Nonlinear Dyn

DOI 10.1007/s11071-014-1355-8

ORIGINAL PAPER

Piezoelectric energy harvesting from concurrent

vortex-induced vibrations and base excitations

H. L. Dai · A. Abdelkefi · L. Wang

Received: 4 December 2013 / Accepted: 7 March 2014

© Springer Science+Business Media Dordrecht 2014

Abstract We investigate the potential of using a

piezoelectric energy harvester to concurrently harness

energy from base excitations and vortex-induced vibra-

tions. The harvester consists of a multilayered piezo-

electric cantilever beam with a circular cylinder tip

mass attached to its free end which is placed in a uni-

form air flow and subjected to direct harmonic exci-

tations. We model the fluctuating lift coefficient by a

van der Pol wake oscillator. The Euler–Lagrange prin-

ciple and the Galerkin procedure are used to derive a

nonlinear distributed-parameter model for a harvester

under a combination of vibratory base excitations and

vortex-induced vibrations. Linear and nonlinear analy-

ses are performed to investigate the effects of the elec-

trical load resistance, wind speed, and base acceleration

on the coupled frequency, electromechanical damp-

ing, and performance of the harvester. It is demon-

strated that, when the wind speed is in the pre- or post-

synchronization regions, its associated electromechan-

ical damping is increased and hence a reduction in the

H. L. Dai · L. Wang (B)

Department of Mechanics, Huazhong University of Science

and Technology, Wuhan 430074, China

e-mail: wanglindds@mail.hust.edu.cn

H. L. Dai · L. Wang

Hubei Key Laboratory for Engineering Structural Analysis

and Safety Assessment, Wuhan 430074, China

A. Abdelkefi

Department of Engineering Science and Mechanics,

MC 0219, Virginia Tech, Blacksburg, VA 24061, USA

harvested power is obtained. When the wind speed is

in the lock-in or synchronization region, the results

show that there is a significant improvement in the

level of the harvested power which can attain 150 %

compared to using two separate harvesters. The results

also show that an increase of the base acceleration

results in a reduction in the vortex-induced vibrations

effects, an increase of the difference between the res-

onant excitation frequency and the pull-out frequency,

and a significant effects associated with the quenching

phenomenon.

Keywords Energy harvesting · Piezoelectric

material · Vortex-induced vibrations · Combined

loadings · Nonlinear analysis · Quenching phenomenon

1 Introduction

Energy harvesting through conversion of wasted

mechanical energy to electrical power has received sig-

nificant attention over the past decade [1–5]. These

energy harvesting systems have been proposed to oper-

ate self-powered electronics, such as wireless and

health monitoring sensors, data transmitters, medical

implants, and cameras [6–9]. These harvesters are also

designed to replace small batteries that have a finite life

span and would require expensive or time consuming

replacement. Of the different options of transduction

mechanisms which include electrostatic [3], electro-

magnetic [3], and piezoelectric [1,3], the piezoelectric

123

H. L. Dai et al.

one has received the most attention because of its ease

of application and it can effectively be placed in small

volumes. In addition, the piezoelectric option does not

require an external input voltage.

Several concepts have been introduced in order to

harvest energy from unused mechanical energy avail-

able in the environment. The most common concept

consists of a piezoelectric cantilever beam subjected

to external excitations applied at its base. When the

excitation frequency matches the natural frequency of

the harvester, resonant motions are obtained and then

the piezoelectric material strains and produces an elec-

trical charge and hence an output voltage is obtained

through an electrical load resistance. Anton and Sodano

[3] presented different strategies to enhance the per-

formance of these harvesters including the use of effi-

cient piezoelectric materials, use of distinct mode cou-

pling, and optimization of the power conditioning cir-

cuitry. For the same purpose, different techniques have

been employed, such as beam geometry [10,11], mul-

timodal systems [12,13], magnetic coupling [14–16],

and bistable configurations [17–19].

There has also been recent interest in the concept

of harvesting energy from aeroelastic or flow-induced

vibrations, such as flutter of airfoil sections [20–24],

vortex-induced vibrations (VIVs) of circular cylinders

[25–28], galloping of prismatic structures [29–33], and

wake galloping of parallel cylinders [34,35]. In this

concept, the piezoelectric energy harvester is placed in

a flow field and excited to undergo large limit-cycle

oscillation amplitudes that can be converted to usable

electrical power. Depending on the operating wind

speed, these piezoaeroelastic systems can be designed

and deployed in different locations including ventila-

tion outlets, rivers, bridges, structure’s surface, etc.

