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Poling-architected graded piezoelectric energy harvesters: On exploiting
inevitable rotational speed variability
Soumyadeep Mondal, Tanmoy Mukhopadhyay†, Susmita Naskar‡
University of Southampton, Southampton, SO16 1BJ, United Kingdom
Abstract
Different components of rotating machines inevitably experience variations in angular speed due to
intermittent activation of the driving power while controlling the speed at a target level. The effect of
such fluctuation in high-speed rotating structural components is proposed to be exploited here for the
dual purpose of energy harvesting and speed sensing through mounting piezoelectric elements. The
voltage output is enhanced through the coupled effect of poling angle tuning and introducing
functionally graded materials based on power and sigmoid laws. An efficient, yet insightful theoretical
framework is developed on the basis of active Euler-Bernoulli beam theory and conservation of energy
principles, which are further validated using separate finite element results and qualitative
experimental characterization. We develop an effective computational mapping among the output
open-circuit voltage, rotational speed, functional gradation and poling orientation, wherein the output
voltage, voltage sensitivity, and charge sensitivity are used as critical metrics to quantify the
performance of the energy harvesters. The concept of exploiting angular speed variation along with
poling-architected piezoelectric functionally graded configurations will be crucial for designing self-
powered optimized electronic sensors and devices for a range of aerospace and mechanical
applications such as helicopter blades, aircraft engines, wind turbines and other rotating machines.
Keywords: Piezoelectric energy harvesters; Poling angle; Functionally graded material; Angular
speed variation; Self-powered sensors
1. Introduction
Piezoelectric energy harvesting through various means has garnered significant research
interest since the inception of this smart material concept in the field of green energy. It has direct
applications in the area of self-powered low-energy sensor applications for sustainable, continuous
and battery-less power supply [1–4]. The central theme of this article focuses on the concept of
exploiting the inevitable angular speed variation in rotation machines along with poling-architected
piezoelectric functionally graded configurations to enhance the energy harvesting capability.
Doctoral student, School of Engineering, Burgess Road, Southampton, UK
†Faculty, School of Engineering, Burgess Road, Southampton, UK, Email: S.Naskar@soton.ac.uk
‡Faculty, School of Engineering, Burgess Road, Southampton, UK
The initial concept of direct piezoelectricity was put forward experimentally in by Pierre and
Jacques Curie, where they catalogued a few non-centrosymmetric crystals, displaying an accumulation
of electric charge on their surface under applied mechanical stress [5]. Later Lippmann [6] highlighted
mathematically the converse effect of piezoelectricity (i.e., generation of mechanical stress/force under
a given electrical field in such crystals) with the help of fundamental laws of thermodynamics. Since
then, the use of the fundamental equation of piezoelectricity has evolved through different stages in
different areas, and energy harvesting (EH) /power conversion (PC) is one such prominent area.
Because of the coupled electrical and mechanical properties, piezoelectric materials have been used
extensively as actuators and sensors for controlling and monitoring applications [7–9].
Generally, piezoelectric energy harvesters [2] involve multilayer stacks, attached by adhesives.
The homogeneous piezoelectric materials can be classified into two broad categories which are
piezoelectric polymers (such as Polyvinylidene fluoride (PVDF)) [10] and piezoelectric ceramics
(such as Lead zirconate titanate (PZT)) [11–13]. In the present study, lead and lead-free piezoelectric
ceramics have been chosen over polymer due to the high values of electromechanical coupling
coefficients, capability to be rendered into various shapes and other advantages [14,15]. However, they
provide a few disadvantages like brittleness, poor conductivities, low tensile strength. To mitigate such
issues in energy harvesters of pure piezoceramics, a third category is introduced with layered
composite of metal and piezoceramics [13,16]. Such piezoelectric material configurations have also
shown other useful properties like corrosion resistance. However, the problem with layered composites
is that they are prone to delaminate under excessive inter-laminar stresses [17]. Later, to address these
issues, functionally graded piezoelectric materials (FGPMs), which are composed of two or more
constituent phases (often metal and piezoceramic) with continuously and smoothly varying
composition from one surface to another, are employed instead of typical laminated composites. The
initial concept of functionally graded materials (FGM) was put forward by Japanese material scientists
in 1984 [18]. Later extensive numerical and experimental studies have been reported on FGM
structures, embedded with piezoelectrical materials within them [19]. Based on non-uniform property
variations in FGM system, many types of gradations are reported in the literature such as power-law
functions (P-FGM), exponential functions (E-FGM), or sigmoid functions (S-FGM) [20]. Zhu and
Meng [21] reported a comparison between analytical and experimental values of bending deflection
of a PNN-PZT piezoelectric ceramics-based FGPM actuator, manufactured using powder mould
stacking press method. After that, a large number of experimental fabrications of FGPM structures
have been reported in the existing literature [22–27]. Naskar et al. [28] explored single-term extended
Kantorovich method (EKM) and Ritz method based novel semi-analytical frameworks for graphene-
reinforced piezoelectric functionally graded nanocomposites including the effect of residual surface
stresses, elastic and piezoelectric surface modulus, and direct flexoelectric effects. Reddy and Cheng
3
Fig. 1: (A) Schematics of the rotating beam based multi-level energy harvesting system with primary
support and secondary piezoelectric beams, rotating at a base angular speed,
. (B)(I) Mechanical
strain distribution in beam section. (II) Electric field distribution for series connection. (C) Detailed
view of one secondary beam. The leftmost layer is referred to as the ‘top’ layer, while the rightmost
layer is referred to as the ‘bottom’ layer in this paper. (D)(I) Conventional poling orientations of
secondary beam (left one) and transformed poling orientations (
1
,
2
) of the secondary beam (right
one). (II) Ideal case of piezoelectric layer with applied electric field and poling direction.
[29] have reported an analytical analysis on the bending of 3D FGM rectangular plate, attached to PZT
actuator employing the transfer matrix and asymptotic expansion methods. Yang and Xiang [30]
present a theoretical investigation based on the Timoshenko beam theory and differential quadrature
(DQ) method on monomorph and multimorph FGPM beams under thermos-electro-mechanical
loadings. Lezgy-Nazargah et al. [31] analyzed the static and transient behavior of FGPM beams with
the help of efficient three-nodded beam finite elements. Ajitsaria et al. [32] reported three
mathematical models (based on an equivalent electrical circuit, traditional beam theories and the
conservation of energy) to predict the amount of electrical energy generated in PZT bimorph sensor,
under mechanical base excitation. Maruani et al. [33] carried out a numerical study on the sensing,
actuation efficiency and active vibration control of a single-layered FGPM PZT4-Aluminium beam,
incorporating ROM and Maxwell-Garnett rule for material homogenization.
