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Finite element modeling and characterization of a magnetoelastic broadband energy
harvester
Alcala-Jimenez, L.R.; Lei, A.; Thomsen, E.V.
Published in:
Sensors and Actuators A: Physical
Link to article, DOI:
10.1016/j.sna.2020.112104
Publication date:
2020
Document Version
Peer reviewed version
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Citation (APA):
Alcala-Jimenez, L. R., Lei, A., & Thomsen, E. V. (2020). Finite element modeling and characterization of a
magnetoelastic broadband energy harvester. Sensors and Actuators A: Physical, 312, [112104].
https://doi.org/10.1016/j.sna.2020.112104
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Journal Pre-proof
Finite Element Modeling and Characterization of a Magnetoelastic
Broadband Energy Harvester
L.R. Alcala-Jimenez, A. Lei, E.V. Thomsen
PII:
S0924-4247(20)30463-5
DOI:
https://doi.org/10.1016/j.sna.2020.112104
Reference:
SNA 112104
To appear in:
Sensors and Actuators: A. Physical
Received Date:
28 March 2020
Revised Date:
8 May 2020
Accepted Date:
23 May 2020
Please cite this article as: L.R. Alcala-Jimenez, A. Lei, E.V. Thomsen, Finite Element
Modeling and Characterization of a Magnetoelastic Broadband Energy Harvester,
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© 2020 Published by Elsevier.
Finite Element Modeling and Characterization of a
Magnetoelastic Broadband Energy Harvester
L. R. Alcala-Jimenez, A. Lei and E. V. Thomsen
Department of Micro- and Nanotechnology, DTU Nanotech,
Technical University of Denmark, Building 345E, DK-2800 Kgs. Lyngby, Denmark
Abstract
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of
Piezoelectric-based vibration energy harvesters have been extensively developed to the end of replacing low-power
batteries. However, matching the frequency of the ambient vibration is not always possible and to broaden the frequency
response of the harvesters, di↵erent proof-of-concept devices have been developed. This work presents a miniaturized
device intended for magnetoelastic broadband vibration energy harvesting. The device consists of a silicon beam where
ferromagnetic foils act as proof mass and interacts with an external pair of permanent magnets. The interaction is
first simulated using a Finite Element Method (FEM) model for di↵erent distances between the magnets and beam
(a) and between the magnets (b). This is done for both an attractive and a repulsive magnet configuration and the
calculation is performed for a set of a and b values. Both spring softening and hardening e↵ects are observed for
the two magnet configurations. The attractive configuration has a monostable potential energy landscape with an
associated spring softening for a large range of a and b values which makes this configuration very usefull for energy
harvesting applications. The attractive configuration is experimentally investigated by impedance measurements. These
measurements are performed for a 2 [400 µm, 2500 µm] and b 2 [320 µm, 3140 µm] and a region that allows for broadband
harvesting is found experimentally. Compared to the linear case the largest spring softening e↵ect yielded a decrease
in the e↵ective spring constant of 74%, a decrease in the resonant frequency of 49%, and the coupling coefficient was
increased with a factor of 2.6.
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Keywords: MEMS, Energy harvester, FEM, Piezoelectric, Magnetoelastic, Broad bandwidth.
