The potential of electrical impedance on the performance of galloping systems for energy harvesting and control applications

Abstract

Performances of galloping-based piezoelectric systems for energy harvesting and control applications when considering complex electrical impedance are investigated. The aeroelastic system is composed of a unimorph piezoelectric cantilever beam with a square cylinder attached at its tip and subjected to a uniform flow speed. A quasi-steady representation is used to model the aerodynamic force. A nonlinear distributed-parameter model is developed when considering various scenarios of connections between electrical resistance, capacitance, and inductance. Theoretical strategies are developed in order to determine the relation between the onset speed of galloping and the components of the electrical impedance. The results show that the presence of the electrical capacitance and inductance is not beneficial in terms of improving the levels of the harvested power crossing the load resistance. On the other hand, it is shown that the inclusion of these electrical components may be useful for energy harvesting purposes when charging/discharging batteries. One of the important findings of this research study is that including an electrical inductance in connection to a load resistance is very beneficial for control purposes because a significant increase in the onset speed of instability can be obtained for well-defined values of the electrical components. Analytical predictions of these optimum values of the electrical inductance and resistance are determined and compared with numerical simulations. It is also demonstrated that supercritical Hopf bifurcations take place at this controlled optimal configuration without having any hysteresis and jumps when increasing and decreasing the wind speeds.

Full text

The potential of electrical impedance on the performance
of galloping systems for energy harvesting
and control applications
H. Abdelmoula, A. Abdelkefi
n
Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88003, USA
article info
Article history:
Received 11 August 2015
Received in revised form
18 January 2016
Accepted 20 January 2016
Handling Editor: W. Lacarbonara
Keywords:
Galloping
Electrical impedance
Control
Energy harvesting
Nonlinear dynamics
abstract
Performances of galloping-based piezoelectric systems for energy harvesting and control
applications when considering complex electrical impedance are investigated. The aero-
elastic system is composed of a unimorph piezoelectric cantilever beam with a square
cylinder attached at its tip and subjected to a uniform flow speed. A quasi-steady repre-
sentation is used to model the aerodynamic force. A nonlinear distributed-parameter
model is developed when considering various scenarios of connections between electrical
resistance, capacitance, and inductance. Theoretical strategies are developed in order to
determine the relation between the onset speed of galloping and the components of the
electrical impedance. The results show that the presence of the electrical capacitance and
inductance is not beneficial in terms of improving the levels of the harvested power
crossing the load resistance. On the other hand, it is shown that the inclusion of these
electrical components may be useful for energy harvesting purposes when charging/
discharging batteries. One of the important findings of this research study is that
including an electrical inductance in connection to a load resistance is very benefi cial for
control purposes because a significant increase in the onset speed of instability can be
obtained for well-defined values of the electrical components. Analytical predictions of
these optimum values of the electrical inductance and resistance are determined and
compared with numerical simulations. It is also demonstrated that supercritical Hopf
bifurcations take place at this controlled optimal configuration without having any hys-
teresis and jumps when increasing and decreasing the wind speeds.
Published by Elsevier Ltd.
1. Introduction
Offshore structures, heat exchanger cylinders, and bridges are generally surrounded by fluids that can be the source of
flow-induced vibrations including flutter in airfoil sections, vortex induced vibration (VIV) in circular cylinders, galloping in
prismatic structures, and wake galloping in parallel cylinders. Flow-induced vibrations are harmful for large scale systems as
they may lead to fatigue, fretting wear, and damage. For these reasons, these structures are usually controlled by using
different techniques which need external power sources or external structures. On the other hand, flow-induced vibrations
phenomena have been used in the past few years in small scale applications in order to convert unused mechanical
Contents lists available at ScienceDirect
journal ho mepage: www.elsevier.com/locate/jsvi
Journal of Sound and Vibration
http://dx.doi.org/10.1016/j.jsv.2016.01.037
0022-460X/Published by Elsevier Ltd.
n
Corresponding author. Tel.: þ575 646 6546; fax: þ575 646 6111.
E-mail address: abdu@nmsu.edu (A. Abdelkefi).
Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Please cite this article as: H. Abdelmoula, & A. Abdelkefi, The potential of electrical impedance on the performance of
galloping systems for energy..., Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.037i