The majority of the literature concerning various EH systems incorporating piezoelectric
materials has considered beam sensor/actuator of cantilever types with base/support excitation and
wind-generated vibrations [34]. Also, they are mostly based on the (transverse) bending mode. The
sensing and energy harvesting performances of EH systems are crucial factors that need to be
monitored and optimized. Wang et al. [35] presented a comparative analytical study on the clamping
effect of elastic shim layer on dielectricity, voltage and charge sensitivity of three non-FGPM
piezoelectric benders (unimorph, bimorph and triple layer bender) under external mechanical
influences. Ng and Liao [36] discussed the sensing efficiencies of non-FGPM bending sensors and a
triple-layer type beam-mass based on experimental and analytical investigations. Mo et al. [37]
developed a theoretical model of non-FGPM unimorph benders with an interdigitated electrode pattern
and studied the sensing performances under quasi-static forces. Kebaili et al. [38] presented an
experimental study on PZT-5A sample where they poled the piezoelectric sample at different angles
by partially coating the opposite faces of the sample with silver paste under an applied electric field of
1 kV/mm. Kiran et al. [39] performed a finite element simulation study to analyze the performance of
piezoelectric elements with linearly graded poling angles along the thickness. Singh et al. [40]
discussed the active vibration control of poling-tuned piezoelectric laminated cantilever-type beams
employing a lumped parameter approach. They highlighted the notion of on-demand property
modulation by analyzing the electromechanical responses of multilayered piezoelectric composites in
terms of piezoelectric pole tuning. In summary, the literature review presented here highlights that
most studies explore the transverse bending mode for energy harvesting and sensing applications,
while there exists a tremendous potential for exploiting the shear and axial modes as well through
poling-architected configurations.
In traditional piezoelectric EH systems, the output energy responses are a direct function of
input external electro-mechanical influences. Different components of rotating machines inevitably
experience variations in angular speed due to intermittent activation of the driving power while
controlling the speed at a target level. The effect of such fluctuation in high-speed rotating structural
components is proposed to be exploited here in energy harvesting through mounting piezoelectric
elements. The voltage output would further be enhanced through the coupled effect of tuning the poling
angle and introducing functionally graded structures based on power-law and sigmoid rules. Apart
from the commonly used transverse bending mode (
31
d
), there are two conventionally unused modes
of operation in piezoelectric materials, involving shear (
15
d
) and longitudinal (
33
d
) modes [40]. A
detailed comparative study between the aforementioned three modes has been reported in the literature
[41]. In any traditional piezoelectric material/system, the poling is done along the thickness direction
of the sample (generally, denoted as direction 3). By the present poling tuning method, this poling
orientation can be reoriented at a certain angle, other than with the longitudinal axis of the material.
5
Subsequently, performance of the piezoelectric EH system would depend on the absolute values of
piezoelectric coupling coefficients (
31 15 33
,,d d d
), wherein the value of
15
d
is generally much higher
than the other two coefficients. In
31
d
mode-type applications, with the help of poling tuning technique,
the performance of piezoelectric sample can be increased significantly by incorporating not only the
influence of
31
d
but also
15
d
and
33
d
. This will lead to an enhanced energy harvesting performance
while exploiting the fluctuation in rotational or angular speed, as proposed in this paper.
The underlying concept behind angular speed variation is that there will always be inevitable
variation in the base rotational speed when a rotating system is operated at a certain high angular speed
[42,43]. We propose to exploit such fluctuations in piezoelectric energy harvesting through
experimental and analytical investigations (refer to Figure 1). This novel approach of energy
harvesting from a hitherto untapped potential will be crucial for designing self-powered low-energy
electronic sensors and devices for a range of aerospace and mechanical applications. Further, it is
possible to exploit the proposed system as a speed sensor by relating the output voltage and rotational
speed. In the following sections, we derive the mathematical formulation and present numerical results
concerning the development of computational mapping of the rotational speed with output voltage and
the structural parameters including the gradation scheme, poling angle, material and geometric
properties.
2. Multi-level computational framework
The multi-level energy harvesting system as described in Figures 1 and 2 consists of primary
and secondary beams. In the following subsections, we describe the mathematical models for the
piezoelectric secondary beams (along with poling architecture and functionally graded material
configurations) under centrifugal forces that act as the primary source of energy generation.
2.1. Modelling of piezo-active beams under rotation
Each active beams consist of three layers as shown in Figure 1: two oppositely poled
piezoelectric layers and a sandwiched substrate elastic (slim) layer to lessen fragility. The material
property of the piezoelectric layer is assumed to be varying along its thickness direction i.e., the layers’
properties are functionally graded in nature. Perfectly conductive films (of negligible thickness)
affixed on the top and bottom surfaces of the secondary beams are employed as electrodes.
The secondary beams are modelled here in the form of bending mode sensors with cantilever
configurations due to the higher sensitivities than other modes of sensor [44]. The centrifugal forces
due to the angular rotation of the system will act on each secondary beam. Centrifugal force comes
under the category of body force, wherein the present force is assumed to be a point load (quasi-static
after being saturated at
base
), acting at the centroid of each beam (refer to Figure 1(C)). The total
centrifugal force
F
acting over the body of the secondary beam can be expressed as follows:
( )
2
2
2
sec sec
() 2
L
F m r
=+
(1a)
where
sec
m
represents the mass of the respective secondary beam element,
represents base angular
speed and
sec
r
represents the radial distance of the neutral axis of the beam from the center of rotation.
It is important to note that here length (and mass) of the secondary piezoelectric hybrid beam is
considerably smaller than that of the primary full anchor (blade), i.e.,
sec 2rL
( and )
(refer to Figure 10). Based on this observation, the expression for the centrifugal force
F
can be
simplified as follows:
2
sec sec
()F m r
=
(1b)
Before going to direct formulation, the following assumptions are taken into consideration to capture
the stress states and their effect on electrical responses:
▪ Each secondary beam acts as a thin Euler–Bernoulli beam (length-to-thickness ratio > 10) to
neglect the shear deformation.
