency

changes. The transfer function characterization also

clearly demonstrates the gain in the output signal due

to velocity ampliﬁcation.

In the second part of the paper, an optimization

model has been proposed, that is capable of predicting

the most appropriate coil wire diameter for maximum

voltage and/or power. The model employs a linear ap-

proximation, and it has been veriﬁed through compar-

ison with three sets of data from different coils. The

experimental tests were carried out using three levels

of acceleration, and it was veriﬁed that the optimal

coil conﬁguratondid not change substantially with the

Figure 9: Comparison between recently published FoM

(Ashraf et al., 2013a; Ashraf et al., 2013b; Galchev et al.,

2011; Galchev et al., 2009; Renaud et al., 2009; Beeby

et al., 2007; Ayala et al., 2009; Berdy et al., 2009; Ching

et al., 2002; Sardini and Serpelloni, 2011; Zhu et al., 2012;

Kulkarni et al., 2008; Yang and Lee, 2010) . The diamond

is our ﬁrst prototype.

frequency shifting that occurs due to the nonlinearity

of the system.

In Fig.9 the comparison with some recently re-

ported energy harvesters in the literature is shown. A

high Figure of Merit (FoM) (Mitcheson et al., 2008;

Ashraf et al., 2013a) demonstrates the effectiveness of

velocity ampliﬁcation in electromagnetic energy har-

vesting. Until now, there has been no evidence of a ﬁ-

nite lifetime of the analysed harvester due to mechan-

ical failure, even with high accelerations. Future work

will focus on improving the model accuracy through

the introduction of a method to capture the effects of

the nonlinearity of the multi-mass system. Moreover,

the scaling down of the system will be addressed, in

order to enable future integration on vibrating systems

such as bridges or rotating machines.

ACKNOWLEDGEMENTS

The authors acknowledge the ﬁnancial support

of Science Foundation Ireland under Grant No.

10/CE/I1853and the Irish Research Council (IRC) for

funding under their Enterprise Partnership Scheme

(EPS). This work was ﬁnancially supported by the In-

dustrial Development Agency (IDA) Ireland.

REFERENCES

Ashraf, Khir, Dennis, and Baharudin (2013a). Improved

energy harvesting from low frequency vibrations by

OptimizationofCoilParametersforaNonlinearTwoDegree-of-Freedom(2DOF)Velocity-amplifiedElectromagnetic

VibrationalEnergyHarvester

127

resonance ampliﬁcation at multiple frequencies. In

Sensors and Actuators A: Physical. ELSEVIER.

Ashraf, Khir, Dennis, and Baharudin (2013b). A wide-

band, frequency up-converting bounded vibration en-

ergy harvester for a low frequency environment. In

Smart Materials and Structures. IOP Publishing.

Ayala, Zhu, Tudor, and Beeby (2009). Autonomous tunable

energy harvester. In PowerMEMS, 2009.

Beeby, Torah, Tudor, Glynne-Jones, O’Donnell, Saha, and

Roy (2007). A micro electromagnetic generator for

vibration energy harvesting. In Journal of Microme-

chanics and Microengineering. IOP Publishing.

Berdy, Srisungsitthisunti, Xu, Rhoads, Jung, and Peroulis

(2009). Compact low frequency meandered piezoelec-

tric energy harvester. In PowerMEMS, 2009.

Bouendeu, Greiner, Smith, and Korvink. (2011). Design

synthesis of electromagnetic vibration-driven energy

generators using a variational formulation. In Journal

of Microelectromechanical Systems. IEEE.

Ching, Wong, Li, Leong, and Wen (2002). A laser-

micromachined multi-modal resonating power trans-

ducer for wireless sensing systems. In Sensors and

actuators A: Physical. ELSEVIER.

Cottone, Frizzell, Goyal, Kelly, and Punch (2014). En-

hanced vibrational energy harvester based on velocity

ampliﬁcation. In Journal of Intelligent Material Sys-

tems and Structures. SAGE.

Cottone, Vocca, and Gammaitoni (2009). Nonlinear energy

harvesting. In PhysRevLett, vol.102. American Physi-

cal Society.

Galchev, Cullagh, Peterson, and Najaﬁ (2011). Harvesting

trafﬁc-induced vibrations for structural health moni-

toring of bridges. In Journal of Micromechanics and

Microengineering. IOP Publishing.

