Advances in Theoretical and Applied Mechanics, Vol. 9, 2016, no. 1, 43 - 54
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/atam.2016.634
Piezoelectric Vibration Energy Harvesting
Characteristics of Barium Titanate Laminates
Fumio Narita*, Xiaolong Zhu and Yasuhide Shindo
Department of Materials Processing, Graduate School of Engineering, Tohoku
University, Aoba-yama 6-6-02, Sendai 980-8579, Japan
*
Corresponding author
Copyright © 2016 Fumio Narita, Xiaolong Zhu and Yasuhide Shindo. This article is
distributed under the Creative Commons Attribution License, which permits unrestricted use,
distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
This paper studies the vibration energy harvesting characteristics of poled barium
titanate (BT) unimorph cantilevers both analytically and experimentally. The
unimorph cantilever consists of the BT layer and Cu shim, and the BT layer has
sensing, grounding and driving electrodes. The output voltage of the cantilever
excited by bending vibration was predicted by three dimensional finite element
analysis (FEA). The output voltage was also measured, and a comparison was
made between prediction and experiment. The effects of sensing electrode
geometry and load resistance on the dynamic electromechanical fields and output
power were then discussed in detail.
Keywords: Piezomechanics, Finite element method, Material testing, Barium
titanate laminates, Electromechanical field concentrations, Energy harvesting
1.
Introduction
Piezoelectric ceramics and composites have been used in a wide range of
sensor and actuator applications, and many studies have been conducted on the
performance of piezoelectric devices [2, 3, 8]. On the other hand, the direction of
research has changed slightly from just sensors and actuators to an interest in
extracting and storing energy from the environment [5, 9], and Shindo and Narita
[7] have recently investigated the output voltage and power of S-shaped
piezoelectric energy harvester excited by bending/torsion vibration. Some
investigations have also focused on the design of self-powering, self-sensing and
44
Fumio Narita et al.
self-controlling devices. Wang and Inman [11] examined the concept and design
of a multifunctional composite sandwich structure for simultaneous energy
harvesting and vibration control. They proposed the glass fiber reinforced
composites with lead zirconate titanate (PZT) harvester, sensor and actuator, and
studied the configurations, locations and operating modes of PZT materials for
optimal power generation.
Lead oxide-based ferroelectrics, especially PZTs, have high piezoelectric
properties. Due to lead toxicity, however, the introduction of legislation in Europe
to limit the usage of lead in automotive and electronic products has led to a
worldwide search for lead-free compounds [6]. BT (BaTiO3) is a well known
lead-free material. Although many investigators are developing the
multifunctional composites using PZT materials, few of the BT-based
multifunctional composites have been studied.
In this paper, we discuss the vibration energy harvesting characteristics of
poled BT unimorph cantilevers with sensing, grounding and driving electrodes. A
three-dimensional FEA was carried out to predict the output voltage in the BT
unimorph cantilevers subjected to bending vibration. The output voltage was also
measured, and numerical results were compared with measured values. The
dynamic electromechanical fields and output power were then examined for
various sensing electrode geometries and load resistances.
2. Analysis
2.1 Basic equations
Equations of motion and Gauss’ law are given by
 ji , j  ui ,tt
Di i  0
(1)
(2)
where  ij is the component of stress tensor, ui is the component of
displacement vector,  is the mass density, Di is the component of electric
displacement vector, and a comma denotes partial differentiation with respect to
the coordinates xi (i  1 2 3) or the time t. We have employed Cartesian tensor
notation and the summation convention for repeated tensor indices. Constitutive
relations can be written as
 ij  sijkl kl  dkij Ek
(3)
Di  dikl kl  єTik Ek
(4)
where  ij is the component of strain tensor, Ei is the component of electric
field intensity vector, and sijkl  dkij and єikT are the elastic compliance, direct
piezoelectric coefficient and permittivity at constant stress, which satisfy the
following symmetry relations:
Piezoelectric vibration energy harvesting characteristics
sijkl  s jikl  sijlk  sklij 
d kij  d kji 
єijT  є Tji
45
(5)
The relation between the strain tensor and the displacement vector is given by
1
 ij  (u j i  ui  j )
(6)
2
and the electric field intensity vector is
Ei  i
(7)
where  is the electric potential. The constitutive relations (3) and (4) for
piezoelectric ceramics poled in the x3 -direction are given in Appendix A.
2.2 Finite element method
Fig. 1(a) shows a unimorph cantilever constructed of BT layer and Cu shim.
Let the coordinate axes x = x1 and y = x2 be chosen such that they coincide with
the interface and the z = x3 axis is perpendicular to this plane. The origin of the
coordinate system is located at the center of the bottom left side of BT layer, and
the edge at x = 0 is clamped. Fig. 1(b) shows a geometry and dimensions of the
unimorph cantilever. BT layer of length l = 50 mm, width w = 25 mm and
thickness 1 mm is added to the upper surface of Cu shim of length 50 mm, width
25 mm and thickness 0.5 mm. The total thickness is 1.5 mm.
BT layer has sensing, grounding and driving electrodes. The sensing and
driving electrodes are located at the center and outer parts of the top surface of BT
layer, respectively. In order to suppress coupling, there is a grounding electrode
between these two electrodes [10]. On the other hand, only a whole grounding
electrode is located on the opposite side. The dimensions of the electrodes are
shown in Fig. 1(b). The length and width of the sensing electrode is ls and ws,
respectively.