1. Introduction
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The demands on power sources limit the applicability
of wireless miniature sensor systems for use in o↵-the-grid
applications where it is difficult or impossible to change
batteries. Two of the main requirements for miniatur- 30
ized systems used as power sources are physical dimensions
in the millimeter-scale range and long lifetime. Conventional batteries imply continuous and sometimes expensive replacement cost. Furthermore, they impose a limit
on system miniaturization. This means that for some applications a replacement for conventional batteries must 35
be found. Vibration energy harvesters (VEHs), which can
harvest energy from ambient vibrations and convert it into
electrical energy automatically, have become a promising
alternative for a range of applications. There are three
main approaches to harvesting energy from vibrations, 40
which involve the use of either electrostatic [1–6], electromagnetic [7, 8] or piezoelectric [9–19] principles. As a general rule, the main challenge in vibrational energy harvesting is that the maximum system performance is achieved
when the resonant frequency of the VEH matches the ex- 45
ternal vibration source. The key parameters for VEHs
are the output power and the frequency bandwidth. The
output power can be increased by optimizing the coupling
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Preprint submitted to Sensors & Actuators: A. Physical
factor, the design or the energy conversion scheme. Most
of the available published material presents studies of systems under resonant excitation. However, most ambient
vibration sources have a frequency shift over time [20]. For
that reason, either continuously tuning the resonant frequency using for example passive or active methods [15, 21]
or widening the frequency bandwidth of the VEHs has become of utmost importance before their practical implementation.
The frequency bandwidth of the VEH can be increased
using non-linear e↵ects [22–26]. These non-linear e↵ects
can be introduced by using permanent magnets and ferro
electric materials [22–24], for example by having a ferromagnetic cantilever and one or more permanent magnets
located close to the cantilever [22, 23, 27]. This will e↵ectively change the potential energy landscape. Due to dimensional constraints, the magnetoelastic method is more
suitable for small systems than the approach where several
harvesters operate in parallel [28].
Piezoelectric-based VEHs have attracted much attention because it is feasible to produce them in millimeterscale whilst having large power densities [29, 30]. They
typically consist of a cantilever structure with piezoelectric layers on top. For such a device, the non-linear e↵ect
May 5, 2020
a)
N
S
S
N
Attractive
b)
N
S
N
S
Table 1: Dimensional and material parameters used for the simulations.
Repulsive
Device parameters
Beam length, l
Beam thickness, h
Beam width
Foil length, lf
Foil thickness, hf
Foil width
Magnet length, lm
Magnet thickness, hm
Magnet width
Magnetization, Mmag
Si Young modulus
Foil relative permeability
=1 mm
= 1 mm
= 6.5 mm
= 3.25 mm
Fe
Si
= 150 m
c)
60
65
70
75
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can be accomplished by externally implementing a magnetic set-up and adding ferromagnetic material to the tip 90
of the cantilever, as illustrated in Fig. 1. Commonly, the
external magnetic set-up consists of either one or a pair
of magnets. It has previously been demonstrated that the
latter one leads to increased generated power when compared to the former set-up [31]. Therefore, the two-magnet 95
configuration shown in Fig. 1 will be studied in this work.
A similar, however much larger system, has been previously studied [32]. It consisted of a beam with a length of
about 11 cm placed vertically and two cylindrical magnets.
In that work, an analytical model was developed. There,
the force acting on the cantilever beam was directly calculated from an expression of the magnetic field. However,100
the method presented in that study does not directly apply to smaller system dimensions where tiny square magnets are used. A numerical study of a non-linear oscillator
for broadband energy harvesting was presented by [33].
Nonetheless, the study did not consider any external mag-105
netic setup, therefore lacking any dimensional analysis. On
the other hand, it has been found that for a given set-up,
like the one under study in this work, the parameters that
determine whether the cantilever presents a softening effect or a bi-stable behavior are the distance between the110
magnets and the tip of the cantilever, a, and the distance
between the magnets, b [15, 16, 34]. The position of the
magnets with respect to the cantilever beam is illustrated
in Fig. 1. When a pair of magnets is used they can be positioned in either an attractive magnetic configuration, Fig.115
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85
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80
1a, or a repulsive magnetic configuration, Fig. 1b. As a
consequence, it becomes of utmost importance to find the
dimensional parameters, a and b, that allows for a VEH
that presents a softening e↵ect.
The aim of this work is to find the positions of the magnets, a and b, that lead to a VEH having a softening effect. These positions are found by studying the potential energy of the system using finite element calculations.
