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1
Design
and
Simulation
of
Unimorph
Piezoelectric
E
nergy
H
arvesting
S
ystem
E.varadrajan
1
,M.Bhanusri
2
.1.
Research and Innovation Centre (RIC), IITM Research Park, Chennai, Tamil Nadu, India, evaradarajan@ric.drdo.in2.Department of Physics,Indian Institute of Technology (IIT) Madras, Chennai, Tamil Nadu, India
Abstract
—In this paper we made an attempt to maximizethe power output in the different piezoelectric materials in aunimorph cantilever beam conﬁguration. In this research, amacro -scale uni-morph piezoelectric power generator prototypesconsists of an active piezoelectric layer, stainless steel substrateand titanium proof mass was designed for frequencies 60 Hz- 200 Hz[. An analytical model of a micro power generatoris used to obtain displacement, voltage and generated powerwhich are the ﬁgures of merit for energy harvesting. Thismodel is presented for three different piezoelectric materials like,PbZrTiO3 (PZT), PVDF and PMN-PT. The designed unimorphpiezo energy harvesting system was modeled using COMSOLmulti physics and the observed parameters are compared withanalytical results.
I. I
NTRODUCTION
:Energy harvesting is used for capturing minute energyfrom surrounding sources, accumulating them and storingthem.With recent advancements in wireless technology,energyharvesting is highlighted as alternative for conventional bat-tery. While there are different ways through which energyis harvested, piezoelectric devices shows a great promise.Piezoelectric materials have the property of producing elec-trical charge when strained. This is called direct piezoelectriceffect.On the other hand,these materials undergo deformationwhen an electric ﬁeld is applied.This is called converse piezo-electric effect.This property of piezoelectric materials is usedin converting vibrational energy to electrical energy whichmay be stored and used as an alternative power source forportable electronics.In recent advancements,energy harvestinghave attracted considerable attention as an energy source forwireless sensor networks beacuse batteries cause a series of inconviences like limited operating life,size and contaminationissues.Solar energy provides some solutions but it is limitedin dark conditions. Piezoelectric devices are proved to be thepotential source for power generation[1].Therfore they serveas a good alternative for conventional batteries.
A. The Piezoelectric cantilever conﬁguration:
There are two types of piezoelectric materials, piezoceram-ics like Lead Zirconate Titanate(PZT) and piezopolymers likePolyvinylidene Fluoride(PVDF).When piezoelectric materialsare deformed or stressed, voltage appears across the material.The mechanical and electrical behavior can be modeled bytwo constitutive equations[2 ]
S
=
s
E
T
+
d
t
E
(I.1)
D
=
d
t
T
+
T
E
(I.2)where S-mechanical strain, T-applied mechanical stress, E-Electric ﬁeld, D-Electric displacement,
s
E
- matrix of elasticityunder conditions of constant electric ﬁeld,d-piezoelectric coef-ﬁcient matrix,
T
=permittivity matrix at constant mechanicalstrain.A cantilever type vibration energy harvesting has verysimple structure and can produce large deformation underdeformation. The cantilever model can be used in two differentmodes,33 mode and 31 mode. The 33 mode(compressivemode) means the voltage is obtained in the 3 direction parallelto the direction of applied force. The 31 mode(Transversemode) means the voltage is obtained in 1 direction perpendicu-lar to the direction of applied force(3). The most useful modein harvesting applications is 31 mode, because an immenseproof mass would be needed for 33 conﬁguration[1].Thevibration spectrum shows that the acceleration decreases[1] forhigher modes of frequency compared to fundamental mode of frequency. Therfore, the design of the cantilever beam focusseson fundamental mode of frequency.II. G