▪ For plane stress condition, no stresses, except for
x
, are assumed to exist, that is,
0
y z xy yz zx
= = = = =
and also, the electric field strength
0
xy
EE==
. This can be
presumed only when the domain is allowed to expand freely along
z
- axis and the beam is
long and slender. The out-of-plane thickness is considered to be much thinner than its
longitudinal dimension.
▪ Perfect bonding between the layers is assumed so that the extensional strain,
x
in the present
composite beam is continuous in the thickness direction and linearly varying (refer to Figure
1B(I)).
▪ The radius of curvature of the deformed beam is much higher than beam thicknesses.
Due to applied rotation, the generated flexural stress and strain within the beam (and its layers) can be
summarized by the following constitutive relations (refer to Figure 1).
Top piezoelectric layer:
11 31
( ) ( )
EU
x x z
s z d z E
=−
31 33
() Є ( )
UU
z x z
D d z z E
= − +
(2a)
Middle Elastic Layer:
11
m mid
xx
s
=
(2b)
Bottom piezoelectric Layer:
11 31
( ) ( )
EL
x x z
s z d z E
=+
(2c)
7
31 33
() Є ( )
LL
z x z
D d z z E
=+
Here the parameters for the top (i.e. upper) piezoelectric layer are indicated with the suffix 'U,' while
those for the bottom (i.e. lower) layer are marked with the suffix 'L'.
11
E
s
is the elastic compliance of
the piezoelectric layer which is inverse to its Young Modulus. The superscripts E and σ on the
piezoelectric material parameters represent measurements taken at constant electric field and constant
stress [45], respectively. The piezoelectric coefficients
31
d
and
33
Є
are the transverse piezoelectric
coupling coefficient and permittivity of the piezoelectric material at constant stress [45], respectively.
Note that the directions {1, 2, 3} typically refer to the coordinate system of piezoelectric materials
(where '3' denotes the direction of the electric field and '1' denotes the direction of the induced
mechanical strain) according to the standard nomenclature for piezoelectric coefficients. In contrast,
the directions {x, y, z} are used to describe the coordinate system associated with the geometry of the
secondary beam (see Figure 1(C)). For piezoelectric materials with negative
31
d
(lead-zirconate
titanates, barium titanate and zinc oxide), the piezoelectric domain will tend to contract along its two
in-plane directions (e.g. 1- and 2-directions), perpendicular to the out-of-plane polarization direction
(3-direction) if the applied electric field is parallel to this polarization direction (whereas it will expand
if electric field is opposite to its poling direction) [46]. The coefficient
31
d
has been considered to have
two opposite signs in the top and bottom layers in Equation (2) and it is due to the opposite poling
directions (explained in section 2.3) and the values of electric fields in each piezoelectric layer should
be the same [14].
z
E
is a dummy quasi-static electrical field, applied across the thickness direction of
the piezoelectric beam. In the present study (shown in Figure 1(B)(II)) based on the assumptions of
,2
mp
w t t L+
, the uniform electric field,
2
zp
E V t=
in each piezoelectric layer is considered,
where
V
is the voltage across the beam thickness. The total applied voltage
V
is equally divided into
two piezoelectric sections due to the series connection of the terminals.
The strain
x
can be expressed in terms of the curvature (
) using the elementary definition
of normal stress-strain and geometry of the deformation (assuming a downward curvature of the
beam):
2
2
x
z d z
zz
dx
= =
(3)
Here
z
is the distance of the respective fiber from the neutral axis of the beam and
is the radius of
curvature. The curvature of the neutral plane,
, can be calculated from the bending moment equation
of the composite beam as explained below.
Fig. 2: Detailed flowchart of the present study showing the development of analytical model for the
multi-level energy harvesting system and systematic investigation of the effect of fluctuation of
rotational speed, poling architecture and functional gradation.
2 2 2
2 2 2
() m m m
p
m m m
p
t t t
t
L mid U
x x x
t t t
t
M x wzdz wzdz wzdz
−+
− + −
= + +
(4)
where,
m
t
and
p
t
are the thicknesses of the middle and piezoelectric layer, respectively whereas,
w
is
the width of the secondary beam. Substituting Equations (2) and (3) into Equation (4), the following
expression of
can be obtained.
2
1
() z
M x wM E
wM
−
=
(5)
Here the constants
1
M
and
2
M
can be given as:
9
2 2 2
2 2 2
1
11 11 11
2 2 2
( ) ( ) ( )
m m m
p
m m m
p
t t t
t
t t t
E m E
t
z z z
M dz dz dz
s z s z s z
−+
− + −
= + +
(6a)
22
31 31
2
11 11
22
( ) ( )
( ) ( )
mm
p
mm
p
tt
t
tt
EE
t
d z d z
M zdz zdz
s z s z
+−
−+
=−
(6b)
For present load
()F
, the maximum moment will be in the middle of the beam (Equation 7). The
bending moment
()Mx
can be obtained from the elementary bending moment diagram as follows:
( ) ( ) 2
L
M x F x
=−
02
L
x
(7a)
( ) 0Mx=
2
LxL
(7b)
where
L
is the total length of the beam. Due to the influence of mechanical (here load
F
) and
electrical (here field
z
E
) loads, the stored internal energy density within an infinitesimally small
volume element of beam can be given as:
Top piezoelectric layer:
3
11
22
U U U
x x z
du D E
=+
(8a)
Middle Elastic Layer:
1
2
m mid
xx
du
=
(8b)
Bottom piezoelectric Layer:
3
11
22
L L L
x x z
du D E
=+
(8c)
Now the total energy stored in one whole secondary beam can be obtained by
2 2 2
0 0 0
2 2 2
m m m
p
m m m
p
t t t
t
L L L
L m U
Total t t t
t
U du wdxdz du wdxdz du wdxdz
−+
− + −
= + +
(9)
Substituting Equations (2), (3), (5), (7) and (8) into Equation (9), the following expression for the total
internal energy (
Total
U
) of the system is obtained as follows:
22
( ) ( ) ( )
Total U m L U m L z U m L z