Experiments are performed to validate the proposed device. Since one of the requirements for powering wireless sensor systems, as described previously, is dimensions
in the millimeter-scale range, the focus of this study is
on small-size cantilevers with lateral dimensions no longer
than 10 mm.
The article is organized as follows: First, the simulation
methods used are explained. Secondly, the experimental
methods are presented, together with a fabrication process
description. Thirdly, the simulation results are presented
followed by the experimental findings. Finally, the article
ends with conclusions.
re
Figure 1: Side-view of the vibrational energy harvester system showing how the two magnets are placed and a definition of the magnet configurations studied in this work. a) Shows the two magnets
placed in an attractive magnetic configuration. b) The two magnets are placed in a repulsive magnetic configuration. c) The energy
harvester consists of a silicon cantilever beam with integrated iron
foils acting as proof mass. The magnets are placed in the attractive
magnetic configuration.
ro
of
Fe
-p
= 40 m
Values
6.5 mm
40 µm
6 mm
3.25 mm
150 µm
6 mm
1 mm
1 mm
5.5 mm
750 kA/m
150 GPa
µr = 4000
2
2. Simulation Methods
The system studied in this work is depicted in Fig. 1
and relevant dimensions are given in Tab. 1. It consists of
a silicon cantilever beam with a pair of iron foils attached
on either side and two external neodymium magnets. The
two magnets can be placed in either an attractive or a
repulsive magnetic configuration as shown in Fig. 1a and
Fig. 1b, respectively.
The materials used in the simulations are silicon for
the cantilever structure, with a length of l = 6.5 mm,
see Tab. 1; neodymium for the external magnets, which
are modeled as squares with a constant magnetization of
Mmag = 750 kA/m; and iron for the ferromagnetic foils,
which are modeled as hysteresis free magnetic soft materials with a magnetic relative permeability of µr = 4000.
FEM studies were performed using COMSOL multiphysics 5.2 [35] combining the Solid Mechanics and the
Magnetic Fields modules. Each simulation was divided
125
130
contacts to the electrodes, Fig. 2b. To analyze the nonlinear response of the system, a pair of NdFeB magnets,
obtained from Supermagnete [36], were mounted on a positioner in the attractive magnetic configuration, Fig. 2c.
In order to have a precise control over the positioning of
the magnets in the three spatial dimensions, a XYZ 300
TR stage model from Quater Research and Development
was used, Fig. 2a.
The distance between the magnets and the beam, a,
and the distance between the magnets, b, were measured by using a Dino-Lite edge digital microscope model
AZ7915MZTL together with the DinoCapture 2.0 software
[37]. The distance between the magnets and the cantilever,
a, was varied by moving the stage controlling the position
of the magnets. The distance between the magnets, b, was
varied using spacers made out of brass.
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120
into two steps. The first step was a calculation performed170
in the mechanical domain. A uniformly distributed input
force was applied on the whole cantilever structure and the
position of the center of mass for the cantilever was found.
The next step was a calculation in the magnetic domain
where the magnetic force on the iron foils mounted on the175
beam was found. The position of the cantilever found in
the mechanical domain was fixed during this calculation,
i.e. the total force as function of cantilever deflection could
be determined. These steps were repeated for di↵erent input forces, from which the potential energy as a function180
of deflection was calculated. This procedure was then repeated for di↵erent positions, a and b, of the magnets.
The e↵ective spring constant, ke↵ , and the linear spring
constant, k0 , were calculated from the total force as the
negative of the derivative of the force with respect to po-185
sition.