Galchev, Kim, and Najaﬁ (2009). A parametric frequency

increased power generator for scavenging low fre-

quency ambient vibrations. In Procedia Chemistry,

Volume 1, Issue 1, September 2009, Pages 14391442.

ELSEVIER.

Grasselli and Pelinovsky (2008). Numerical Mathematics.

Jones & Bartlett Learning.

Jang, Rustighi, Brennan, Lee, and Jung (2011). Design of a

2dof vibrational energy harvesting device. In Journal

of Intelligent Material Systems and Structures. SAGE.

Jang, Rustighi, Brennan, Lee, and Jung (2012). Vibration

energy harvesting from random force and motion ex-

citations. In Journal of Intelligent Material Systems

and Structures. SAGE.

Kulkarni, Koukharenko, Torah, Tudor, Beeby, O’Donnell,

and Roy (2008). Design, fabrication and test of in-

tegrated micro-scale vibration-based electromagnetic

generator. In Sensors and Actuators A: Physical. EL-

SEVIER.

Leadenham and Erturk (2014). M-shaped asymmetric non-

linear oscillator for broadband vibration energy har-

vesting: Harmonic balance analysis and experimental

validation. In Journal of Sound and Vibration,vol.333,

n.23, pag.6029-6223. Elsevier.

Mitcheson, Yeatmann, Rao, Holmes, and Green (2008).

Energy harvesting from human and machine motion

for wireless electronic devices. In Proceedings of the

IEEE. IEEE.

Mizuno and Chetwynd (2003). Investigation of a resonance

microgenerator. In Journal of Micromechanics and

Microengineering 13, 209-216. Institute of Physics

Publishing.

Muller and Massarani (2001). Transfer-function measure-

ment with sweeps. In J. Audio Eng. Soc. AES.

Nico, O’Donoghue, Frizzel, Kelly, and Punch (Septem-

ber 2014). A multiple degree-of-freedom velocity-

ampliﬁed vibrational energy harvester part b: Mod-

elling. In ASME 2014 International Conference on

Smart Materials. SMASIS.

O’Donoghue, Nico, Frizzel, Kelly, and Punch (September

2014). A novel velocity ampliﬁed vibrational energy

harvester: Experimental analysis. In ASME 2014 In-

ternational Conference on Smart Materials. SMASIS.

Poulin, Sarraute, and Costa (2004). Generation of electrical

energy for portable devices: Comparative study of an

electromagnetic and a piezoelectric system. In Sen-

sors and Actuators A Physical. ELSEVIER.

Rebeiz, Regehr, Rutledge, Savage, and Jr, L. (1987).

Submillimeter-wave antennas on thin membranes.

In International Journal of Infrared and Millimeter

Waves. Springer.

Renaud, Fiorini, Schaijk, V., and Hoof, V. (2009). Har-

vesting energy from the motion of human limbs: the

design and analysis of an impact-based piezoelectric

generator. In Smart Materials and Structures. IOP

Publishing.

Sardini and Serpelloni (2011). An efﬁcient electromagnetic

power harvesting device for low-frequency applica-

tions. In Sensors and actuators A: Physical. ELSE-

VIER.

Waters, Chisum, Jazo, and xFralick (2008). Development of

an electro-magnetic transducer for energy harvesting

of kinetic energy and its applicability to a mems-scale

device. In Nanopower Forum 2008.

Wheeler (1928). Simple inductance formulas for radio

coils. In Proceedings of the I.R.E.

Wheeler (1942). Formulas for the skin effect. In Proceed-

ings of the I.R.E.

Wu, Tang, Yang, and Soh (August 21, 2012b). A novel two-

degree-of-freedom piezoelectric energy harvester. In

Journal of Intelligent Material Systems and Struc-

tures. SAGE.

Wu, Tang, Yang, and Soh (July 22, 2012a). Development of

a broadband nonlinear two-degree-of-freedom piezo-

electric energy harvester. In Journal of Intelligent Ma-

terial Systems and Structures. SAGE.

Yang and Lee (2010). Non-resonant electromagnetic wide-

band energy harvesting mechanism for low frequency

vibrations. In Microsyst Technol (2010) 16:961966.

Springer-Verlag 2010.

Zhu, Beeby, Tudor, and Harris (2012). Vibration energy

harvesting using the halbach array. In Smart Materials

and Structures. IOP Publishing.

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