U A A A F B B B FE C C C E= + + + + + + + +
(10)
where
( )
3 2 2 2
2 2 2
2
1 11 11 11
2 2 2
, , , ,
48 ( ) ( )
m m m
p
m m m
p
t t t
t
U m L t t t
E m E
t
L z z z
A A A dz dz dz
M w s z s s z
+−
− − +
=
(11a)
( )
22 2 2
2 2 2
2
2
1 11 11 11
2 2 2
, , , ,
8 ( ) ( )
m m m
p
m m m
p
t t t
t
U m L t t t
E m E
t
LM z z z
B B B dz dz dz
M s z s s z
+−
− − +
=
(11b)
2 2 2 2 2
22 1 11 33 1 31
2
1 11
2
( ( )Є ( ) ( ) )
2 ( )
m
p
m
tE
t
UtE
Lw M z M s z z M d z
C dz
M s z
+
+−
=
(11c)
22
22
2
1 11
22
m
m
t
mtm
LwM z
C dz
Ms
−
=
(11d)
2 2 2 2 2
22 1 11 33 1 31
2
1 11
2
( ( )Є ( ) ( ) )
2 ( )
m
m
p
tE
LtE
t
Lw M z M s z z M d z
C dz
M s z
−
−+
+−
=
(11e)
From the concept of piezoelectric capacitance, the electric charge generated due to the combined effect
of load (
F
) and electric field (
2
zp
E V t=
) can be obtained by the partial derivative of the total internal
energy with respect to the voltage:
2
( ) ( )
22
Total U m L U m L
pp
U F B B B V C C C
QV t t
+ + + +
= = +
(12)
The dielectric open-circuit free capacitance of the piezoelectric layer can be derived as:
2
()
2
U m L
p
p
C C C
Q
CVt
++
==
(13)
As in the present study, the secondary beams are acting as the sensor, the applied voltage
V
will be
zero. Moreover, for an open-circuit condition
z
E
is equal to zero since there is no applied external
electric field which eventually gives the following Equation of
Q
:
( )( )
2
U m L
p
F B B B
Qt
++
=
(14)
The open voltage or Thevenin voltage (sensor voltage)
open
V
across the piezoelectric layer (appearing
on the electrode) due to the applied centrifugal force i.e., rotation of the primary beam can now be
calculated as follows:
2
sec sec
() ()
( ) ( )
U m L p U m L
open p
p U m L U m L
F B B B t B B B
Q
V m r t
C C C C C C C
++ ++
= = =
+ + + +
(15)
Here the mass of the system
sec
m
can be given as:
sec ( ) ( )
middle m top bottom
top bottom
m Lt w z Lwdz z Lwdz
= + +
11
Fig. 3: Variation of elastic properties in different FGPM beams (A) Non-FGPM (B) P-FGPM (C) S-
FGPM.
where,
(.) ()z
is the mass densities of respective layers. As each piezoelectric layer is supposed to act
like a capacitor, the electric energy generated and stored in the piezoelectric layers can be determined
as follows:
( )
2
2
2
sec sec
()
11
2 4 ( )
U m L
cp open
U m L
B B B
E QV m r C C C
++
== ++
(16)
From Equation (15), the present FGM beam can be used as a (angular) speed sensor. As here we are
considering the effect of angular speed change only on the centrifugal force acting on the beam, the
charge and voltage sensitivity w.r.t angular speed can be found by taking the derivative of
V
and
Q
with respect to
. The voltage and charge sensitivities of the secondary piezoelectric beams due to
rotation can be expressed as follows:
sec sec
()
2()
open
VU m L
p
U m L
VB B B
S m r t C C C
++
==
+ +
(17a)
sec sec
()
22
QU m L
p
B B B
Q
S m r t
++
==
(17b)
Note that the units of charge and voltage sensitivity are: C.s/rad (Coulomb. second/radian) and V.s/rad
(Voltage. second/radian).
2.2. Modelling of functionally graded piezoelectric material
In the present study, the middle substrate layer is assumed to be isotropic, homogeneous, and
non-piezoelectric in nature. The material properties of functionally graded piezoelectric layer are
varied along the thickness direction (z). The variation in volume fractions can be used to define such
functionally graded material (FGM). To characterize volume fractions, most researchers employ three
fundamental mathematical laws which are the power-law function, exponential function, or sigmoid
function [20]. It results in three different FGM systems i.e., P-FGM, E-FGM, and S-FGM and among
these, P-FGM and S-FGM are considered in the current formulation. E-FGM cannot take care of all
the properties of the systems, especially of piezoelectric coupling coefficient and permittivity as the
values of these two are zero in the middle shim layer. Later rule of mixture (ROM) is used to determine
the effective properties of the medium.
▪ P-FGPM layer:
1
2
( ) ( ) 2
k
pm
p m m
p
tt
z
P z P P P
t
+
−
= − + +
when
( )
/ 2 ( / 2)
m p m
t z t t +
(18a)
▪ S-FGPM layer:
12
( ) ( ) 1 2
2
k
m
p
p m m
p
t
tz
P z P P P
t
+−
= − − +
when
( )
( ) / 2 ( / 2)
p m p m
t t z t t+ +
(18c)
13
12
( ) ( ) 2
2
k
m
p m m
p
tz
P z P P P
t
−+
= − +
when
( )
/ 2 ( ) / 2)
m p m
t z t t +
Here
p
P
and
m
P
are the material properties (elastic properties, piezoelectric properties) of piezoelectric
and middle layer respectively.
k
is the non-negative material index parameter. In Figure 3, the material
property distributions of P-FGPM, S-FGPM and Non-FGPM systems are depicted.
2.3. Poling tuning of functionally graded piezoelectric layer
As depicted in Figure 1(D)(I) (left), in both the top and bottom FGPM layer, the inherent
material poling directions are conventionally taken perpendicular to the beam’s longitudinal axis (i.e.
along the z-direction) which results in the activation of
31
d
mode under the electric field
z
E
. In the
present subsection, a poling tuning technique for FGPM system is introduced to tune the performance
of secondary beams by modifying the values of piezoelectric coefficients i.e.,
31()dz
and
33
Є ( )z
. The
piezoelectric layers are poled at an angle
1
and
2
with the z-axis (Figure 1(D)(I) (right)), and
subsequently the constitutive relations i.e., Equation (2) will be transformed to the following form:
Top piezoelectric layer:
11 31 1
( ) ( , )
E U eff
x x z
s z d z E
=−
31 1 33 1
( , ) Є ( , )
U eff U eff
z x z
D d z z E
= − +
(19a)
Middle Elastic Layer:
11
m mid
xx
s
=
(19b)
Bottom piezoelectric Layer:
11 31 2
( ) ( , )
E L eff
x x z
s z d z E
=+
31 2 33 2
( , ) Є ( , )
L eff L eff
z x z
D d z z E
=+
(19c)
where,
31
eff
d
and
33
Єeff
are the effective piezoelectric coefficients for the present
31
d
mode.