Almost all previous research studies have consid-

ered a single excitation source, either base or aeroelas-

tic. There are only three studies which investigated the

idea of energy harvesting from a combination of both

sources of excitation. Bibo and Daqaq [36,37] inves-

tigated theoretically and experimentally the response

behavior of a flutter-based piezoaeroelastic energy

harvester when subjected to a combination of vibra-

tory base excitations and aerodynamic loadings. They

demonstrated that the efficiency as well as the power

density of the harvester are significantly improved

when the wind speed is either below or above the linear

flutter speed. Furthermore, there is only one theoreti-

cal study investigating the influence of both vibratory

base and aeroelastic excitations on a galloping-based

piezoaeroelastic energy harvester [38]. Yan et al. [38]

investigated the effects of the base acceleration, wind

speed, and electrical load resistance on the performance

of the harvester and the associated nonlinear phenom-

ena that take place.

Based on the authors’ knowledge, no research study

has considered the potential of integrating both vibra-

tory base and aeroelastic excitations on a VIV-based

piezoaeroelastic energy harvester. In this work, we aim

to investigate the transduction of piezoelectric energy

harvesting under a combination of vibratory base exci-

tations and vortex-induced vibrations. To this end, a

multilayered piezoelectric cantilever beam with a cir-

cular cylinder-tip mass attached to its free end is con-

sidered. The design of the proposed harvester and the

derived distributed-parameter model are presented in

Sect. 2. In Sect. 3, the derived model is validated

with previous experimental measurements when only

vortex-induced vibrations are present. Linear and non-

linear analyses are performed in Sects. 4 and 5, respec-

tively, in order to investigate the effects of the electri-

cal load resistance, wind speed, and base acceleration

on the coupled frequency, electromechanical damping,

and performance of the harvester. Summary and con-

clusions are presented in Sect. 6.

2 Mathematical model

2.1 Model formulation

The piezoaeroelastic energy harvester under consid-

eration consists of a bimorph piezoelectric cantilever

beam with a circular cylinder attached to its free end,

as shown in Fig. 1. This harvester is placed in a uni-

form air flow with mean flow speed, U0, and subjected

to an external harmonic base motion, zb = z0cos(Ωt),

where z0 is the amplitude of the base displacement and

Ω is the base excitation frequency. In addition to the

base motion, when the shedding frequency is close to

the coupled frequency of the harvester, the harvester

can undergo steady-state limit-cycle oscillations in the

cross flow direction. The piezoelectric sheets are placed

on both sides of the cantilever beam. The two piezo-

electric sheets are bounded by two in-plane electrodes

of negligible thickness connected in series to an elec-

trical load resistance, R. The thickness of the beam is

assumed to be small compared to its length so that the

123

Piezoelectric energy harvesting from vibrations and excitations

L b

Lp

m0

U0

D y

zx

o

( )w

tx1zb

Fig. 1 Schematic of the piezoelectric energy harvester under

concurrent loadings

shear deformation and rotary inertia can be neglected.

The geometric and material properties of the system

are presented in Table 1.

To develop a reduced–order model of the considered

energy harvester, we apply the Galerkin discretization

to the Lagrangian (L = T − U ) and nonconservative

work (Wnc). To this end, we first calculate the total

potential energy U , kinetic energy T , and the noncon-

servative work. The total potential energy of the mul-

tilayered cantilever beam is given by:

U =1

2

⎡

⎢

⎣

∫

Vb

σ bx εb

x dVb+

∫

Vp

σp

x εpx dVp−

∫

Vp

E3 D3dVp

⎤

⎥

⎦,

(1)

where Vb and Vp are, respectively, the volumes of the

substrate and the piezoelectric sheets. ρb and ρp are the

densities of the substrate and piezoelectric sheets. σx

and εx are the stress and strain in the axial direction.