4. Simulation Results
150
155
160
165
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The piezo electric energy harvesters were fabricated by
standard silicon micro- and nano fabrication methods us-190
ing a (100) oriented n-type Si substrate with a resistivity
of 0.0015 ⌦cm. This very low resistivity allows the silicon substrate to be used as a bottom electrode for the
piezoelectric material. The process proceeds as follows:
First, a 3000 nm thick oxide layer was grown on both195
sides of the wafer. Holes were opened in the oxide using lithography and HF etching, and KOH etching was
used to define a plate. After stripping the masking layer,
the cantilever’s anchoring point was rounded at the corner to increase the mechanical robustness of the energy200
harvester. This was done by growing a 1000 nm wet oxide layer at 1100 C, which rounds the sharp corner obtained after KOH etching. The oxide was then removed
in HF. The piezo electric material and top and bottom
electrodes were deposited by evaporation and sputtering205
of Pt/AlN/Pt/Ti with thickness of 50 nm/400 nm/120 nm,
and 10 nm, respectively. Lithography and wet etching were
used to define the geometry of the piezoeletric layer and
the electrodes. Once this was done the cantilever was released from the plate using a STS ICP Advanced Silicon210
Etcher system and photoresist as masking material. After
stripping the photoresist the energy harvesters were diced
out using a Disco Automatic dicing saw, model DAD321.
Finally, ferromagnetic foils were glued on both sides of the
beam using a pick-and-place operation.
215
The magnetoelastic behaviour of the cantilevers was
studied using impedance spectroscopy. The experimental
set-up is shown schematically in Fig. 2. The impedance
measurements were performed using an Agilent E4990A
Impedance Analyzer equipped with an Agilent 4294A220
impedance probe, Fig. 2a. The impedance measurements
were performed using a test signal level (AC) of 50 mV.
The cantilever was mounted on a sample holder having
pogo pins to both secure the beam and provide electrical
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135
The results obtained from the simulation study, explained in Section 2, are described in the this section.
Fig. 3 shows the calculated potential energy landscapes
for the linear configuration and three magnetoelastic cases
(a = 1000 µm, a = 564 µm and, a = 200 µm, respectively)
where the magnets are mounted in the attractive configuration, and extracted results are summarized in Tab. 2
. In all cases, the distance between the magnets was
b = 500 µm. The dashed curve in Fig. 3 corresponds to
the linear case, for which a spring constant k0 = 120.7 N/m
was found. For the longest distance between magnets and
cantilever tip, that is for a = 1000 µm, a monostable landscape was found. The e↵ective spring constant for this
case is ke↵ = 90.8 N/m, i.e. compared to the linear case a
spring softening has occurred. The black curve in Fig. 3
represents the results obtained for a distance a = 564 µm,
which leads to a rather ”flat” potential energy landscape
around zero deflection. The e↵ective spring constant corresponding to zero deflection is 2.2 N/m. Compared to the
linear case a significant spring softening e↵ect has taken
place. Such a flat region in the potential energy landscape
is attractive seen from a broadband energy harvesting perspective.
Finally, when the magnets are positioned at a = 200 µm,
a bistable potential energy landscape was obtained, i.e.
two stable positions, located at deflections of ±399 µm,
are found. The e↵ective spring constants corresponding to the local potential energy minima are, based on
symmetry, expected to be the same and the calculated
value is 434.3 N/m. For zero deflection the e↵ective spring
constant becomes negative and a value of 439.6 N/m is
found.
The potential energy landscape for a repulsive magnet
configuration having b = 500 µm is shown on Fig. 4. The
figure shows the potential energy as function of deflection
for the linear system (i.e. without magnets) and for three
magnetoelastic cases: monostable (a = 1000 µm), flat potential (a = 384 µm), and a bistable system (a = 200 µm).