vibrations to usable amount of electrical power. These autonomous energy harvesters can be used to power or substitute
small batteries that have fi nite life span or would need difficult and expensive maintenance. Various transduction
mechanisms have been used by several researchers including electromagnetic [1], electrostatic [2], and piezoelectric [2–5].
The last one has received considerable interests in energy harvesting applications (direct effect) because of its suitability to
be placed in small volumes and its ease of application [6,7]. The piezoelectric option has also been used in the past decades
to control vibrational structures by imposing a well-defined voltage (converse effect) [8–10].
For energy harvesting purposes, energy harvesters subjected to galloping oscillations are expected to be more effi cient
and suitable compared to VIV- and flutter-based piezoelectric energy harvesters. Indeed, the design of a galloping energy
harvester is easier than the flutter one because it can be designed as a one degree of freedom system [4]. Furthermore, the
presence of rotational modes (pitch and/or flap) are not beneficial when using the piezoelectric transduction mechanism [4].
Also, the galloping energy harvester is expected to be more efficient than the VIV one. This is due to the fact that the
galloping harvester is useful for any speed higher than the onset speed of galloping. However, in the VIV case, only wind
speeds in the synchronization region (narrow range) can give resonant amplitudes. Several investigations have been per-
formed in order to study the influences of the cross-section geometry [11,12], Reynolds number [13], and base excitation
[4,5] on the efficiency of galloping-based piezoelectric energy harvesting systems. However, all available research studies in
galloping energy harvesting have only considered an electrical load resistance in the harvester's circuitry. One of the
objectives of this work is to determine the potential of including electrical inductance and/or capacitance on the onset speed
of instability and the performance of galloping-based piezoelectric energy harvesters.
For control purposes, many research studies have focused on reducing or suppressing flow-induced vibrations in large
scale applications including overload transmission lines [14,15], bridges [16,17], prismatic tall buildings [18–20], aircraft
[21,22], and slender seabed risers and pipelines [22,23]. Different passive, semi-active, and active control strategies have
been utilized for this purpose. As for passive controllers, nonlinear energy sinks (NES) [24,25] and tuned-mass dampers
(TMD) [26,27] ha
ve been proposed in many investigations. Semi-active controllers have been used by Abdel-Rohman and
John [28] and Johnson et al. [29] in which they combined a linear velocity feedback controller and a tuned-mass damper to
increase the onset speed of instability of a bridge. Using active control strategies, Abdel-Rohman and John [28] investigated
the impacts of a linear velocity feedback controller on the onset speed of instability of a bridge. He demonstrated that this
active controller is very useful in order to increase the onset speed of galloping to higher and higher values. Mehmood et al.
[30] studied the efficiency of linear and nonlinear velocity feedback controllers on the response of a circular cylinder
subjected to vortex-induced vibrations. For very small controlled values of the cylinder's oscillations, it was indicated that
the nonlinear velocity feedback controller is more efficient than its linear counterpart. As another type of active controllers,
linear and/or nonlinear time delay feedback controllers have been proposed in several research studies in order to reduce/
suppress vortex-induced vibrations in circular cylinders [31] and galloping oscillations in prismatic structures [32]. Other
active and passive control strategies have been considered to avoid or decrease possible oscillations in aeroelastic systems,
such as adaptive fuzzy sliding mode [33], flow blowing and suction [34], and cylinder shape controls [35].
Generally, the direct effect of piezoelectricity has been used in order to convert any unused vibrations to electrical
energy. This direct effect has been utilized in some investigations to control the response of such aeroelastic systems. In fact,
the presence of an electrical impedance in the circuitry part of the harvester results in the presence of a passive shunt
damping which can enhance the coupled damping of the electromechanical system. It was demonstrated in previous studies
[13] that there is an optimum value of the electrical load resistance where the coupled damping in the system is maximum
and hence higher onset speed of instability is obtained. Almost all performed investigations have considered a simple
electrical load resistance in the circuitry part of the harvester. Only few research studies have determined the influences of
the electrical impedance on the stability of structures under aeroelastic vibrations. Kim et al. [36] used the shunted pie-
zoelectric to increase the structural damping in order to make the blades of a helicopter stable in most flight conditions by
adding electrical inductance to the electrical circuit. To investigate its efficiency, they showed experimentally that the shunt
damping effect due to inductive–resistive series configuration is able to increase the damping of the blades by 217 percent.
Agneni et al. [37] proved that some values of the electrical inductance which is mounted in parallel to an electrical resis-
tance can increase the speed of instability of a two-degree-of-freedom wing. They indicated that this result is due to
resistive–inductive shunt damping effect. In a recent study and for the purpose of energy harvesting, De Marqui et al. [38]
considered an electrical inductance connected in series to an electrical load resistance in order to study its effects on the
performance of flutter-based piezoelectric energy harvesters. They reported that the presence of an electrical inductance
results in an increase of the coupled damping of the system and hence the linear flutter speed.
In this research study, our objective is to investigate th e potential of includi ng el ectrical capacitance and/or
inductance on the performance of galloping-based piezoelectric systems for both energy harvesting and control
applications. Comprehensive mathematical strategies are used to determine analytically the variations of the onset
speed of galloping as functions of the e lectrical load resistance, capacitance, and inductance. For the purpose of energy
harvesting, our goal is to reduce the o nset speed of galloping and enhance the levels of the harvested power for dif-
ferent optimal configurations of the capaci tance and in ductance. For control purposes, we aim to determine the opti-
mum values of the electrical resistance , capacitance, and indu ctance in order to increase the onset speed of gallop ing to
very high values and hence suppressing any possible vibrations. The rest of this work is organized as follows: in
Section 2, the modeling of galloping-based piezoelectric systems is presented using both Hamilton's principle and
Euler–Lagrange equations. The influences of the electrica l resistance, capacitance, and inducta nce on the onset speed of
Please cite this article as: H. Abdelmoula, & A. Abdelkefi, The potential of electrical impedance on the performance of
galloping systems for energy..., Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.037i
H. Abdelmoula, A. Abdelkefi / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎2

galloping are determined through a linear analysis in Section 3.InSection 4, bifurcation diagrams and performances of
galloping-based piezoelectric systems for energy harvesting and control applications are investigated and deeply
discussed. Summary and conclusions are finally presented in Section 5 .
2. Galloping system's modeling with resistive, capacitive, and inductive shunt damping effects
2.1. Representation of the galloping system
The aeroelastic system under investigation consists of a unimorph cantilever beam attached to a square cross-section
bluff body at its free end, as shown in Fig. 1. The cantilever beam is composed of a substrate and a piezoelectric layer which
is connected to an electrical impedance. Different electrical circuit configurations are considered in this research study, as
presented in Fig. 1. The considered aeroelastic system is subjected to an uniform cross-flow. At a well-defined flow speed,
this system starts to oscillate in the transverse direction due to the presence of a galloping force.
To determine the equations of motion and boundary conditions of these galloping systems, Euler–Bernoulli beam
assumptions are used. The bluff body is modeled as a rigid body with mass M
t
and mass moment of inertia I
t
. The extended
Hamilton's principle is then used which consists of:
Z
t
2
t
1
ðδT δΠ þδW
nc
Þ dt ¼0 (1)
where T,
Π
and W
nc
, respectively, the kinetic energy, potential energy, and the nonconservative work in the system.
The total potential energy
Π
of each of the galloping systems is composed of the potential energy of the substrate beam
and the piezoelectric layer which is given by:
Π ¼
1
2
Z
L
s
0
Z
A
s
σ
s
11
ε
11
dA
s
dxþ
Z
L
p
0
Z
A
p
σ
p
11
ε
11
dA
p
dx
Z
L
p
0
Z
A
p
E
3
D
3
dA
p
dx
"#
(2)
where dA
s
and dA
p
are the cross-section areas of differential substrate and piezoelectric beam elements, respectively. Using
the linear constitutive equations of piezoelectricity and Hooke's law, the strains and stresses in the substrate and piezo-
electric layers are, respectively, related by:
σ
s
11
¼E
s
ε
11
and σ
p
11
¼E
p
ε
11
e
31
E
3
where E
s
and E
p
denote Young's moduli at a constant electric field and E
3
is the electric field in the poling direction. The
generated voltage between the piezoelectric electrodes V(t) is related to the electric fi eld by E
3
¼
VðtÞ
h
p
. In addition to that, D
3
represents the electric displacement which is related to the strain and electric field by:
D
3
¼e
31
ε
11
þϵ
33
E
3
(3)
where e
31
is the piezoelectric stress coefficient and
ϵ
33
is the permittivity at constant strain.
The kinetic energy T of each of the galloping systems, shown in Fig. 1(a) and 1(b), can be expressed as:
T ¼
1
2
Z
L
p
0
m
1
_
ν
2
dxþ
Z
L
s
L
p
m
2
_
ν
2
dx
"#
þ
1
2
M
t
_
ν
L
þ
D
t
2
_
ν
L
0