To derive the closed-form expressions, we have considered an ideal case (shown in Figure
1(D)(II)) where the applied electric field and conventional poling direction are along
z
-axis, and the
poling angle is
(measured from
z
-axis). A full
d
matrix of the piezoelectric system (of tetragonal
Symmetry) needs to be considered here. In the new inclined material coordinate system, the sensing
law of a typical piezoelectric system can be written as follows:
. . ( ) Є ( )
new old T old old
zz
D T D T d z z E
= = +
(20)
where stress and electric field can be written in terms of the new coordinate system as follows:
1.
old new
T
−
=
(21a)
1.
old new
E T E
−
=
(21b)
Here
T
and
T
are the respective transformation matrices [47,48]. Substituting Equation (21) into
(20), the following final form of sensing law in the new coordinate system is obtained.
11
( ) . Є ( ) . ( ) . Є ( ) .
eff eff
new T new new T new new
z
D T d z T T z T E d z z E
−−
= + = +
(22)
Now from the above transformed piezoelectric coefficients matrix and dielectric permittivity matrix,
31 ()
eff
dz
and
33
Є ( )
eff z
can be obtained as follows:
( )
2 2 2
31 31 33 15
( , ) cos ( )cos ( )sin ( )sin
eff
d z d z d z d z
= + −
(23a)
( )
22
33 33 11
Є ( , ) Є ( )cos Є ( )sin
eff z z z
=+
(23b)
From Equation (23), it is evident that
31 ()
eff
dz
becomes dependent on
31 33 15
,,d d d
and the poling angle,
.
3. Results and discussion
In this section, the performance of a triple-layer bender comprised of FGPM layers in open-
circuit conditions and its dependency on different material and geometric parameters are investigated.
The harvested energy is subsequently optimized by varying material properties and tuning poling
directions. Lastly, the effect of angular speed variation on overall output responses is discussed. Thus,
the results are presented for a single FGPM beam under static deformation first. Subsequently, it is
extended to the dynamic analysis of the multi-level energy harvesting system (consisting of primary
and secondary beams as shown in Figure 1(A), functional gradation, and poling architectures). As an
integral part of this section, we also present a qualitative experimental proof of concept study to show
that fluctuation in rotational speed can obtain energy output, before progressing to the dynamic
analysis of the secondary beams and the multi-level energy harvesting system.
3.1. Energy output of a single FGPM beam
Finite element simulation (FEA) has been conducted using a commercially available finite
element package (COMSOL Multiphysics 5.6) to validate present analytical formulations considering
60L=
mm,
3w=
mm,
0.3
m
t=
mm and
0.2
p
t=
mm [49]. Material properties of the layers are
summarized in Table 1. Here one piezoceramic material (lead-zirconate titanates/PZT) with negative
31
d
have been taken into FEM modelling whose elasticity matrix can be given as follows:
15
10
12.72 8.02 8.47 0 0 0
8.02 12.72 8.47 0 0 0
8.47 8.47 11.74 0 0 0 10
0 0 0 2.29 0 0
0 0 0 0 2.29 0
0 0 0 0 0 2.35
PZT
E
s
=
Pa
For the sake of validation, a static load (0.2 N) is considered acting on the system in place of the current
centrifugal load,
()F
and it is applied along the centroidal axis of the beam. The supports (anchor)
to each secondary beam, given by the primary beam are taken into consideration by fixing the
secondary beam at the left clamped end. Variation of open-circuit voltage with respect to load and
other geometric parameters has been plotted and compared with the existing literature and FEM
solutions, as discussed in the following paragraphs.
Table 1 Elastic and material properties of the hybrid beam system
Geometrical properties
Piezoelectric constants (pC/N)
L
31
PZT
d
w
33
PZT
d
p
t
15
PZT
d
m
t
Dielectric constant (relative permittivity)
(
12
0
Є 8.854 10−
=
F/m)
Elastic properties of non-piezoelectric layer
11 0
Є / Є
PZT
3128.5
Al
1920 kg/m3
22 0
Є / Є
PZT
3128.5
Al
E
70 GPa
33 0
Є / Є
PZT
3398.5
Before extracting the necessary output responses, it is always recommended to cross-check the
dependency of finite elements i.e., meshes on the results. The same has been checked through a mesh
convergence study of the current FEM model, as presented in Figure S1 of the supplementary material.
It is observed that the magnitude of voltage generation starts saturating at around 16.68 V. To ensure
enough solution accuracy and minimize computational intensiveness (solution time), the entire finite
element model is discretized by quadratic serendipity tetrahedra (‘tets’) elements with finer mesh
element size (average element size of 1.8 mm). Figure S1 depicts (inset) the present discretized model
with a total of 64709 elements and 401781 degrees of freedom (DOF), giving a converged and
optimized output voltage response. The output responses (open-circuit voltage) are plotted in Figure
4(A-C) as the function of the width and thicknesses of the hybrid beam, and it is indicated that initially
the magnitudes of voltage decrease with the increasing width and thicknesses of both shim and
piezoelectric layers and achieve a saturation point at a lower value of voltage. The open-circuit voltage
increases continuously with the applied mechanical load and is zero at zero load i.e., in the absence of
any rotation (shown in Figure 4(D)). In general, the results obtained from finite element analysis and
the current analytical model agree well, ascertaining the validity of the proposed framework.
Fig. 4: Comparison of analytical and finite element method (FEM) simulation results of output voltage
response (A) Effect of shim layer thickness (B) Effect of piezoelectric layer thickness (C) Effect of
beam width (D) Effect of applied mechanical load.
17
In this section, the dependency of material parameters on the output voltage responses of the
FGPM system is investigated. The parameters include the gradient index in the analytical expressions,
thickness and Young’s modulus of the shim layer. The material properties of the piezoelectric layer
and overall dimensions are kept constant as mentioned before. Figure 5 shows the variation of output
open circuit voltage with the applied angular speed of the system. In P-FGPM system, the present
model shows two different trends in the output responses based on values of the gradient index (
k
).