-p
3. Experimental methods
3
Impedance probe
Impedance probe
Microscope
Stage for beam
Magnets
Microscope
Beam holder
Stage for magnets
a)
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Beam holder
b)
c)
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Figure 2: Experimental set-up used for characterizing the electrical properties of the energy harvesters. a) The cantilever beam is placed
in a sample holder which can be moved by the stage for the beam. The impedance probe is connected to the two electrodes of the energy
harvester. The magnets are placed in a fixture attached to the stage for the magnets. The distance between the magnets and the beam, a,
can be varied using the two stages. b) Details of the experimental set-up used for the electrical characterization showing the sample holder
with integrated pogo-pins securing the beam to the sample holder. One pogo-pin is used to contact the top electrode of the harvester allowing
impedance measurements to be performed. The microscope used for measuring distances is shown as well. c) Holder for the magnets with
the rectangular magnets located at the center of the fixture. The distance between the magnets, b, can be adjusted by placing brass spacers
between the two magnets.
Table 2: E↵ective spring constants, ke↵ , and the location, xmin , of the minimum in potential energy for selected potential energy landscapes.
The results are for di↵erent values of the beam to magnet distance, a, as indicated in the table. The distance between the magnets is
b = 500 µm
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for all cases considered. Results for both a repulsive and an attractive configuration of the magnets are shown together
with the results obtained for the linear configuration (i.e. without the magnets).
Linear Repulsive
Attractive
Parameter Unit Bistable
Bistable Flat Monostable Bistable
Bistable Flat Monostable
ke↵
N/m 120.7
281.6
-268.7
4.3
136.6
434.3
-439.6
2.2
90.8
a
µm
200
200
384
1000
200
200
564
1000
xmin
µm
0
±311
0
0
0
±399
0
0
0
4
Attractive configuration, b=500 m
25
Linear (without magnets)
Monostable, a=1000 m
Flat, a=564 m
Bistable, a=200 m
20
Repulsive configuration, b=500 m
30
Linear (without magnets)
Monostable, a=1000 m
Flat, a=384 m
Bistable, a=200 m
25
10
5
0
-5
-10
-15
-800
-600
-400
-200
0
200
400
600
20
15
10
5
0
800
Deflection ( m)
-5
-800
Figure 3: Potential energy as function of deflection for the linear
system (i.e. without magnets) and for three magnetoelastic cases:
monostable (a = 1000 µm), flat potential (a = 564 µm), and a
bistable system (a = 200 µm). The simulation was performed for
an attractive magnet configuration. The distance between the magnets was b = 500 µm for all cases considered.
240
245
250
255
0
200
400
600
800
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For this configuration the monostable case shows a spring
hardening where the e↵ective spring constant is increased
to 136.6 N/m, i.e. slightly higher than for the linear case
where the spring constant is 120.7 N/m. For the flat potential energy landscape the e↵ective spring constant is 4.3
N/m corresponding to a significant spring softening. Finally, the bistable potential energy landscape has its minima located at deflections of ±311 µm and the e↵ective
spring constant is 281.6 N/m.
Both the attractive and repulsive magnet configurations
lead to flat potential energy landscapes for certain values of a. In order to find a range of magnet positions
that lead to a flat potential energy landscape a sweep over
both the a and b parameters was performed and the effective spring constant for zero deflection was calculated.
The results are shown in Fig. 5, where the normalized effective spring constant, ke↵ /k0 , is plotted as function of a
and b for an attractive magnet configuration. The contour
line for ke↵ /k0 = 0 determines the contour between the
monostable and bistable cases. The sign and value of the
normalized e↵ective spring constant at zero deflection can
be used to classify the system. Spring hardening occurs for
ke↵ /k0 > 1, spring softening occurs for 0 < ke↵ /k0 < 1,
flat potential occurs for ke↵ /k0 = 0, and bistable behavior
occurs for ke↵ /k0 < 0.
The contour plot of the normalized e↵ective spring constant, Fig. 5, shows three distinct regions named I, II and
III. Both regions I and II present monostable potential
energy landscapes with spring softening and spring hardening e↵ects taking place, respectively. Region III corresponds to a bistable regime. For the attractive configura-
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-200
Figure 4: Potential energy as function of deflection for the linear
system (i.e. without magnets) and for three magnetoelastic cases:
monostable (a = 1000 µm), flat potential (a = 384 µm), and a
bistable system (a = 200 µm). The simulation was performed for
a repulsive magnet configuration. The distance between the magnets was b = 500 µm for all cases considered.