We used a commercial finite element package ANSYS with
three-dimensional (3D) eight-node elements to perform the analysis. For
simplicity, the electrode layers were not incorporated into the model. This is
because the thickness of the electrode layer is much smaller than the thickness of
the BT layer. The imposed base excitation was given by the displacement
uz0exp(it) of the clamped end (x = 0 plane), where uz0 is amplitude of applied
displacement and  is angular frequency. The top and bottom grounding electrode
surfaces are connected to the ground, so that  = 0. The output voltage of sensing
electrode was then solved. The dynamic electromechanical fields for the
cantilever were also predicted. Here, damping parameters were not included in the
model for simplicity [4].
3.
Experimental procedure
The unimorph cantilever was fabricated using BT layer (NEC/Tokin Co.
Ltd., Japan) and Cu substrate. Electrodes were coated on both sides of BT layer,
46
Fumio Narita et al.
and the BT layer was bonded to the upper surface of Cu shim by conductive
bonding as shown in Fig. 1. Fig. 2 shows the unimorph sample.
The material properties of BT layer are listed in Table 1. Elastic
T
compliances s11 s33 , direct piezoelectric coefficients d31 d33 , permittivity є33
and mass density  can be found in the published data, while the remaining
properties are assumed to be the same as those of BT ceramics reported by Jaffe
and Berlincourt [1]. The elastic compliance, Poisson’s ratio and mass density of
Cu shim are 7.69  10 -12 m2/N, 0.34 and 8920 kg/m3, respectively.
The fabricated cantilever was mounted on the vibration shaker (ET-132,
Labworks Inc., USA). The imposed displacement vibration was applied with the
shaker, and the acceleration of the imposed displacement uz02 was measured
with a laser displacement sensor (LK-G10, KEYENCE, Co. Ltd., Japan). Output
voltage Vout of the sensing electrode was also measured for the cantilever using an
oscilloscope (GDS-1062A, Good Will Instrument Co., Ltd., Japan). Then, a
resistive load was connected to the cantilever, and the output voltage Vout of the
sensing electrode was measured. Moreover, the output power Pout from the
cantilever can be calculated using the output voltage Vout and load resistance R.
4. Results and discussion
Here, the numerical and experimental results under the applied
displacement amplitude uz0 = 0.1 mm are presented. Fig. 3 shows the measured
output peak voltage Vout of the BT layer as a function of frequency f = ω/2π for
the unimorph cantilever (ls/l = 0.8 and ws/w = 0.16) at open circuit condition (R →
∞). The solid line represents the value of the output peak voltage predicted by the
FEA, and the solid circle denotes the average of two measured data. For
comparison, the predicted output peak voltage by the FEA for PZT C-203
unimorph cantilever (Fuji Ceramics Co. Ltd., Japan) is also shown as dashed line.
The material properties of C-203 are listed in Table 2. Although the damping
parameters are not included in the model, the results of the FEA are in agreement
with measured data. It can be seen that the output peak voltage of the BT layer is
smaller than that of C-203 layer.
Fig.4 shows the predicted output peak voltage Vout versus the ratio of
sensing electrode width to cantilever width ws/w for the cantilever (ls/l = 0.8)
under f = 50 Hz at R → ∞. It is noted that as the sensing electrode width decreases,
the output peak voltage increases. Fig.5 shows the predicted output peak voltage
Vout versus the ratio of sensing electrode length to cantilever length ls/l for the
cantilever (ws/w = 0.16) under f = 50 Hz at R → ∞. As the sensing electrode
length decreases, the output peak voltage increases. It is interesting to note that
the effect of sensing electrode length on the output voltage is remarkably larger
than that of sensing electrode width.
The output peak voltage of the BT unimorph cantilevers under f = 50 Hz at
R → ∞ for three types of sensing electrode geometry, obtained from the FEA, is
shown in Fig.6. Although the areas of sensing electrode are the same, the output
voltage for ls = 2 mm, ws = 8 mm is larger than that for ls = 8 mm, ws = 2 mm.
Piezoelectric vibration energy harvesting characteristics
47
The variation of normal stress σxx along the length direction is calculated for
the BT unimorph cantilever (ls/l = 0.8, ws/w = 0.16) at a chosen point (y = 0 mm
and z = 0 mm here) under f = 50 Hz at R → ∞ and the result is shown in Fig.7.
The normal stress near the clamped end in the cantilever is larger than that near
the free end.
Fig.8 shows the output power for BT unimorph cantilever (ls/l = 0.8, ws/w =
0.16) under f = 50 Hz. The output power tends to increase with load resistance
reaching a peak and then to decrease in magnitude. It is found that the optimal
resistance for BT unimorph cantilever is about 0.6 MΩ. Moreover, it is expected
that smaller sensing electrode length or width leads to larger output power, since
the output peak voltage increases as the sensing electrode length or width
decreases (see Figs. 4 and 5).
5. Conclusion
BT unimorph cantilevers with sensing, grounding and driving electrodes
were proposed, and a numerical and experimental study was conducted to discuss
the effect of sensing electrode geometry on the piezoelectric vibration energy
harvesting behavior. The output voltage of BT unimorph cantilever increases as
the area of sensing electrode decreases, and the output power of BT unimorph
cantilever under an applied displacement amplitude of 0.1 mm can achieve about
5.2 nW at a frequency of 50 Hz. Our present study is useful in designing new
piezoelectric energy harvesters, and provides a basis for refining the real device
design in order to increase output power.
Appendix A
For piezoelectric ceramics which exhibit symmetry of a hexagonal crystal
of class 6 mm with respect to principal x1 x2 and x3 (poling) axes, the
constitutive relations can be written in the following form:
 11   s11
   s
 22   12
  33 
  s13