Using Hooke’s law and the linear constitutive relations

of piezoelectricity, the strains and stresses in the alu-

minum and piezoelectric layers are, respectively, given

by:

εbx = ε

px = εx = −z

∂2w(x, t)

∂x2, (2)

σ bx = Ebεx , (3)

σp

x = E p (εx − d31 E3) = E pεx − e31 E3, (4)

where w(x, t) denotes the relative displacement of the

beam. Eb and E p are Young’s moduli at constant elec-

tric field of the aluminum and piezoelectric layers,

d31 is the strain coefficient of the piezoelectric sheet,

e31 = E pd31 is the piezoelectric stress coefficient,

and E3 is the electric field developed in the piezoelec-

tric sheets. For series connections of the piezoelectric

sheets, the electric field is related to the voltage, V (t),

by E3 = −V (t)/2tp, where tp is the thickness of the

piezoelectric sheet [39].

In addition, D3 is the electric displacement given by

the following piezoelectric constitutive relation:

D3 = d31 E pεx + ε33 E3 = e31εx + ε33 E3, (5)

where ε33 is the permittivity component at constant

strain.

The kinetic energy of the harvester is the sum of the

kinetic energies of the beam and tip mass. Taking into

account for the moment of inertia of the cylindrical tip

mass, the kinetic energy of the system is expressed as:

T =1

2

⎡

⎢

⎣

∫

Vb

ρb

[

∂w(x, t)

∂t+

∂zb(t)

∂t

]2

dVb

+

∫

Vp

ρp

[

∂w(x, t)

∂t+

∂zb(t)

∂t

]2

dVp

]

Table 1 Dimensions and

properties of the harvester

components

Physical properties PZT elements Beam Cylinder

Length, L p, Lb, L0 (mm) 31.8 267 203

Width, wp, wb (mm) 25.4 32.5 –

Thickness, tp, tb (mm) 0.267 0.635 –

Diameter, D (mm) – – 19.8

Mass dentisy, ρp, ρb(kg/m3) 7800 2730 –

Mass, m p, mb, Mt (g) 1.68 15.8 10

Young’s modulus, E p, Eb (GPa) 66 70 –

Strain coefficient, d31

(

pm V−1)

−190 – –

Permittivity at constant strain, ε33

(

nFm−1)

13.28 – –

Capacitance, C p (nF) 20.3 – –

123

H. L. Dai et al.

+1

2Mt

[

∂w(x, t)

∂t

∣

∣

∣

∣

x=Lb

+∂zb(t)

∂t

+D

2

∂2w(x, t)

∂x∂t

∣

∣

∣

∣

x=Lb

]2

+1

2It

[

∂2w(x, t)

∂x∂t

∣

∣

∣

∣

x=Lb

]2

, (6)

where Mt is the mass of the cylinder at the free end

of the beam and It mass moment of inertia defined as

It = 1/3 Mt (D/2)2, which is considered here because

the radius of the cylinder cannot be neglected.

As for the nonconservative work, three major com-

ponents are present which are the work due to the power

delivered to an electrical load resistance R, the work

done by the aerodynamic forces, F(t), and the work

done by the damping force. Hence, the variation of the

nonconservative work term can be expressed as

δWnc = −V δQ R + F(t)

(

δw(L , t) +D

2δw′(L , t)

)

−

Lb∫

0

c∂w

∂tδw(x, t)dx, (7)

where the prime denotes the derivative with respect

to x and c represents the viscous damping coefficient.

Q R denotes the extracted charge and is related to the

generated voltage through the Ohm’s law as follows:

V (t) = RdQ R(t)

dt. (8)

2.2 Aerodynamic force representation

The representation of the vortex-induced forces is very

challenging [40] and several researchers have focused

on this topic [41–48]. The widely employed approach

is the so-called wake oscillator model, which was first

put forward by Birkhoff and Zarantanello [41]. Bishop

and Hassan [42] were the first to suggest modeling the

lift over a circular cylinder due to vortex shedding by a

nonlinear self-excited oscillator. A Rayleigh oscillator

was used by Hartlen and Currie [43] in order to model

the lift force and they considered a velocity term to

couple it with the motion of the cylinder. Skop and

Griffin [44,45] used a modified van der Pol oscillator

to represent the vortex-induced force and they coupled

it with the motion of the cylinder by a velocity term.