2
þ
1
2
I
t
_
ν
L
02
(4)
Fig. 1. Schematics of galloping systems with an electrical circuit composed of resistance–inductance–capacitance (a) in parallel and (b) in series.
Please cite this article as: H. Abdelmoula, & A. Abdelkefi, The potential of electrical impedance on the performance of
galloping systems for energy..., Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.037i
H. Abdelmoula, A. Abdelkefi / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 3

where L
s
is the length of the substrate beam and L
p
denotes the distance from the clamped side of the cantilever beam to the
ending location of the piezoelectric layer. m
1
and m
2
are the mass of the beam per unit length and depend on the location of
the piezoelectric layer:
m
1
¼b
s
ρ
s
h
s
þb
p
ρ
p
h
p
for 0 r xr L
p
m
2
¼b
s
ρ
s
h
s
for L
p
r xr L
s
where h
s
and h
p
are, respectively, the thicknesses of the substrate beam and piezoelectric layer;
ρ
s
and
ρ
p
represent the
densities of the substrate and piezoelectric layer, respectively; and b
s
and b
p
denote, respectively, the width of the substrate
and piezoelectric layers.
Using these expressions and considering
ε
11
¼yν
″
, the total potential energy of the system can be rewritten as follows:
Π ¼
1
2
Z
L
p
0
EI
1
ν
″2
dxþ
Z
L
s
L
p
EI
2
ν
″
2
dx
"#

Z
L
p
0
e
31
b
p
y
1
þy
2
2
ν
″
VtðÞdx
1
2
ϵ
33
b
p
L
p
h
p
VðtÞ
2
(5)
where EI
1
¼ð1=3ÞE
s
b
1
ðy
3
1
y
3
0
Þþð1=3ÞE
p
b
2
ðy
3
2
y
3
1
Þ and EI
2
¼ð1=12Þb
1
E
s
h
s
. y
1
and y
2
denote the positions of the layers and are
defined with respect to the position of the neutral axis, as shown in Fig. 2, which is defined by:
y ¼
E
p
b
p
h
2
p
þE
s
b
s
h
2
s
þ2E
p
b
p
h
p
h
s
2ðE
p
b
p
h
p
þE
s
b
s
h
s
Þ
(6)
where y
0
¼y, y
1
¼h
s
y, and y
2
¼ðh
s
þh
p
Þy.
The nonconservative work is divided into three parts, namely, the work done by the viscous damping force, the work
done by the aeroelastic force F
g
(t), and the work due to the power delivered to the electrical circuit. Hence, the variation of
the nonconservative work term can be expressed as:
δW
nc
¼
Z
L
s
0
c
_
νδν dx þF
g
tðÞδνL
s
; tðÞþ
D
t
2
ν
0
L
s
; tðÞ

iδλ (7)
where c denotes the viscous damping, i is the generated current between the piezoelectric electrodes and
λ
is related to the
generated voltage by
_
λ ¼ V. F
g
(t) is the galloping force. To model this force, different tools can be used, such as solving the
Navier–Stokes equation using computational fluid dynamics techniques as done by Mehmood et al. [30] for circular
cylinders or using vortex-lattice method as done by Tang et al. [39,40] or utilizing quasi-steady hypothesis which is based on
empirical-experimental representations. The quasi-steady representation can be utilized to evaluate the galloping force
when the lag between the unsteady oscillations and its effect on the aerodynamic loads is negligible which limits the
modeling to very small reduced frequency values k ¼
ω
D
U
where
ω
is the frequency of oscillations and U is the wind speed. In
addition to that, the oscillating frequency should be very small compared to the vortex-shedding frequency. Considering the
given parameters in Table 1, these conditions are justified. Therefore, we use the quasi-steady approximation to represent
the galloping force which is given by [22,41]:
F
g
tðÞ¼
1
2
ρ
a
D
t
L
t
U
2
X
7
j ¼ 1
a
j
α
j
¼
1
2
ρ
a
D
t
L
t
U
2
X
7
j ¼ 1
a
j
1
U
∂νðx; tÞ
∂t
þ
D
t
2U
∂
2
νðx; t Þ
∂t∂x

j
(8)
where
α
denotes the angle of attack; a
j
represent the linear and nonlinear coefficients of the galloping force. These
Fig. 2. Neutral axis position.
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galloping systems for energy..., Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.037i
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coefficients are empirically determined by polynomial fitting of the experimental normal force coefficient versus the angle
of attack. The empirical coefficients a
j
depend on the type of the cross-section geometry [13,41] and the Reynolds number
[11,41,42]. For high Reynolds number scenarios, it was demonstrated by Barrero-Gil et al. [41] that the empirical coefficients
a
j
are independent of the Reynolds number and that the linear and cubic coefficients a
1
and a
3
are sufficient to evaluate the
normal force coefficient. The values of a
1
and a
3
for a square cross-section geometry are given in Table 1.
Using the extended Hamilton principle, the governing equations of motion and boundary conditions of the galloping
systems are expressed as:
EI
∂
4
νðx; tÞ
∂x
4
þc
∂νðx; tÞ
∂t
þm
∂
2
νðx; tÞ
∂t
2
ϑ
p
VtðÞ
dδðxÞ
dx

dδðxL
p
Þ
dx

¼
1
2
ρ
a
D
t
L
t
U
2

X
3
i ¼ 1
a
i
1
U
∂νðx; tÞ
∂t
þ
D
2U
∂
2
νðx; tÞ
∂t∂x

i
δ xL
s
ðÞþ
D
2
dδðxL
s
Þ
dx

(9)

ϑ
p
Z
L
p
0
∂
3
νðx; tÞ
∂t∂x
2

ϵ
33
b
p
L
p
h
p
dVðtÞ
dt
¼i (10)
where
ϑ
p
¼e
31
b
p
y
1
þy
2
2
and the stiffness EI and mass of the beam per unit length m are given by EI ¼ EI
1
and m ¼ m
1
for
0r xr L
p
, and EI ¼EI
2
and m ¼ m
2
for L
p
r x r L
s
. The associated boundary conditions are given by:
νð0; tÞ¼0; ν
0
ð0; tÞ¼0; (11)
EI
2
ν
‴
L
s
; tðÞ¼M
t
€
ν L
s
; tðÞþM
t
D
t
2
€
ν
0
L
s
; tðÞ; (12)
EI
2
ν
″
L
s
; t
ðÞ
¼M
t
D
t
2
€
ν L
s
; t
ðÞ
 I
t
þM
t
D
t
2