As
k
is a non-negative index, it can vary from zero to any positive number. In Figure 5(A), we observe
the generation of negative potential for the case of an even index. Whereas in Figure 5(B), a positive
potential is obtained from the electrode attached to the upper surface of the beam when
k
is odd. For
the even-numbered index (Figure 5(A)), the absolute value of potential decreases with the increase of
k
and the same increases with the increase of angular speed. Here gradient index of zero is equivalent
to a non-FGM system. In Figure 5(B), the opposite trend has been observed i.e., voltage increases with
the increase in
k
as well as angular speed. It is also observed that P-FGPM beams with an odd-
numbered gradient index provide a larger amount of open-circuit voltage in comparison to that of an
even-numbered index. In Figure 5(C), a similar investigation is carried out for S-FGPM beams. We
can observe the same trend of voltage with the angular speeds in even-numbered index. However, the
absolute values of voltage increase for a higher gradient index in this case, unlike P-FGPM. Further,
S-FGPM beams provide a larger amount of electric potential than P-FGPM.
In Figure 5, the dependency of beam response on angular speed is also validated with FEM
results, wherein an excellent agreement between the analytical and the FE solution for the open-circuit
voltage is observed. The extensive validation of the current analytical framework with separate FE
results, as presented in Figures 4 and 5, establishes the confidence for further analyses on energy
harvesting.
In order to study the performance and dependencies of the present FGPM beams quantitively,
we have investigated the charge and voltage sensitivities. In Figure 6, sensitivities (V.s/rad or C.s/rad)
are considered at an angular speed of 500 RPM. Piezoelectric properties are kept constant, whereas
the thickness and elastic properties of the middle layer are varied over a reasonable range. An inverse
proportionate relation between voltage sensitivity and the elastic modulus and thickness has been
observed here (refer to Figure 6(A – B)). The overall trend of the sensitivity is exactly in coherence
with the existing literature on non-FGM sensors [36]. For the case when
k
is odd, with the increase of
index
k
, voltage sensitivity and charge sensitivity of P-FGPM increase and decrease, respectively. In
a similar manner, for the case when
k
is even, voltage sensitivity and charge sensitivity both decrease
with the index
k
. From Figures 6(B) and 6(C), it is observed that some overlapping regions are present
where the aforementioned observations are not being followed due to a complex numerical confluence.
For example, in Figure 6(C), for a certain region of
m
t
and
m
E
, charge sensitivity is negative whereas
voltage sensitivity is still positive. For S-FGPM beams (refer to Figure 6(E-F)), both the sensitivity
(absolute values) increase as
m
t
and
m
E
reduce. The general trend of sensitivity with
k
can be
summarized as follows: in the large elastic modulus zone, sensitivity (absolute values) has an inverse
relationship with
k
, while in the small young modulus region, it has a direct proportionate relationship
with
k
.
Fig. 5: Variation of open-circuit voltage with angular speed for different values of
k
(A) P-FGPM for
the cases when
k
is even (B) P-FGPM for the cases when
k
is odd. (C) S-FGPM.
19
Fig. 6: Variation of sensitivity with shim layer thickness (
m
t
)and Young modulus (GPa) (
m
E
) at 500
RPM (A) P-FGPM for the cases of voltage sensitivity (
k
is odd) (B) P-FGPM for the cases of voltage
sensitivity (
k
is even) (C) P-FGPM for the cases of charge sensitivity (
k
is odd) (D) P-FGPM for the
cases of charge sensitivity (
k
is even) (E) S-FGPM for the cases of voltage sensitivity (F) S-FGPM
for the cases of charge sensitivity.
Table 2 Material Properties of PZT5H and KMLNTS
Materials
Density
()
11
E
s
()
31
d
()
33
d
()
15
d
()
11 0
Є / Є
33 0
Є / Є
PZT5H
7500
16.5
-274
593
741
3130
3400
KMLNTS
4600
8.88
-75.5
121.7
256.8
501.7
878.8
In order to explore the influence of poling tuning on the energy harvesting performances of the
present triple-layer FGM beam model, poling angles in both piezoelectric layers are varied over a
reasonable range. In Figure 7(A-B), the effective transverse piezoelectric coupling coefficient (
31
d
)
and relative permittivity (
33 0
Є / Є
) are varied over a certain range of poling angles, wherein the
contribution of other coefficients in the effective property is depicted. Initially, PZT5H material is
taken for validation purposes and later materials are varied as per the efficacy of output responses. For
example, the material properties of two negative
31
d
materials (PZT5H and KMLNTS) are given in
Table 2 which are used in the later sections. Note that KMLNTS (
0.475 0.475 0.05 0.92 0.05 0.03
()K Na Li Nb Ta Sb
) is lead-free piezoelectric ceramics [50], which might have specific sensitive applications under
stricter environmental, health, and safety regulations aimed at limiting the use of lead worldwide.
In Figure 7(C-D), the output voltage from the present energy harvesting system at 500 base
RPM is checked for different poling angles in each layer of the beam. For that, two different cases are
considered which are as follows: (A) Poling angle at the upper piezoelectric layer is kept normal (zero)
whereas that of the lower piezoelectric layer is varied from zero to 90 (shear mode) (B) Poling angle
at the lower piezoelectric layer is kept normal (zero) whereas that of the upper piezoelectric layer is
varied from zero to 90. The system is taken of PZT5H material and no FGM gradations are
incorporated i.e., non-FGM piezoelectric layers. FEM results are compared with analytical predictions,
showing a good agreement and validity of the analytical framework. The stress generation in the
composite beam due to poling tunings and respective poling directions are demonstrated in Figure 7(E-
F). It is evident that the upper layer is in tension whereas the lower piezoelectric layer is mostly in
compression due to the rotation of the system. As shown in Figure 7(C-D), for PZT5H system the
absolute values of output voltages keep decreasing as we increase the poling angles in both cases. The
same trend has also been verified in the present finite element analysis. To capture the effect of poling
tuning, the same analysis is reinvestigated taking KMLNTS in the place of PZT5H material within
each piezoelectric layer. In case of PZT5H, an absence of optimized poling angles with respect to the
voltage is noticed, whereas in KMLNTS system, at a particular poling angle, the voltage gets
maximized. Similar results for KMLNTS are shown in Figure S2 of the supplementary material. At a
poling angle of in the lower piezoelectric layer, the maximum voltage of 23.42 V is obtained when
the poling direction of the upper piezoelectric layer is kept normal.