Subsequently, the excellent insights into the workings

of wake oscillator model were contributed by Skop and

Balasubramanian [46] and Facchinetti et al. [47]. Skop

and Balasubramanian [46] recommended a stall term

in the description of the cross-flow force. However,

Facchinetti et al. [47] systematically investigated the

pattern of the force induced by the cylinder’s motion

on the wake oscillator. Furthermore, they found that an

acceleration coupling term is better than a velocity term

to model the lock-in region. The representation of the

vortex-induced force and its wake oscillator were then

confirmed by Violette et al. [48] through comparison

with numerical simulation and experiments.

Following Facchinetti et al. [47], the vortex-induced

force is expressed as:

F(t) =CLρ0 DU 2

0 L0

2−

CDρ0 DU0 L0

2

[

∂w(x, t)

∂t

∣

∣

∣

∣

x=Lb

+D

2

∂2w(x, t)

∂x∂t

∣

∣

∣

∣

x=Lb

]

, (9)

where ρ0 is the density of fluid, L0 is the length of

the cylinder, and D is the diameter of the cylinder. The

fluctuating lift coefficient is denoted by CL = CL02

q(t).

CL0 and CD are the steady lift and mean sectional

drag coefficients which can be, respectively, consid-

ered equal to 0.3 and 1.2 in the region of well-developed

wakes [47].

The wake oscillator employed in the present study

is based on the model of Facchinetti et al. [47] which

states:

q + λωs

(

q2 − 1)

q + ω2s q =

A

D

d2w(t)

dt2

=A

D

[

∂2w(x, t)

∂t2

∣

∣

∣

∣

x=Lb

+∂2zb(t)

∂t2

+D

2

∂3w(x, t)

∂x∂t2

∣

∣

∣

∣

x=Lb

]

, (10)

where q(t) describes the behavior of the near wake and

stands for the lift acting on the cylinder. w(t) is the

displacement of the cylinder. The values of λ and A are

constants and are identified from the experimental mea-

surements. ωs is the vortex-shedding frequency, which

is defined as ωs = 2π StU0/D, St being the Strouhal

number which is determined by the Reynolds numbers

range [49] and U0 represents the wind speed.

123

Piezoelectric energy harvesting from vibrations and excitations

2.3 Reduced-order model

The Galerkin procedure is used to develop a reduced-

order model of this energy harvester. Therefore, we

express the displacement of the harvester in the fol-

lowing form:

w(x, t) =

n∑

i=1

ϕi (x)ri (t), (11)

where ri (t) are the modal coordinates of the displace-

ment and ϕi (x) are the mode shapes of a cantilever

beam with a tip mass. Taking into account for the iner-

tial force at the offset D/2 from the free end of the

beam and the rotational moment of inertia of the cylin-

der around the end of the beam, the mode shapes of this

system can be expressed as [50]:

ϕi (x) = C1 [cos (βi/Lbx) − cosh (βi/Lbx)

+ ξi (sin (βi/Lbx) − sinh (βi/Lbx))] , (12)

where

ξi =sin(βi ) − sinh(βi ) + κ1βi [cos(βi ) − cosh(βi )] − κ1κ2β

2i

[sin(βi )+sinh(βi )]

cos(βi ) + cosh(βi ) − κ1βi [sin(βi ) − sinh(βi )] − κ1κ2β2i

[cos(βi ) − cosh(βi )]. (13)

In Eq. (13), κ1 represents the mass ratio of the cylin-

der and the beam system, κ2 indicates the ratio of the

cylinder radius and beam length, and the frequency

parameters βi are the roots of the following frequency

equation

1 + cos(β) cosh(β) + κ1β [cos(β) sinh(β)

− cosh(β) sin(β)] − 2κ1κ2β2 sin(β) sinh(β)

− 4/3κ1κ22 β3 [cos(β) sinh(β) + cosh(β) sin(β)]

+ 1/3κ21 κ2

2 β4 [1 − cos(β) cosh(β)] = 0. (14)

To determine the modal amplitude constant, C1, we

apply the following orthogonality conditions:∫

Vb

ϕi (x)ρbϕ j (x)dVb +

∫

Vp

ϕi (x)ρpϕ j (x)dVp

+ ϕ′i (Lb)Jϕ′

j (Lb) + m0

[

ϕi (Lb) + D/2ϕ′i (Lb)

]

×[

ϕ j (Lb) + D/2ϕ′j (Lb)

]

= δi j

∫

Vb

z2ϕ′′i (x)Ebϕ′′

j (x)dVb

+

∫

Vp

z2ϕ′′i (x)E pϕ′′

j (x)dVp = ω2i δi j ,

(15)

where δi j is the Kronecker delta, defined as being equal

to unity for i = j and equal to zero for i = j . And ωi

is the undamped natural frequency of the i th vibration

mode.