2 23   0
 2 31   0

 

 212 
  0
 D1   0
  
 D2    0
 D  d
 3   31
s12
s11
s13
0
0
0
0
0
d31
s13
s13
s33
0
0
0
0
0
d33
0
0
0
s44
0
0
0
d15
0
0
0
0
0
s44
0
d15
0
0
0   11   0
 
0  
 22   0
 33 
 0
0 
  
0   23   0
0   31   d15
  
s66  
 12 
 0
0
0
0
d15
0
0
d31 
d31 
 E1 
d33   
  E2 
0  
E3
0  

0 
(A.1)
 11 
 
0   22  є11T
 
 

0   33    0

0   23   0
 31 
 

 12 

0
є11T
0
0   E1 
 
0   E2 
T 

є33
  E3 
(A.2)
48
Fumio Narita et al.
where
 23   32 ,
 31  13 ,
12   21
(A.3)
 23   32 ,
 31  13 ,
12   21
(A.4)
s11  s1111  s2222 ,
s44  4s2323  4s3131 ,
s12  s1122 ,
s13  s1133  s2233 ,
s66  4s1212  2s11  s12 
d15  2d131  2d223 ,
d31  d311  d322 ,
s33  s3333 ,
(A.5)
d33  d333
(A.6)
References
[1] H. Jaffe and D. A. Berlincourt, Piezoelectric transducer material, Proceedings
IEEE, 53 (1965), 1372-1386. http://dx.doi.org/10.1109/proc.1965.4253
[2] F. Narita, R. Hasegawa and Y. Shindo, Electromechanical response of
multilayer piezoelectric actuators for fuel injectors at high temperatures, Journal
of Applied Physics, 115 (2014), no. 18. http://dx.doi.org/10.1063/1.4875487
[3] F. Narita, Y. Shindo and K. Sato, Evaluation of electromechanical properties
and field concentrations near electrodes in piezoelectric thick films for MEMS
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http://dx.doi.org/10.1016/j.compstruc.2010.12.008
[4] F. Narita, Y. Shindo and T. Watanabe, Dynamic characteristics and
electromechanical fields of 1-3 piezoelectric/polymer composites under AC
electric fields, Smart Mater. Struct., 19 (2010), no. 7.
http://dx.doi.org/10.1088/0964-1726/19/7/075004
[5] M. Okayasu, D. Sato, Y. Sato, M. Konno and T. Shiraishi, A study of the
effects of vibration on the electric power generation properties of lead zirconate
titanate piezoelectric ceramic, Ceramics International, 38 (2012), 4445-4451.
http://dx.doi.org/10.1016/j.ceramint.2012.02.018
[6] Y. Saito, H. Takao, T. Tani, T. Nonoyama, K. Takatori, T. Homma, T. Nagaya
and M. Nakamura, Lead-free piezoceramics, Nature, 432 (2004), 84-87.
http://dx.doi.org/10.1038/nature03028
[7] Y. Shindo and F. Narita, Dynamic bending/torsion and output power of
s-shaped piezoelectric energy harvesters, International Journal of Mechanics and
Materials in Design, 10 (2014), 305-311.
http://dx.doi.org/10.1007/s10999-014-9247-0
Piezoelectric vibration energy harvesting characteristics
49
[8] Y. Shindo, T. Sasakura and F. Narita, Dynamic electromechanical response of
multilayered piezoelectric composites from room to cryogenic temperatures for