2
!
€
ν
0
L
s
; t
ðÞ
: (13)
2.2. Reduced-order models of the resistive–inductive–capacitive galloping systems
To derive a reduced-order mod el for the considered galloping systems shown in Fig. 1(a) and (b), we express the
displacement
νðx; t Þ using the Galerkin discretization in the form νðx; tÞ¼
P
1
j ¼ 1
Φ
kj
ðxÞη
j
ðtÞ where k ¼1, 2 which depends
on the value of x,
Φ
kj
ðxÞ are the mode shapes, and η
j
ðtÞ denote the modal coordinates. It should be mentioned that the
exact mode shapes are determined by considering two different regions of the beam depending on the placement of the
pie zoelectric layer. More d etails about the derivation of the exact mode shapes and the associated natural frequencies are
provided in [43].
Recently, it was proved by Bibo et al. [5] that only one mode in the Galerkin approach is sufficient to accurately predict
the performance of a galloping-based piezoelectric system. Hence, only one mode in the Galerkin discretization is
Table 1
Geometric and physical properties of the square cross-section galloping systems.
Parameter Description Values
E
s
ðPaÞ Substrate Young's modulus
190  10
9
ρ
s
ðkg m
3
Þ
Substrate density 9873
b
s
ðmÞ
Width of the beam layer
1:4  10
2
h
s
ðmÞ
Beam layer thickness
1:2  10
3
L
s
ðmÞ Length of the beam
14:5  10
2
E
p
ðPaÞ Piezoelectric material Young's modulus
62  10
9
ρ
p
ðkg m
3
Þ
Piezoelectric material density 7800
b
p
ðmÞ
Width of the piezoelectric layer 7e3
h
p
ðmÞ
Piezoelectric layer thickness
0:4  10
3
L
p
ðmÞ Left of the beam to ending of the piezoelectric layer
8:5  10
2
ϵ
33
ðFm
1
Þ
Permittivity component at constant strain
27:3e  10
9
d
31
ðmV
1
Þ
Strain coefficient of piezoelectric layer
320  10
12
D
t
ðmÞ Width of the tip mass
2  10
2
L
t
ðmÞ Length of the tip mass
10:2  10
2
M
t
ðkgÞ
Tip mass
15  10
3
ρ
a
ðkg m
3
Þ
Air density 1.24
a
1
Linear empirical coefficients for a square cross section 2.3
a
3
Nonlinear empirical coefficients for a square cross section 18
ξ Mechanical damping ratio 0.005
Please cite this article as: H. Abdelmoula, & A. Abdelkefi, The potential of electrical impedance on the performance of
galloping systems for energy..., Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.037i
H. Abdelmoula, A. Abdelkefi / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 5

considered. Applying the Euler–Lagrange equations which are given by these relations:
d
dt
dl
d
_
η

dl
d
η
¼
δW
nc
δη
(14)
d
dt
dl
d
_
λ

dl
d
λ
¼
δW
nc
δλ
(15)
where l ¼T 
Π is the Lagrangian.
One obtains the nonlinear distributed-parameter model of the galloping systems under investigation:
€
η þ2ξω
_
η þω
2
ηθV ¼
1
2
ρ
a
D
t
L
t
U
2
X
3
j ¼ 1
a
j
_
η
j
U
j
Φ L
s
ðÞþ
D
t
2
Φ
0
L
s
ðÞ

j þ1
(16)
C
p
_
V þ
θ
_
η ¼i (17)
where Cp ¼
ϵ
33
b
p
L
p
h
p
and θ ¼ϑ
p
Φ
0
ðL
p
Þ.
For different electrical circuit configurations presented in Fig. 1(a) and (b), the nonlinear aeroelectromechanical model for
each galloping system is given by:
Parallel resistive–inductive– capacitive case:
€
η þ 2ξω
n

1
2
a
1
ρ
a
D
t
L
t
U Φ L
s
ðÞ
þ
D
t
2
Φ
0
L
s
ðÞ

2
!
_
η þω
2
n
ηθV 
1
2U
a
3
ρ
a
D
t
L
t
Φ L
s
ðÞ
þ
D
t
2
Φ
0
L
s
ðÞ

4
_
η
3
¼0 (18)
C
p
þC

€
V þ
_
V
R
þ
V
L
þ
θ
€
η ¼0 (19)
Series resistive–inductive–capacitive case:
€
η þ 2ξω
n

1
2
a
1
ρ
a
D
t
L
t
U Φ L
s
ðÞþ
D
2
Φ
0
L
s
ðÞ

2
!
_
η þω
2
n
ηθ
Q
C
þR
_
Q þL
€
Q


1
2U
a
3
ρ
a
D
t
L
t
Φ L
s
ðÞþ
D
t
2
Φ
0
L
s
ðÞ

4
_
η
3
¼0 (20)
C
p
L
€
Q þC
p
R
_
Q þ
C
p
C
þ1

Q þ
θη ¼0 (21)
where R, L, and C are, respectively, the electrical load resistance, inductance, and capacitance. Q denotes the generated
charge between the piezoelectric electrodes and defined by
_
Q ¼ i.
In this research study, as there is an equivalence in the electrical impedance between series and parallel resistive–
capacitive and resistive–inductive configurations and to avoid any repetitions in the presented curves, a particular focus is
paid to the parallel resistive–capacitive and resistive–inductive cases. Similar techniques and procedures can be used in the
series case. The combined effects of resistive–capacitive–inductive can be easily detected after performing their separate
analyses. In the rest of this work, the differences between the parallel and series configurations are mentioned without
presenting the analysis of the series configurations.
3. Influences of the electrical resistance, capacitance, and inductance on the onset speed of galloping
3.1. Capacitive–resistive electrical circuit
Staring with the capacitive–resistive electrical circuit, the inductance is considered in open-circuit configuration (LC 1).
To investigate the impacts of the load resistance and capacitance on the onset speed of instability when considering a
parallel connection, we perform a linear analysis for the coupled equations of motion. Dropping the nonlinear aerodynamic
term and considering the following state variables:
Y ¼
Y
1
Y
2
Y
3
2
6
4
3
7
5
¼
η
_
η
V
2
6
4
3
7
5
(22)
the linear reduced-order model given in Eqs. (18) and (19) can be rewritten in a matricial form as:
_
Y ¼ AðR; CÞY ; (23)
Please cite this article as: H. Abdelmoula, & A. Abdelkefi, The potential of electrical impedance on the performance of
galloping systems for energy..., Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.037i
H. Abdelmoula, A. Abdelkefi / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎6