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-400
Deflection ( m)
I: Monostable,
softening
III: Bistable
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-600
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Potential energy ( J)
Potential energy ( J)
15
II: Monostable,
hardening
Figure 5: Contour plot of the normalized e↵ective spring constant,
ke↵ /k0 , evaluated at zero deflection and for di↵erent a and b values
associated to the attractive magnet configuration.
5
1800
I: Monostable,
softening
Attractive, monostable
1600
Attractive,
bistable
1400
Repulsive, monostable
b, [ m]
1200
III: Bistable
Repulsive, bistable
1000
800
600
400
200
0
0
100
200
300
400
500
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of
II: Monostable,
hardening
600
a, [ m]
Figure 7: The plot shows matching values of the magnet separation parameters, a and b, for which ke↵ /k0 = 0. The two curves
correspond to the repulsive (blue) and attractive (red) magnet configuration, respectively.
Figure 6: Contour plot of the normalized e↵ective spring constant,
ke↵ /k0 , evaluated at zero deflection and for di↵erent a and b values
associated to the repulsive magnet configuration.
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5. Experimental results
The same study was performed for the repulsive magnet290
configuration, resulting in the contour plot shown in Fig.
6. For this configuration region II is more pronounced and
extends over a larger range of a and b values.
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260
285
The static simulations performed in this study allow
to determine the potential energy landscapes. Once this315
has been determined a dynamic study of the performance
of the energy harvester can be performed using Duffing’s
equation as described in refs. [33, 38], among others.
275
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In order to more easily compare both configurations,
295
the contour lines for ke↵ /k0 = 0 from Fig. 5 and Fig. 6
are plotted in Fig. 7. Both configurations have regions
where bistable and monostable behaviour occur. For the
repulsive configuration configuration of the magnets the
shift between the bistable and monostable regions occurs
300
in most cases for smaller values of a and b compared to the
attractive magnet configuration. From a practical point of
view both configurations have a set of a and b values leading to flat potential energy landscapes and they can be
realistically obtained during assembly of the energy har305
vesting system. Comparing the attractive, Fig. 5, and the
repulsive magnet configurations, Fig. 6, it is observed that
the attractive configuration leads to a more simple e↵ective spring constant landscape as the iso-lines are more
vertical and the region where spring softening takes place
310
(Region I) is larger. Therefore, the attractive configuration of the magnets is expected to be easier to implement
in practice and is therefore investigated experimentally.
265
To experimentally investigate the potential energy landscape for the attractive configuration of the magnets,
impedance measurements were performed for di↵erent positions of the magnets, i.e. for di↵erent a and b values,
using the measurement set-up described in Section 3.
As an example of the results obtained the impedance
spectra measured for b = 420 µm are presented in Figs.
8, 9, and 10. A typical impedance measurement is shown
in Fig. 8 where both the measured impedance magnitude
and phase are plotted as function of the frequency, and
in the measurements the position of the magnets were
(a, b) = (1000 µm, 420 µm). The resonant frequency, fr ,
for a given set of a and b values was obtained from the
minimum in the impedance magnitude and a value of
fr = 158 Hz was found. Fig. 9 shows the impedance magnitude for a range of a values corresponding to the data
points shown in Fig. 10. From Fig. 9 it is apparent that not
only the impedance magnitude at the resonant frequency
changes, but all spectra shift towards higher impedance
magnitudes. This is due to the electrical capacitance of
the AlN layer, whose associated impedance scales inversely
with respect to frequency.