The discretization is performed by substituting

Eq. (11) into Eqs. (1), (6), and (7). After that, we use

the Euler–Lagrange equations which are based on the

following relations:

∂

∂t

∂L

∂ ri

−∂L

∂ri

=δWnc

δri

(i = 1, 2, . . . , n), (16)

∂

∂t

∂L

∂λ−

∂L

∂λ=

δWnc

δλ= −

V

R, (17)

where λ = V .

Substituting Eq. (11) in Eq. (10) and simplifying

Eqs. (16) and (17), the reduced-order model of this

energy harvester can be expressed as:

ri +

⎧

⎨

⎩

2ςiωi +CDρ0 DU0 L0

2

[

ϕi (Lb) +D

2φ′

i (Lb)

]

×

n∑

j=1

(

ϕ j (Lb) +D

2ϕ′

j (Lb)

)

⎫

⎬

⎭

r j + ω2i ri − θi V

= f (t)(i = 1, 2, 3 . . . , n) (18)

C p V +V

R+

n∑

i=1

θi ri = 0 (19)

q + λωs(q2 − 1)q + ω2

s q

=A

D

[

n∑

i=1

([

ϕi (Lb)+D

2ϕ′

i (Lb)

]

ri

)

−a cos (t)

]

.

(20)

The external force f (t) is given by:

f (t) = αi q +

⎧

⎪

⎨

⎪

⎩

x1∫

0

m1ϕi (x)dx +

x1+L p∫

x1

m2ϕi dx

+

Lb∫

x1+L p

m1φi dx

+Mt

[

ϕi (Lb) +D

2ϕ′

i (Lb)

]

⎫

⎪

⎬

⎪

⎭

[a cos(t)] ,

(21)

123

H. L. Dai et al.

Fig. 2 Variation of the

maximum generated voltage

as a function of the wind

speed for different modes in

the Galerkin discretization

when R = 2.46 × 106

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.20

5

10

15

20

251st mode

2nd mode

3rd mode

4th mode

5th mode

()

Vp

V

( )0 m sU

where a = zb2 is the base acceleration, m1 =

wbρbtb, m2 = wbρbtb + 2wpρptp, and the rest of the

coefficients are given by:

αi = CL0ρ0 DU 20 L0

[

ϕi (Lb) + D/2ϕ′i (Lb)

]

/4,

(22)

C p =ε33wp L p

2tp

, (23)

θi =e31wp

(

tp + tb)

2(ϕ′

i (x1 + L p) − ϕ′i (x1)). (24)

3 Model validation

In Sect. 2, a reduced-order model for the piezoelec-

tric energy harvester concurrently subjected to base

excitations and vortex-induced vibrations has been

developed. To determine the accuracy of the derived

distributed-parameter model, we first compare our

numerical predictions with the experimental measure-

ments of Akaydin et al. [26]. In their experiments, only

vortex-induced vibrations were present.Therefore, the

base acceleration is considered to be zero in the derived

reduced-order model. To make an accurate comparison,

we plot, in Fig. 2, the predicted numerical values of the

generated voltage as a function of the wind speed for

different number of modes in the Galerkin procedure

when the load resistance is set equal to 2.46×106. We

have to mention that this plot is performed for the same

geometric and physical properties as used by Akaydin

et al. [26]. It follows from the plotted curves in Fig. 2

that one mode, two, or three modes in the Galerkin pro-

cedure are not sufficient to get convergence. Clearly,

four or five modes are required to get an excellent con-

vergence for the generated voltage and hence the har-

vested power. In the rest of this study, the numerical

predictions are performed when using five modes in

the Galerkin discretization.

A comparison between the numerical predictions

of our developed distributed-parameter model and the

experimental measurements of Akaydin et al. [26] is

performed. The plotted curves in Fig. 3a show the vari-

ation of the average harvested power as a function of

the wind speed when the load resistance is set equal to

2.46×106. The average harvested power is calculated

as follows:

Pe =V 2

rms

R, (25)

where Vrms denotes the root mean square (RMS) value

of the generated voltage.