where
AR; CðÞ¼
01 0

ω
2
n
ψθ
0 
θ
C þC
p

1
RðC
p
þCÞ
2
6
6
4
3
7
7
5
with
ψ ¼ 2ξω
n

1
2
a
1
ρ
a
D
t
L
t
UðΦ L
s
ðÞþ
D
2
Φ
0
L
s
ðÞÞ
2
. We should note that when U ¼U
g
(U
g
is the onset speed of galloping), ψðU
g
Þ is
defined as
ψ
g
.Ifψ
g
¼0 then U
g
is equal to the onset speed of short-circuit configuration (U
short
). ψ
g
r 0 implies that
U
g
Z U
short
.
The matrix AðR; CÞ has three eigenvalues. The first two eigenvalues are complex conjugate. The third one is always real
and negative as it is due to the piezoelectric coupling. The plotted curves in Fig. 3(a) show the variations of the onset speed
of galloping as a function of the capacitance when considering different values of the load resistance. It follows from this
plot that the inclusion of the capacitance reduces the onset speed of instability for any value of the resistance. For high
values of capacitance, the onset speed of galloping tends to U
short
.InFig. 3(b), the variations of the onset speed of instability
as a function of the load resistance for various values of the capacitance are plotted. Inspecting this curve, it is clear that an
increase in the capacitance is followed by a decrease in the onset speed of galloping. Moreover, there is a left shift for the
optimum value of the load resistance when increasing the capacitance. This result can be explained due to the fact that an
increase in the capacitance is accompanied by a decrease in the coupled damping of the galloping system and hence a
decrease in the onset speed of instability.
It follows from the plots in Fig. 3 (b) that there is a combination of electrical load resistance and capacitance at which the
onset speed of galloping is maximum with respect to R. From a mathematical point of view, we determine the relation
between the onset speed of instability and the system's parameters and we also determine the corresponding optimum
values of the resistance and capacitance to get maximum cut-in speeds for all possible wind speed values. To this end, we
first define C
eq
¼C
p
þC. When the wind speed U reaches U
g
, the polynomial of the matrix detðAðR; CÞXIÞ can be expressed
in these two forms:
P
A
XðÞ¼X
3
þ ψ
g
þ
1
RC
eq

X
2
þ ω
2
n
þ
θ
2
C
eq
þ
ψ
g
RC
eq
!
X þ
ω
2
n
RC
eq
(24)
P
A
XðÞ¼X
2
þjλ
1
j
2

X 
λ
2

(25)
where det denotes the determinant of a matrix and I represents the identity matrix.
λ
1
, λ
1
, and
λ
2
denote the three distinct
eigenvalues of the system when U ¼ U
g
. At this speed, the real part of
λ
1
is zero and
λ
2
is real and negative. Equating the
terms in Eqs. (24) and (25), one obtains the following relations:

λ
2
¼ψ
g
þ
1
RC
eq
; (26)
j
λ
1
j
2
¼ω
2
n
þ
θ
2
C
eq
þ
ψ
g
RC
eq
; (27)
j
λ
1
j
2
λ
2
¼
ω
2
n
RC
eq
: (28)
Fig. 3. Variations of the onset speed of instability as a function of (a) the capacitance for various values of R and (b) the resistance for various values of C.
Please cite this article as: H. Abdelmoula, & A. Abdelkefi, The potential of electrical impedance on the performance of
galloping systems for energy..., Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.037i
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Because λ
2
r 0 hence, ψ
g
Z 
1
RC
eq
. Substituting Eqs. (26) and (27) into Eq. (28) we find that
ψ
g
is a root for the following
polynomial function:
GyðÞ¼y
2
þ
1
RC
eq
þω
2
n
C
eq
Rþθ
2
R

yþ
θ
2
C
eq
: (29)
This second-order polynomial function has two distinct negative solutions. Considering
ψ
g
Z 
1
RC
eq
, only one of the two
solutions satisfies this condition. Therefore, the analytical solution of
ψ
g
is given by:
ψ
g
¼
C
eq
R
2
ðθ
2
þω
2
n
C
eq
Þþ1
ffiffiffiffi
Λ
p
2C
eq
R
(30)
where
Λ ¼ðθ
2
C
eq
R
2
1 Þ
2
þω
2
n
C
2
eq
R
2
ð2θ
2
C
eq
R
2
þω
2
n
C
2
eq
R
2
þ2Þ.
Hence the analytical expression of the onset speed of galloping for a parallel resistive–capacitive configuration can be
written as:
U
g
¼
1
1
2
a
1
ρ
a
D
t
L
t
Φ L
s
ðÞþ
D
2
Φ
0
L
s
ðÞ

2
2ξω
n
ψ
g

(31)
To determine the optimum values of R and C in order to guarantee maximum values of the onset speed of galloping with
respect to R for different wind speeds, we solve numerically the following system of equations:
∂U
g
∂R




ðC
opt
;R
opt
Þ
¼0; U
g
C
opt
; R
opt

¼U; RZ 0 and C Z 0 (32)
The plotted curves in Fig. 4 show the corresponding values of R
opt
and C
opt
for each possible value of the wind speed U
that gives an optimum onset speed of galloping with respect to the load resistance. Clearly, an increase in the onset speed of
instability is accompanied by an increase in the optimum values of the load resistance and a decrease in the optimum values
of the capacitance, as presented in Fig. 3(b).
As for the seri es configuration, we brieflypresentinFig. 5(a) and (b) the variat ions o f the onse t speed of instability as
functions of the capacitance and resistance, respectively. Clearly, a decrease in the capacitance is followed by a decrease in
the onset speed of gal loping and an increase in the optimum value of the load resistance for maximum onset speeds, as
show n in Fi g. 5(b). Inspecting the curves in Fig. 5(a), it is noted that using high values of the capacitance results in a
resistive case configuration. This is expected because a short-circuit configuratio n in the capacitance cancels its effe ct on
the onset speed of galloping. It should be me ntioned th at simi lar techniques as performed for the parallel resistive–
capacitive case can be done to dete rmine the optimum values of the load resistance and capac itance for maximum onset
speeds with respect to R.
10 15 20 25
10
−9
10
−8
10
−7
10
−6
10
−5
C
opt
(F)
10 15 20 25
10
3
10
4
10
5
10
6
R
opt
(Ω)
U
g
(m/s)
C
opt
R
opt
Fig. 4. Variations of the optimum values of the capacitance and resistance for optimum onset speed of galloping with respect to R when considering a
parallel configuration.
Please cite this article as: H. Abdelmoula, & A. Abdelkefi, The potential of electrical impedance on the performance of
galloping systems for energy..., Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.037i
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3.2. Inductive–resistive electrical circuit
To consider an inductive–resistive electrical circuit, an electrical capacitance open-circuit configuration (C C 0) is utilized.
The following state variables are defined:
X ¼
X
1
X
2
X
3
X
4
2
6
6
6
6
4
3
7
7
7
7
5
¼
η
_
η
V
_
V
2
6
6
6
4
3
7
7
7
5
(33)
which results to this matricial form of the governing Eqs. (18) and (19):
_
X ¼ BðR; LÞX; (34)
where
BR; LðÞ¼
01 0 0

ω
2
n
ψ
θ
m
0
00 0 1
ω
2
n
θ
C
p
ψ
θ
C
p

1
LC
p

θ
2
C
p

1
RC
p
2
6
6
6
6
4
3
7
7
7
7
5
Unlike the resistive and resistive–capacitive cases, the matrix BðR; LÞ has a set of four eigenvalues, which are dependent
on the value of the load resistance and inductance. The first two eigenvalues are complex conjugate which is due the
mechanical contribution. As for the third and fourth eigenvalues (
λ
3
; λ
4
), they strongly depend on L and R. The last two
eigenvalues can be:

Complex and conjugate, which indicates the presence of a second coupled frequency of electrical type.