The resonant frequency corresponding to each a value
was found from the impedance magnitude measurements
presented in Fig. 9. Fig. 10 shows the resonant frequencies,
impedance magnitudes and maximum phases at resonance,
which were obtained for b = 420 µm and for di↵erent values of a. For large values of a the measured resonant
frequency, f0 = 240.8 Hz, corresponds to the natural resonant frequency of the beam, i.e. the magnetic force is not
strong enough at that distance to influence the dynamics
of the beam. At a distance of about a = 3 mm a small shift
re
tion, region I extends to most of the values associated to
a and b, while region II only occurs for small values.
6
-78
240
-80
230
-82
220
-84
210
-86
200
-88
190
155
156
157
158
159
160
161
260
220
-90
162
Frequency [Hz]
! = 5900&m
180
160
335
340
345
350
! = 700 &m
140
120
! = 600 &m
60
150
! = 500 &m
-5 = 240.8 Hz
! = 400 &m
80
200
250
300
350
400
450
500
550
Frequency [Hz]
Figure 9: Impedance magnitude associated to the measurements
used to create the data shown in Fig. 10. The distance between the
magnets was b = 420 µm and each curve corresponds to a di↵erent
value of the beam to magnet distance, a.
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in the measured parameters is observed, indicating that a
spring softening e↵ect takes place. The minimum in resonant frequency, fmin , is obtained at a = 1 mm, where the
resonant frequency has decreased to fmin = 158 Hz which
corresponds to a decrease in frequency of 34% compared
to the resonant frequency of the linear device. For smaller
values of a the resonant frequency begins to increase and
for a < 0.8 mm a spring hardening e↵ect occurs.
The softening and hardening e↵ects observed are in overall agreement with the simulation results presented in Fig.
5, where first a spring softening and then a spring hardening e↵ect is obtained for b = 420 µm as a decreases in
value. However, the exact value of a where these e↵ects
are introduced di↵er most probably due to uncertainties
in the magnetization.
The main results from an analysis of the impedance
spectra measured for a range of b values are shown in Tab.
3. The values of amin , which correspond to the lowest
resonant frequency found for each b value, are also listed
in Tab. 3. The relative spring constant, kr , is calculated
from the measured resonant frequency as kr /k0 = (fr /f0 )2
where k0 is the natural spring constant of the beam. These
values are also shown in Tab. 3. The coupling coefficient
2
2
is calculated for each case as KSYS = (fmax
/fmin
1)(1/2) ,
where fmax represents the antiresonant frequency associated to each case, i.e. the frequency for which the maximum value in impedance magnitude is found. As seen
from Tab. 3, the coupling coefficient, KSYS , increases for
each optimal positioning of the magnets with respect to
the linear case, KSYS,lin , which indicates a widening of
the separation between the resonant and antiresonant frequencies. These higher coupling coefficients lead to higher
power output, as described in [39]. The largest reduction
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! = 800 &m
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325
! = 1250 &m
200
100
Figure 8: Impedance magnitude (red, left y-axis) and phase (blue,
right y-axis) measured for a magnet position that leads to a spring
softening e↵ect. The measurement was performed for (a, b) =
(1000 µm, 420 µm), which corresponds to the lowest resonant frequency, fr = 158 Hz, obtained in Fig. 10, where b is fixed to 420 µm.
320
! = 1000 &m, -min = 158 Hz
240
ro
of
250
Impedance magnitude [k ]
-76
Phase [°]
Impedance magnitude [k ]
260
Figure 10: Plot of the impedance magnitude (red, left y-axis) at the
resonant frequency, the resonant frequency (blue, right y-axis), and
maximum phase (black) for a range of a values. For the experiment
the separation distance between the magnets was b = 420 µm.
7
Table 3: Main results from the experimental study. Minimum resonant frequencies, fmin , measured for a range of b values. The a value,
for which the minimum in the resonant frequency was found, amin , is listed in the second column. The experiment was performed for an
attractive configuration of the magnets. The maximum reduction in resonant frequency compared to the natural resonant frequency of the
beam was 49%.