21
Fig. 7: Effect of poling angle considering PZT5H. (A) Effect on longitudinal piezoelectric coupling
coefficient (B) Effect on relative piezoelectric permittivity (C) Variation of open circuit voltage in
non-FGM system at 500 RPM with Poling angle () when (D) Variation of open circuit
voltage in non-FGM system at 500 RPM with Poling angle () when (E) Bending stress
contour plot across thickness in PZT5H when and (F) Bending stress contour plot
across thickness in PZT5H when and .
Fig. 8: (A) Variation of open circuit voltage with poling angles in P-FGPM KMLNTS system when
k
is 1 (odd) (B) Variation of open circuit voltage with poling angles in P-FGPM KMLNTS system when
k
is 2 (even) (C) Variation of voltage sensitivity with poling angles in P-FGPM when
k
is 1 (odd) (D)
Variation of voltage sensitivity with poling angles in P-FGPM when
k
is 2 (even) (E) Variation of
23
charge sensitivity with poling angles in P-FGPM when
k
is 1 (odd) (F) Variation of charge Sensitivity
with poling angles in P-FGPM when
k
is 2 (even).
Fig. 9: (A) Variation of open circuit voltage with poling angles in S-FGPM KMLNTS system (B)
Variation of voltage sensitivity with poling angles in S-FGPM KMLNTS system (C) Variation of
charge sensitivity with poling angles in S-FGPM KMLNTS system.
The effect of simultaneous changes in poling angles (both upper and lower layers) has also
been extended for each FGPM system in terms of voltage, voltage sensitivity and charge sensitivity of
the present rotating structure. The poling angles in both layers are varied over a reasonable range i.e.,
0 to 90. Figure 8 shows the responses of PFGPM beams for even and odd gradation index (
1,2k=
).
In Figure 8(A), the output voltage in odd-numbered PFGPM system gets maximized at poling angle
of
85
in upper and lower piezoelectric layers both. We can call this particular point as optimal point
for output voltage. In Figure 8(B), the same is investigated for even numbered PFGPM where the
optimal point is obtained at the poling angle of
50
in both layers. The same nature of 3D correlation
with poling angles has been observed for voltage and charge sensitivities of the PFGPM beams (refer
to Figures 8(C – F)). Figure 9 shows the variation in voltage, voltage sensitivity and charge sensitivity
with variation in poling angles of both the upper and bottom layers simultaneously for SFGPM beams.
The optimal angles for voltage output change as we change the values of the gradient index,
k
. For
instance, an optimal point is obtained for
1k=
at poling angles and . The values
of optimised parameters that we get are as follows: 24.05 V, 0.92 V. s/rad for output voltage and
voltage sensitivity, respectively. However, the optimal point shifts to poling angles and
with respect to the charge sensitivity of the sensor (refer to Figure 9(C)). In the SFGPM beams,
the output responses are more sensitive to the change in than as a rapid change in the parameters
with is observed in Figure 9. A similar analysis of poling tuning technique for PZT5H material is
included in the supplementary material (refer to Figures S3 and S4).
Fig. 10: (A) Physical prototype of a representative rotating piezoelectric beam system and the
schematic representation of the experimental set-up. Note that we consider a special case of non-graded
secondary beams for the purpose of qualitative proof of concept. The simplified electrical circuit (to
capture the voltage variation on oscilloscope) corresponding to the rotating frame of the physical
prototype is given here as an inset. (B) Results from the prototype showing the fluctuation in rotational
speed of the primary beam (C) Results from the prototype showing the voltage output due to
fluctuation in rotational speed.
25
3.2. Energy output of the multi-level dynamic energy harvesting system
So far in this article, we have discussed the static voltage output due to rotation which is
generated due to the centrifugal force. However, the central theme of this article is to focus on the
time-variant fluctuation in rotational speed so that a continuous voltage output can be obtained in a
dynamic regime over a period of time. In this section, we will extend the investigation to understand
the effect of minor time variation (perturbation) on the angular speed of the primary support (refer to
Figure 1) due to external inevitable sources along with poling tuning and material property tailoring.
In this context, it may be noted that the concept of harvesting piezoelectric energy from the
variation of rotational speed in high-speed rotating blades (helicopters, ship propellers) was initially
introduced in our previous work [51], taking commercially available traditional piezoelectric
bimorphs. In the earlier work, different circuital parameters (number of normal piezoelectric bimorphs
attached on rotating blades, different electrical connections between them, and the load resistances)
were varied to investigate the electrical response. In the present paper, we have developed a novel and
insightful theoretical framework based on Euler Bernoulli beam theory in conjunction with the energy
method, exploiting the poling architecture and different FGM gradations at the material level. This
leads to fully customized piezoelectric beams to optimize harvested piezoelectric energy from such
rotating platforms. Thus, the major contributions of the current work lie in (1) proposing the concept
of optimal architecture involving poling angle and functionally graded materials, and (2) the
development of an insightful computational framework thereof.
Based on the underlying theme of this paper, in engineering applications concerning high
angular speed such as rotating blades, turbine blades and similar machines, the base angular speed is
not maintained precisely constant (i.e. there exists non-uniform speed over a period of time) as an
inevitable minimal arbitrary fluctuation in the form of vibration enters into that base speed
[42,43,51,52]. This minimal variation can be utilized for harvesting piezoelectric energy. As a proof
of the proposed concept, a physical prototype is developed [51], as described in Figure 10(A). The
prototype consists of rotating blades at the ends of which the piezoelectric bimorph beams are
connected. The blades are rotated using a brushless DC electric motor. The DC motor is placed at the
base of the prototype set-up. There is a slip-ring that is placed just above the rotating blade. It is used
to make electrical connections from a stationary object to a rotating assembly. Two terminals from the
slip-rings are connected to the oscilloscope. The output voltage waveforms are obtained in the
oscilloscope CRT display, while the RPM output measurement from the incremental encoder is
displayed and monitored on the display screen. The current consumption of BLDC motor is relatively
high. For this purpose, SMPS (Switched-mode power supply) is used. There is an incremental optical
encoder that is connected at the top of the slip-ring. It is used to measure RPM of the rotating shaft.