Real and negative, which is due to high damping values in the electrical part.
In both scenarios, we have the following inequity:
λ
3
þλ
4
r 0; (35)
λ
3
λ
4
Z 0: (36)
For the resistive–inductive electrical circuit, the characteristic polynomial of the matrix BðR; LÞ can be expressed as:
P
B
XðÞ¼det BXIðÞ¼X
2
 λ
1
þλ
1

X þ
λ
1
λ
1

X
2
þμ
1
X þμ
2
2

where
μ
1
¼λ
3
λ
4
and μ
2
2
¼λ
3
λ
4
.
When U is equal to U
g
, the characteristic polynomial P
B
can be expressed in two distinct forms as:
P
B
ðXÞ¼ðX
2
þjλ
1
j
2
ÞðX
2
þμ
1
X þμ
2
2
Þ (37)
Fig. 5. Variations of the onset speed of instability as a function of (a) the capacitance for various values of R and (b) the resistance for various values of C for
resistive–capacitive series configuration.
Please cite this article as: H. Abdelmoula, & A. Abdelkefi, The potential of electrical impedance on the performance of
galloping systems for energy..., Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.037i
H. Abdelmoula, A. Abdelkefi / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 9

P
B
XðÞ¼X
4
þ ψ
g
þ
1
RC
p

X
3
þ ω
2
n
þω
2
e
þ
θ
2
C
p
þ
ψ
g
RC
p
!
X
2
þ ψ
g
ω
2
e
þ
ω
2
n
RC
p

X þ
ω
2
n
ω
2
e
(38)
where
ω
e
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1=LC
p
p
represents the electrical natural frequency of a resistive–inductive circuit.
From Eqs. (37) and (38), we get the following system of equations:
μ
1
¼ψ
g
þ
1
RC
p
; (39)
j
λ
1
j
2
þμ
2
2
¼ω
2
n
þω
2
e
þ
θ
2
C
p
þ
ψ
g
RC
p
; (40)
j
λ
1
j
2
μ
1
¼ψ
g
ω
2
e
þ
ω
2
n
RC
p
; (41)
j
λ
1
j
2
μ
2
2
¼ω
2
n
ω
2
e
: (42)
Substituting Eqs. (39)–(41) into Eq. (42) and doing some calculations in the system of equations given in Eqs. (39)–(42),
ψ
g
is
a solution for the following polynomial function D(x):
DðxÞ¼A
1
x
3
þB
1
x
2
þC
1
xþD
1
(43)
where A
1
¼ω
2
e
C
2
p
R
2
, B
1
¼θ
2
ω
2
e
C
2
p
R
3
þðω
2
n
þω
2
e
ÞC
p
R, C
1
¼θ
2
C
p
R
2
ðω
2
n
þω
2
e
Þþðω
2
n
ω
2
e
Þ
2
C
2
p
R
2
þω
2
n
, and D
1
¼θ
2
ω
2
n
R.
In order to determine the sign of
ψ
g
, we draw the monotony table of the polynomial function D(x). As all the coefficients
of D
0
ðxÞ are positive, so its roots are either negative or complex. Considering the scenario when the two roots of D
0
ðxÞ are
negative, the corresponding monotony table of D(x) is shown in Table 2. It follows from this table that D(x) is always higher
than Dð0Þ¼D
1
. In the scenario when the two roots of D
0
ðxÞ are complex conjugate, D(x) is then increasing with x and hence
DðxÞZ Dð0Þ¼D
1
, as shown in Table 3.
For both scenarios shown in Tables 2 and 3, DðxÞ4 Dð0Þ¼D
1
4 0 when x4 0. As
ψ
g
is a root of D(x)so
ψ
g
must be
negative. Consequently, the onset speed of galloping of the system when including an inductance in its electrical circuit
cannot go lower than U
short
. This result is clear in the plotted curves in Fig. 6(a) and (b) when the variations of the onset
speed of instability as functions of the load resistance and inductance are depicted. It follows from Fig. 6(a) that high values
of the onset speed of galloping can be obtained for specific values of the electrical inductance and load resistance. Clearly,
there is an optimum value of the inductance when the onset speed of instability is maximum for all values of the electrical
load resistance. However, the peaks of the onset speed of instability for some load resistance do not appear in Fig. 6
(a) because these load resistance values make the electrical circuit in short-or open-circuit configurations. Inspecting Fig. 6
(b), it is noted that depending on the value of the inductance, the maximum onset speed of instability may decrease or
increase compared to its resistive case counterpart. For the purpose of energy harvesting, the values of inductance that
result in a decrease in the onset speed of galloping are desired. On the other hand, values of inductance that cause an
increase in the onset speed of galloping are required for control purposes.
It should be mentioned that the effects of the inductance is totally different than the electrical capacitance on the onset
speed of instability. In fact, the capacitance always decreases the onset speed of galloping, however, the presence of the
inductance in the electrical circuit may increase or decrease U
g
compared to the resistive electrical circuit. Moreover, for
high values of the onset speed of galloping, it is clear from the plotted curves in Fig. 6(a) that multiple combinations of R and
L can be obtained to have the same onset speed. For onset speed values lower than the optimum resistive onset speed of
galloping, there are only one optimum combination of R and L, as shown in Fig. 7. These values of R
opt
and L
opt
for maximum
Table 3
Monotony table of the function D(x) when D
0
ðxÞ has two complex
conjugate roots.
Table 2
Monotony table of the function D( x) when D
0
ðxÞ has two negative roots.
Please cite this article as: H. Abdelmoula, & A. Abdelkefi, The potential of electrical impedance on the performance of
galloping systems for energy..., Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.037i
H. Abdelmoula, A. Abdelkefi / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎10

onset speed of galloping with respect to R are determined by numerically solving the following system's of relations:
∂U
g
∂R