✓0
[ ]
-63.1
-63.1
-62.7
-64.0
-64.5
-63.0
-62.2
-63.9
-64.2
-62.6
-63.3 ± 0.8
fmin
[Hz]
169.5
157.9
177.3
198.5
182.6
158.9
129.2
173.2
122.2
122.8
–
✓min
[ ]
-76.5
-76.1
-74.4
-75.3
-78.5
-74.1
-75.9
-72.5
-74.5
-74.2
–
in resonant frequency, compared to the natural resonant
frequency of the beam, was 49%. This value was found for
a position where a = 550 µm and b = 2950 µm. For this
position the spring constant showed a decrease by 74% and
the coupling coefficient increased by a factor of 2.6 compared to the linear device. For this case, the phase angle
was ✓min = 74.5 , i.e. a decrease of of 10.2 compared
to the linear device where the phase was ✓0 = 64.2 .
Fig. 11 shows a contour plot of the relative spring
constant, kr /k0 , for a 2 [400 µm, 2500 µm] and b 2
[320 µm, 3140 µm]. The experimental results presented in
Fig. 11 show the two distinct regions, I and III, that are
also visible in Fig. 5, i.e. the softening region for the monostable case and the hardening region corresponding to the
bistable regime. Spring softening is seen for a large range
of a and b values, which is in good agreement with the
findings shown in Fig. 5. As explained previously in this
study, once a bistable regime is obtained a hardening e↵ect
occurs and the beam bends towards one of the magnets.
These measurements demonstrate that experimentally it
is possible to find positions leading to a significant spring
softening, which is attractive seen from an energy harvesting point of view.
fmin /f0
–
0.704
0.656
0.736
0.825
0.759
0.660
0.536
0.720
0.508
0.510
–
KSYS,lin
[%]
7.3
7.4
7.5
7.5
7.5
7.3
7.2
7.5
7.5
7.3
7.4 ± 0.1
re
360
f0
[Hz]
240.9
240.8
240.7
240.7
240.7
240.7
240.8
240.6
240.6
240.7
240.7 ± 0.1
380
385
b [mm]
lP
6. Conclusion
ur
na
375
kr /k0
–
0.49
0.43
0.54
0.68
0.58
0.44
0.29
0.52
0.26
0.26
–
6
0.5
0.6
0.8
0.7
!"
= 0.9
!#
5.5
5
2.5
2
5
1.5
4
3
1
4.5
4
I: Monostable,
softening
3.5
3
0.9
2.5
1
0.6
2
2
0.7
1.5
0.8
1
6
0.5
0.5
0.5
1
1.5
2
2.5
a [mm]
Jo
370
KSYS /KSYS,lin
–
2.01
2.15
1.79
1.65
2.12
2.03
2.67
1.75
2.60
2.69
–
3
III: Bistable - hardening
365
KSYS
[%]
14.7
15.8
13.4
12.4
16.0
14.9
19.3
13.1
19.6
19.6
–
-p
355
amin
[µm]
1004
1000
1250
1050
1250
1100
800
700
550
400
–
ro
of
b
[µm]
320
420
520
910
1170
1870
2270
2790
2950
3140
–
A cantilever-based structure intended for broadband
magnetoelastic vibrational energy harvesting was presented. The device was made using silicon micro fabrication techniques. Ferromagnetic foils were placed on either side of the beam and served both as proof mass and
to provide interaction with an external pair of permanent
magnets such that a magnetoelastic behavior is obtained.
FEM simulations were performed for two di↵erent magnet configurations: repulsive and attractive, with sweeps
over di↵erent a and b values. Both the repulsive and attractive magnetic configurations showed regions that allowed for a spring softening e↵ect. These regions can
Figure 11: Contour plot of the measured normalized spring constant,
kr /k0 , for di↵erent a and b values. This was measured using the
attractive magnet configuration.
8
400
405
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425
430
435
440
445
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re
lP
420
ur
na
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10