The motor speed controller is connected to an external pulse width modulation (PWM) signal
modulator that is operated by a potentiometer. The RPM of the primary beam of the rotating beam
system is varied manually by potentiometer. It is attempted to keep it varying about a base high RPM.
Due to such variation in the RPM, a generation of electric voltage is observed in the oscilloscope
screen. Figure 10(B) depicts one such instance where the RPM is varied about base 800 RPM and due
to this, a voltage is being generated in the piezoelectric energy harvester (refer to Figure 10(C)). Such
experimental observation verifies the viability of the present method of harvesting energy from a high-
speed rotating beam system. Note that the purpose of the experimental study here is not to carry out
any quantitative investigation and comparative assessment, rather we show a qualitative proof of
concept, experimentally, concerning the possibility of energy harvesting through angular speed
fluctuation.
To incorporate speed fluctuation in the present analytic model, the following expression of
angular speed
()t
is introduced to the system:
( ) sin(2 )
base low
t f t
= +
(24)
Here
base
is the base angular speed, considered in the prior sections whereas
is the amplitude of
the fluctuation with a frequency of
low
f
. In practice, such variation can be irregular (arbitrary),
depending on the sources but here for the sake of simplicity, the sinusoidal variation is considered.
More complex variational patterns can be incorporated following the current framework. A load
resistance
L
R
(10 kohm) is attached between the extreme most surfaces of the beam and the output
sensing voltage (closed circuit voltage) is monitored over time. Considering the effect of time
dependency of
along with material property variation and poling tuning in the mathematical model
(section 2), the following two expressions of output current and voltage are obtained.
()
( ) ( , )
( ) ( ) 2
U m L
gen
p
B B B
Q t dF t
i t Q t t t dt
++
= = =
(25a)
()
( , )
( ) ( ) 2
U m L
close L gen L
p
B B B dF t
V t R i t R t dt
++
==
(25b)
It may be noted that we have utilized a simplified mechanical method to derive the power. For instance,
from equation (25), it is evident that the calculated power ( ) will be simply in
direct proportionate relationship with . Here we mainly address the proof of concept (from a
structural point of view) concerning the scope of exploiting material-level architectures in varying the
output voltages (refer to Figure 11). However, a more detailed study can be performed in the future
incorporating the electrical parameters (e.g. internal capacitance of current source terms,
electromechanical frequency response functions etc.) [53].
27
Fig. 11: Output closed circuit voltage with time considering different FGM gradation schemes at 500
RPM base angular speed with
10
RPM speed fluctuation. (A) P-FGPM PZT5H system when
k
is
even (B) P-FGPM PZT5H system when
k
is odd (C) S-FGPM PZT5H system (D) Electrical circuit
consisting of RL, as used in the analysis.
In the current study, each FGPM beam is acting as an individual current source (refer to Figure
11(D)). In this section, the results are extracted at a base rotating speed of 500 RPM. To demonstrate
the concept, ±10 RPM speed perturbation (sinusoidally) about the base speed with 0.7 Hz frequency
is considered (
10w=
;
0.7
low
f=
). Note that the numerical results on energy output presented in this
section are a direct extension of the FGPM beam-level analysis carried out in section 2. Since the
single FGPM beam-level results are extensively validated with finite element results, it is imperative
to rely on the results of Figures 11 and 12, as discussed in the following paragraphs.
Fig. 12: Output power with different poling angles at 500 RPM base angular speed with RPM
speed fluctuation. (A) P-FGPM KMLNTS system (B) S-FGPM KMLNTS system.
In Figure 11, the influence of FGM material property variation is examined in terms of output
voltage where the sense of poling directions is kept normal to the longitudinal axis of the beam in both
the piezoelectric layers. Like the prior sections, the behaviour of PFGM is discussed by dividing it into
even and odd-numbered indexes. In both cases, with the increase in indexes’ values, the voltage keeps
decreasing whereas, in SFGM, it exhibits an opposite trend. From an analytical point of view, these
voltages can be increased further by increasing the value of
base
. However, after a certain threshold
high value, the aspect of structural integrity and prospective failure needs to be accounted for.
In Figure 12, the effect of poling tuning is shown by plotting the average output power (over a
time span of 5 sec) with the poling angles of piezoelectric layers (considering KMLNTS for numerical
demonstration). It is observed that the maximum power (optimal) of 0.891 pW and 0.131 pW are
obtained from PFGPM and SFGPM systems respectively when the poling direction is aligned at the
angle of
30
in both the piezoelectric layers. Thus, the results presented in Figures 11 and 12
demonstrate the voltage output of the proposed multi-level energy harvesting system (consisting of
primary and secondary beams, functional gradation and poling architectures) under fluctuating
rotational speed can be optimized by considering appropriate gradation and poling architecture.
In this paper, all the results are shown considering only one extreme most secondary beam
(refer to Figure 1(A)). The total output voltage sensed from the energy harvesting system due to the
29
rotation of primary support can be obtained directly by implementing the concept of parallel and series
connections between each secondary beam.
4. Conclusions and perspective
The inevitable effect of speed fluctuation in high-speed rotating structural components is
proposed to be exploited here for the dual purpose of energy harvesting and sensing through the
development of multi-level poling-architected energy harvesting systems. The voltage output is
enhanced and optimized through the coupled effect of tuning the poling angle and introducing
functionally graded materials based on power and sigmoid laws. An efficient, yet insightful theoretical
framework for output voltage is developed at the beam level first under an equivalent centrifugal force
on the basis of conservation of energy principles, which is extensively validated using separate finite
element results. Subsequently, it is extended to the dynamic analysis of the multi-level energy
harvesting system (consisting of primary and secondary beams, functional gradation and poling
architectures). As a proof of concept, we present a qualitative experimental study to show that
fluctuation in rotational speed can obtain energy output, before progressing to the dynamic analysis of
the secondary beams and the multi-level energy harvesting system. We have presented an effective
computational mapping among the output voltage, rotational speed, functional gradation and poling
orientation, wherein the output voltage, voltage sensitivity, and charge sensitivity are used as critical
metrics to quantify the performance of the energy harvesters. The concept of exploiting angular speed
fluctuation along with poling-architected piezoelectric functionally graded configurations will lead to
designing a wide range of self-powered electronic sensors and devices.
Data availability
The dataset generated during the study is available from the corresponding author upon reasonable
request.
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