ðL
opt
;R
opt
Þ
¼0; U
g
L
opt
; R
opt

¼U; RZ 0 and LZ 0 (44)
From Fig. 7, we note that obtaining higher values of the onset speed of galloping (U
short
o U
g
o 25 m=s), high values of the
inductance L
opt
are required and hence a synthetic inductor is needed to get these values which is not beneficial for energy
harvesting purposes because an input power is required. For non-synthetic inductor values Lo 0:1H, the presence of the
electrical inductance results in a short-circuit configuration and hence U
g
U
short
for all possible values of R which is useful
for energy harvesting applications.
AscanbeseenfromFig. 6(a), for well-defined values of the inductance, the onset speed of gal loping is much higher
than the optimum one in the resistive case. Therefore, the inductance gives the opportunity to increase the cut-in speed
and can be an excellent controller for the system. It should be mentioned that having high values of the inductance
results to the need of a synthetic inductor which requires very small amount of external power and can be implement ed
in a very small volume. To determine analytically the optimum values of the resistance and inductance that give
maximum possible onset speed of galloping (U
g
¼U
g max
)andbecauseψ
g
o 0andU
g
is related to
ψ
g
by Eq. (31),
ψ
g
should be minimized. Starting from Eqs. (39)– (42) which are represented in the first line of the chart presented in Fig. 8
and noted by (a– d), Eq. (35) and (a) are then used to find the given relation in (e). Using some mathematical relations
between the first line components (a–d) in the chart, the components of the second line in the chart are then obtained
and noted by (e– h). After that, conside ring two different scenario s depending on the relation between the ele ctrical and
mechanical natural frequencies (
ω
n
r ω
e
and ω
n
Z ω
e
) and using the given relations in (e–h), it is demonstrated that
ψ
g
should be larger than Rθ
2
and 
1
RC
p
. To determine the exact minimum values of
ψ
g
, we consider the case when
Fig. 7. Variations of the optimum values of the electrical inductance and resistance for maximum onset speed of galloping with respect to R when
considering a parallel resistive–inductive configuration.
Fig. 6. Variations of the onset speed of galloping as a function of (a) electrical inductance for various values of R and (b) electrical load resistance for various
values of L .
Please cite this article as: H. Abdelmoula, & A. Abdelkefi, The potential of electrical impedance on the performance of
galloping systems for energy..., Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.037i
H. Abdelmoula, A. Abdelkefi / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 11

ω
n
¼ω
e
. Consequently, the expression of D(x) can be rewritten as :
DðxÞ¼
ω
2
n
ðRC
p
xþ1Þ
2
ðRθ
2
þxÞ (45)
It follows from the last equation that R
θ
2
or/and 
1
RC
p
are the roots of D(x) and hence the minimum value of
ψ
g
should
be equal to min R
θ
2
;
1
RC
p

. Therefore, the maximum possible onset speed of galloping can be determined by the fol-
lowing relation:
U
g max
¼
min Rθ
2
;
1
RC
p

þ2
ξω
n
1
2
ρ
a
D
t
L
t
a
1
Φ L
s
ðÞ
þ
D
2
Φ
0
L
s
ðÞ

2
(46)
It is clear that the possible maximum U
g max
is strongly dependent on the electrical resistance and independent of the
inductance. This result can be explained by the equality between the natural frequencies of mechanical and electrical types
and hence the optimum values of the inductance that gives maximum onset speeds of galloping is given by L ¼ L
max
¼
1
ω
2
n
C
p
.
The optimum value of the load resistance that gives maximum value for the onset speed of instability is associated to the
case when R
max
θ
2
is equal to
1
R
max
C
p
and hence R
max
¼
1
∣
θ
∣
ffiffiffiffi
C
p
p
.InFig. 9, we plot the variations of the onset speed of instability as
a function of the load resistance when considering our analytical predictions and the classical numerical results (when the
real part of one of the eigenvalues becomes zero). Excellent agreements between different methods are obtained which
justify the correctness of our used strategy. Clearly, including the inductance is very beneficial for increasing the onset speed
of galloping of prismatic structures to higher and higher values. It is noted that there is a discontinuity when the load
resistance becomes equal its optimum value (R
max
).
In Fig. 1 0, we plot the variations of the real part of the eigenv alues as a function of the wind speed for different values of the load
resistance and when L ¼L
max
¼
1
ω
2
n
C
p
.WhenR ¼ 10
5
Ω and R ¼ 5  10
5
Ω, only one of the real parts of the eigenv alues becomes
positive and the other one is always negative for all considered wind speeds values. When R ¼R
max
¼
1
j
θ
j
ffiffiffiffi
C
p
p
, the real parts of the
two eigenvalues become zero at the onset speed of instability and then one of them becomes higher than zero and the other one
stays negative for all wind speeds higher than the onsetspeed.Itisclearthat,atthisoptimumconfiguration of R
max
and L
max
,the
two coupled damping terms become equal and hence higher wind speed is needed to get self-ex cit ed oscillations. When
R ¼ 10
6
Ω, the two real parts of the eigenv alues become zero at the onset of instability and stay positive for wind speed values little
bit higher than the onset speed of instability and then one of them becomes negative for very high wind speeds.
Fig. 8. Determination of the possible minimum values of ψ
g
.
Please cite this article as: H. Abdelmoula, & A. Abdelkefi, The potential of electrical impedance on the performance of
galloping systems for energy..., Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.037i
H. Abdelmoula, A. Abdelkefi / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎12

To briefly show the effects of the inductance when connected in series to a load resistance on the onset speed of
instability, we plot the variations of the onset speed of galloping as functions of these parameters in Fig. 11 . Clearly, the main
differences between the series and parallel resistive–inductive configurations are:

In the parallel configuration, for very small values of L, the galloping system behaves as a short-circuit resistive
configuration. However, for high values of L, the system behaves as a resistive configuration.
Fig. 9. Variations of the onset speed of galloping as a function of the electrical load resistance when L ¼ L
opt
¼
1
ω
2
n
C
p
.
Fig. 10. Variations of the real parts of the eigenvalues when L ¼ L
max
¼
1
ω
2
n
C
p
and (a) R ¼ 10
5
Ω, (b) R ¼ 5  10
5
Ω, (c) R ¼ R
max
¼
1
jθj
ffiffiffiffi
C
p
p
, and (d) R ¼ 10
6
Ω.
Please cite this article as: H. Abdelmoula, & A. Abdelkefi, The potential of electrical impedance on the performance of
galloping systems for energy..., Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.037i
H. Abdelmoula, A. Abdelkefi / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 13