OVERNING EQUATIONS AND THEORY
:
A. A simply supported cantilever beam:
The resonant frequency of an cantilever without a proof mass for a simply supported cantilever beam is given by
f
n
=
ν
2
n
2
π
EI
12
AL
4
(II.1)where, E-Young’s modulus,I-Moment of inertia,A-Area,L-Length of the cantilever beam
ν
n
=1.875 for fundamental mode,
ν
n
=4.694 for secondmode.The simulation is done in comsol and both the frequen-cies are compared. Different modes are shown below:
B. Unimorph cantilever conﬁguration
Cantilever beam piezoelectric generator has three typesunimorph, bimorph series and parallel conﬁgurations.Whenthe beam has only one piezolelectrical layer attached tothe substrate,the device is known as unimorph.On the otherhand,if a metal shim is sandwiched between two piezoelectriclayers,the device is known as bimorph. For energy harvesting,an unimorph structure is chosen. One of the most importantdesign parameter in designing a vibration energy harvestingdevice is resonant frequency. The power density would bemaximum when the vibration frequency matches the resonant
Excerpt from the Proceedings of the 2013 COMSOL Conference in Bangalore
2
frequency of piezoelectric generator. It has been proved thatpower density decreases when resonant frequency deviatesfrom the vibration frequency[1]. The frequency range of common environmental vibrations is between 60 Hz and 200Hz[1].Moroever acceleration decreases with higher modes of frequencies[1].Therfore fundamental mode is considered indesigning the cantilever.The unimorph cantilever conﬁgurationlooks as in Fig0.2.2.
Figure II.1. Unimorph cantilever
The frequency of an unimorph cantilever is given by[3]
f
=
ν
2
n
2
π
0
.
236
D
p
w
(
l
−
l
m
2
)
3
(
m
e
+
m
p
)
(II.2)
ν
n
= 1
.
875
m
=
ρ
p
t
p
+
ρ
s
t
s
(II.3)
m
e
= 0
.
236
mw
(
l
−
l
m
2 ) +
mwl
m
2
(II.4)
D
p
=(
E
2
p
t
4
p
+
E
2
s
t
4
s
+ 2
E
p
E
s
t
p
t
s
(2
t
2
p
+ 2
t
2
s
+ 3
t
p
t
s
))12(
E
p
t
p
+
E
s
t
s
)
(II.5)where,
E
p
=
Young’s modulus of piezoelectric material,
E
s
= Young’s modulus os substrate,
l
m
=
Length of proof mass,
l
=
l
b
= Length of the beam,
w
=
w
b
=
w
m
=
Width of the beam,
t
p
= Thickness of piezoelectric material,
t
s
= Thickness of substrate,
m
p
=
Proof mass,
ρ
p
= Densityof piezoelectric material,
ρ
s
= Density of substrate material.:The dimensions of a cantilever are chosen such that thefrequency range is between 60Hz and 200Hz.The dimensionsand parameters of cantilever are shown in table below:
Table ID
IMENSIONS OF CANTILEVER
l
b
(
cm
)
w
b
(
cm
)
t
b
(
cm
)
l
m
(
mm
)
w
m
(
cm
)
t
m
(
mm
)
6 3 0.1 12 3 3.5
The parameters of the cantilever are shown below:
Table IIP
ARAMETERS OF CANTILEVER
E
p
(
Mpa
)
E
s
(
Gpa
)
t
p
(
mm
)
t
s
(
mm
)
ρ
p
(
Kg/m
3
)
ρ
s
(
Kg/m
3
)
2450 205 0.11 1 1770 7850
C. Energy parameters of unimorph cantilever[6]:
Q
=
−
3
d
31
s
s
s
p
t
s
(
t
s
+
t
p
)
l
2
F B
(II.6)
s
s
= 1
E
s
,s
p
= 1
E
p
(II.7)
s
h
=
s
s
t
p
+
s
p
t
s
(II.8)
B
=
s
2
s
t
4
p
+4
s
s
s
p
t
s
t
3
p
+6
s
s
s
p
t
2
s
t
2
p
+4
s
s
s
p
t
p
t
3
s
+
s
p
t
4
s
(II.9)
V
=
−
3
d
31
s
s
s
p
t
s
t
p
(
t
s
+
t
p
)
lF ε
T
33
wB
(1 + (
3
s
2
p
s
s
t
p
t