fuel injector applications, ASME Journal of Engineering Materials and
Technology, 134 (2012), 1-7. http://dx.doi.org/10.1115/1.4006504
[9] D. Vatansever, R. L. Hadimani, T. Shah and E. Siores, An investigation of
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[10] N. Wakatsuki, H. Yokoyama and S. Kudo, Piezoelectric actuator of LiNbO3
with an integrated displacement sensor, Jpn. J. Appl. Phys., 37 (1998), 2970-2973.
http://dx.doi.org/10.1143/jjap.37.2970
[11] Y. Wang and D. J. Inman, Simultaneous energy harvesting and gust
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http://dx.doi.org/10.1177/0021998312448677
Table 1. Material properties of BT
Elastic compliance
Direct piezoelectric coefficient
Permittivity
(×10-12 m2/N)
(×10-12 m/V)
(×10-10 C/Vm)
Mass
density
(kg/m3)
s11
s33
s44
s12
s13
d31
d33
d15
T
є 11
T
є 33

8.85a
8.95a
22.8
-2.7
-2.9
-60a
140a
260
128
102a
5400a
a
NEC/Tokin’s product data sheets
Table 2. Material properties of C-203
Elastic compliance
Direct piezoelectric coefficient
Permittivity
(×10-12 m2/N)
(×10-12 m/V)
(×10-10 C/Vm)
Mass
density
(kg/m3)
s11
s33
s44
s12
s13
d31
d33
d15
T
є 11
T
є 33

13.9
16.7
42.7
-4.1
-6.4
-144
325
522
131
129
7700
50
Fumio Narita et al.
y
z
Grounding electrode
Driving electrode
Sensing electrode
x
O
Barium Titanate layer
Copper shim
l = 50
ls
4
1
11
(b)
0.5
1
1
ws
w = 25
(a)
unit: mm
Fig.1 Schematic drawing of (a) BT unimorph cantilever and (b) dimensions
Piezoelectric vibration energy harvesting characteristics
51
Fig.2 BT unimorph sample
3
FEA Test
Vout (V)
BT
PZT C-203
2
uz0 = 0.1 mm
ls / l = 0.8
ws / w = 0.16
1
0
R 
50
f (Hz)
100
Fig.3 Output peak voltage versus frequency for unimorph cantilevers
52
Fumio Narita et al.
0.24
uz0 = 0.1 mm
f = 50 Hz
Vout (V)
ls / l = 0.8
R 
0.23
FEA
BT
0.22
0.2
0.4
ws / w
0.6
Fig.4 Output peak voltage versus ratio of sensing electrode width to cantilever
width
0.8
uz0 = 0.1 mm
f = 50 Hz
Vout (V)
0.6
ws / w = 0.16
R 
0.4
0.2
FEA
BT
0.2
0.4
ls / l
0.6
0.8
Fig.5 Output peak voltage versus ratio of sensing electrode length to cantilever
length
Piezoelectric vibration energy harvesting characteristics
53
Fig.6 Output peak voltage for unimorph BT cantilevers with three types of
sensing electrode geometry
uz0 = 0.1 mm
f = 50 Hz
xx (MPa)
3
ls / l = 0.8
ws / w = 0.16
2
R 
1 FEA
BT
y = 0 mm, z = 0 mm
0
10
20
30
40
50
x (mm)
Fig.7 Normal stress distribution along the length direction for the BT unimorph
cantilever
54
Fumio Narita et al.
Fig.8 Output power versus load resistance
Received: March 24, 2016; Published: September 19, 2016