In the series configuration, when L goes to zero, the galloping-based piezoelectric system has the same characteristics as
the resistive configuration. On the other hand, when L goes to infinity, the characteristics of an open-circuit resistive
configuration is obtained. Same mathematical tools and strategies can be applied to determine the optimum values of the
load resistance R
max
and the inductance L
max
in order to maximize the onset speed of galloping.
Fig. 11. Variations of the onset speed of galloping as a function of (a) the inductance for distinct values of R and (b) the load resistance for various values of
L for series configuration.
Fig. 12. Bifurcation diagrams of the (a) tip displacement, (b) generated voltage, (c) output current, and (d) harvested power for various configurations of
R
opt
and C
opt
.
Please cite this article as: H. Abdelmoula, & A. Abdelkefi, The potential of electrical impedance on the performance of
galloping systems for energy..., Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.037i
H. Abdelmoula, A. Abdelkefi / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎14

4. Nonlinear responses and performances of resistive–capacitive–inductive galloping-based piezoelectric systems
4.1. Energy harvesting purposes
In this section, the effects of including electrical capacitance and inductance on the performance of a galloping-based
energy harvester are investigated. To this end, we plot in Figs. 12 and 13 the bifurcation diagrams of the tip displacement,
generated voltage, output current, and harvested power crossing the electrical load resistance when considering different
scenarios of the electrical capacitance and resistance. For a resistive configuration, it was demonstrated in the literature that
maximum amplitudes of the harvested power can be observed when the load resistance matches its optimum value [13].In
the plots in Fig. 12, we consider different configurations of R
opt
and C
opt
in such a way different optimum onset speeds of
galloping with respect to the electrical load resistance are obtained, namely, 10 m/s, 15 m/s, and 20 m/s. It should be
mentioned that all these values of onset speeds are less than the onset speed of the optimum resistive case. It follows from
these plots that decreasing the onset speed of galloping to 10 m/s, the harvester's oscillates for any speed higher than 10 m/s
compared to the other configurations. Although, for high wind speeds than 15 m/s, the second configuration (U
g
¼15 m=s)
gives higher levels of the harvested power, the first configuration (U
g
¼10 m=s) is very beneficial for speeds between 10 m/s
and 15 m/s by having important levels of harvested power that can operate many commercial devices.
From Fig. 3(b), it can be observed that, for purely resistive case, there are two values of R (R
1
¼4  10
4
and R
2
¼4  10
6
)
that give an onset speed U
g
¼11 m=s. Finding the corresponding R
opt
and C
opt
that gives the same onset speed of galloping,
we plot in Fig. 13 the bifurcation diagrams of the harvester's for the three mentioned configurations that give the same onset
speed of galloping in order to determine if the capacitive–resistive case is enough efficient than its resistive counterpart. It
follows from Fig. 13(d) that the levels of the harvested power are almost the same for the considered three configurations.
However, there is a big change in the output current crossing the load resistance and generated voltage which may be
Fig. 13. Bifurcation diagrams of the (a) tip displacement, (b) generated voltage, (c) output current, and (d) harvested power for different configurations that
give an onset speed of galloping equal to 11 m/s.
Please cite this article as: H. Abdelmoula, & A. Abdelkefi, The potential of electrical impedance on the performance of
galloping systems for energy..., Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.037i
H. Abdelmoula, A. Abdelkefi / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 15

beneficial when charging/discharging the battery. Therefore, depending on the type of the used battery, the value of the
electrical capacitance can be determined in order to have limit-cycle oscillations at the operating wind speed.
For energy harvesting purposes, the effects of the inductance on the performance of a galloping-based piezoelectric
energy harvester are very similar to the obtained results for the resistive–capacitive configuration. As an example, in Fig. 14,
we plot the bifurcation diagrams of the tip displacement, generated voltage, output current, and harvested power crossing
the load resistance for different values of L
opt
and R
opt
in such a way different optimum values of the onset of instability with
respect to R are obtained. It is clear that the inclusion of the electrical inductance is useful in order to decrease the onset
speed of galloping compared to the optimum resistive case. It can be concluded that the electrical capacitance and
inductance are not beneficial in terms of enhancing the level of harvested power compared to the classical resistive case.
However, they may be useful for energy harvesting applications in terms of charging/discharging different types of batteries.
4.2. Control purposes
As demonstrated in the performed linear analysis, the electrical inductance is very beneficial for control purposes by
increasing the onset speed of galloping to very high values. It is also shown that the electrical capacitance is not useful for
control purposes because its presence decreases the onset speed of instability and hence self-excited motions can be
observed for low wind speeds. As for the inductance, we show in Fig. 15 the bifurcation diagrams of the tip displacement for
various cases of the load resistance when setting the electrical inductance equals to its optimum value L
max
in order to get
maximum onset speed of galloping. The corresponding linear analysis curves of these cases are presented in Fig. 10. Clearly,
when R ¼ R
max
, the onset speed of galloping is increased by almost 20 times compared to its uncontrolled counterpart. In
addition to that, it is noted that the considered galloping-based piezoelectric system has a supercritical Hopf bifurcation and
hence is not affected by the initial conditions or possible gusts. In fact, when the wind speed is increased little bit higher
Fig. 14. Bifurcation diagrams of the (a) tip displacement, (b) generated voltage, (c) output current, and (d) harvested power for various configurations of
R
opt
and L
opt
.
Please cite this article as: H. Abdelmoula, & A. Abdelkefi, The potential of electrical impedance on the performance of
galloping systems for energy..., Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.037i
H. Abdelmoula, A. Abdelkefi / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎16

than the onset speed of instability, a smooth increase is observed in the tip displacement without the presence of any jumps
or hysteresis regions.
5. Conclusions
In this work, the potential of the electrical capacitance and inductance on the performance of galloping-based piezo-
electric systems for energy harvesting and control purposes have been determined. To this end, a nonlinear distributed-
parameter model has been developed. A quasi-steady approximation was used to model the galloping force. It was shown
that the inclusion of the electrical capacitance always decreases the onset speed of galloping compared to the classical
resistive configuration. The results showed that the capacitance is not beneficial in terms of enhancing the levels of the
harvested power for energy harvesting purposes and increasing the onset speed of instability for control purposes. However,
for energy harvesting applications, it was indicated that the presence of the capacitance may be useful when charging/
discharging batteries because it can control the generation of voltage and current crossing the load resistance. As for the
inclusion of the electrical inductance, it was demonstrated that its presence is very beneficial for control purposes by
increasing the onset speed of galloping to higher and higher speeds. For energy harvesting, it was shown that including an
inductance has similar impacts as including capacitance.
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Fig. 15. Bifurcation diagrams of the tip displacement when L ¼ L
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Please cite this article as: H. Abdelmoula, & A. Abdelkefi, The potential of electrical impedance on the performance of
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H. Abdelmoula, A. Abdelkefi / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 17

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Please cite this article as: H. Abdelmoula, & A. Abdelkefi, The potential of electrical impedance on the performance of
galloping systems for energy..., Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.037i
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