2
s
(
t
p
+
t
s
)
2
s
h
B
−
1)
K
231
)
(II.10)
U
=
−
9
d
231
s
s
s
2
p
t
2
s
t
p
(
t
s
+
t
p
)
l
3
F
2
ε
T
33
wB
2
(1 + (
3
s
2
p
s
s
t
p
t
2
s
(
t
p
+
t
s
)
2
s
h
B
−
1)
K
231
)
(II.11)III. D
ESIGN OF UNIMORPH CANTILEVER USING COMSOL
The different modes of a simply supported cantilever beamare shown below:
Figure III.1. First modeFigure III.2. Second mode
Excerpt from the Proceedings of the 2013 COMSOL Conference in Bangalore
3
Figure III.3. Third modeFigure III.4. Fourth modeTable IIIC
OMAPRISION OF SIMULATED AND ANALYTICAL FOR DIFFERENT MODES
Analytical Simulated error(%)First 1290.015 1328.695 2.9Second 8084.96 8271.757 2.3
The model is designed in comsol.A 3 dimensional unimorphcantilever is used for the simulation in comsol. Piezopolymermaterial PVDF is used as a piezoelectric and stainless steelis used as substrate.It has been proved that cantilever beamwith higher effective mass and less damping factor gives highoutput power.The proof mass not only increases effective massbut decreases damping.So the cantilever beam with proof masshas the power 10 times of the power of the cantilever beamwithout proof mass[4].Therfore, a proof mass made of tita-nium is used.The power is maximum when non-piezoelectriclength and piezoelectric lengths are equal[5].So the lengthsof substrate and piezomaterial are made equal.Using solidmechanics module ,one end of the model is ﬁxed and the otherend is made to move freely.The eigen frequency analysis isdone.The frequency of 153.22Hz is designed using comsol.Theanalytical and simulated results vary by 1.82%.The designedmodel is shown below.The model consists of non piezoelectricmaterial made of steel,piezoelectric material made of pvdf andproof mass made of titanium.
Figure III.5. Designed model in comsol
1) Meshing::
The model is meshed with physics contolledmesh amd element size ﬁne.The meshed model looks asfollows:
Figure III.6. Meshed model
2) Model shape::
The study of the model is carried out withthe eigenfrequency step.The model shape looks as below:
Excerpt from the Proceedings of the 2013 COMSOL Conference in Bangalore
4
Figure III.7. Designed model of frequency 153.22 Hz
IV. R
ESULTS AND DISCUSSIONSFigure IV.1. Variation of frequency with the length of the beamFigure IV.2. Variation of frequency with the width of the beamFigure IV.3. Variation of frequency with the thickness of the beam
The variation of frequency with length,width and thicknessof beam are shown above.The thickness of the beam hasgreat impact on the frequency of the cantilever.It is concludedfrom the graph that the frequency is directly proportionalto the thickness of beam.As the thickness increases,stiffnessincreases which inturn increases the frequency.The width of the beam has no signiﬁcant effect on the frequency comparedto length and thickness of beam.The frequency increased from100 to 180 Hz as width increased from 0.01 to 0.05 metre.Thelength is inversely proportional to the frequency.After somepoint, the change in frequency is reduced.The desired fre-quency can be obtained for the unimorph cantilever structureconsidering these variations in design parameters.
A. Sensitivity of a unimorph cantilever:
The design parameters of an cantilever would affect thecharge,voltage and energy produced by an unimorph can-tilever.The variations are shown below.
Figure IV.4. Variation of charge with length of beam
Excerpt from the Proceedings of the 2013 COMSOL Conference in Bangalore

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