It follows from the plotted curves in Fig. 3a that there

is a very good agreement between the numerical predic-

tions of the derived model and the experimental mea-

surements of Akaydin et al. [26]. Clearly, the synchro-

nization region is well-predicted by the derived model

and particularly its starting and its peak. In fact, the

maximum value of the average power is the same and

equal to 0.1 mW when the wind speed is U0 = 1.20 m/s.

However, in the post-synchronization region, there is

an underestimation in the average harvested power

123

Piezoelectric energy harvesting from vibrations and excitations

0.5 1 1.2 1.5 2 2.3

0

0.02

0.04

0.06

0.08

0.1

0.12Simulation

ExperimentAkaydin et al.(2012)

()

mW

eP

( )0 m sU

0.5 0.8 1.1 1.4 1.7 2 2.3

0

0.05

0.1

0.15

0.2

0.25

()

mW

eP

( )0 m sU

(b)(a)

Fig. 3 a Comparison between the numerical predictions of the

derived distributed parameter model and the experimental mea-

surements of Akaydin et al. [26] and b Variation of the average

harvested power as a function of the wind speed when consider-

ing the geometric and physical parameters presented in Table 1

when considering the derived model for wind speed

values larger than 1.4 m/s. This result is probably due

to the accuracy of the wake oscillator model to the pre-

synchronization and synchronization regions.

We plot, in Fig. 3b, the variation of the average har-

vested power as a function of the wind speed when

considering the geometric and physical parameters pre-

sented in Table 1. In this plot, the electrical load resis-

tance is set equal to 106 and the base acceleration is

set equal to zero. It follows from the plotted curves in

Fig. 3(b) that the synchronization region takes place

when the wind speed is near U0 = 1.4 m/s.

4 Linear analysis: effects of the load resistance

and wind speed on the coupled frequency

and electromechanical damping

To investigate the effects of the electrical load resis-

tance and wind speed on the coupled frequency and

electromechanical damping, a linear analysis is per-

formed in which the base excitation terms as well

as the nonlinear terms in Eqs. (18) and (20) are

dropped. Because piezoaeroelastic systems are gener-

ally designed to harvest energy at low wind speeds and

hence at low shedding frequencies (ωs = 2π StU0/D),

the linear analysis is performed near the fundamen-

tal natural frequency of the harvester (ω1) and conse-

quently only one mode in the Galerkin discretization

is considered. Hence, the equations of motion can be

expressed as:

r1+

[

2ς1ω1+CDρ0 DU0 L0

2

(

ϕ1(Lb)+D

2ϕ′

1(Lb)

)2]

r1

+ ω21r1 − θ1V − α1q = 0 (26)

C p V +V

R+ θ1r1 = 0 (27)

q − λωs q + ω2s q −

A

D

(

ϕ1(Lb) +D

2ϕ′

1(Lb)

)

r1 = 0.

(28)

Then, we introduce the following state variables:

X =

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎣

X1

X2

X3

X4

X5

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎦

=

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎣

r1

r1

V

q

q

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎦

. (29)

The Eqs. (26), (27), and (28) can be rewritten as:

X = B X, (30)

123

H. L. Dai et al.

102

103

104

105

106

107

108

109

1010

22.5

23

23.5

24

24.5

25

( )R Ω

()

Fre

qu

ency

ra

d s

102

103

104

105

106

107

108

109

1010

1

2

3

4

5

( )R Ω

()

2C

ouple

d d

ampin

g r

atio

10

−

(b)(a)

Fig. 4 Variations of the a coupled frequency and b electromechanical damping as a function of the electrical load resistance when

U0 = 0 m/s

where

B =

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

0 1 0 0 0

−ω21 −

(

2ς1ω1 +CDρ0 DU0 L0

2ψ2

1

)

θ1 α1 0

0 − θ1C p

− 1RC p

0 0

0 0 0 0 1

− AD

ψ1ω21 − A

Dψ1

(

2ς1ω1 +CDρ0 DU0 L0

2ψ2

1

)

AD

ψ1θ1AD

ψ1α1 − ω2s λωs

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

(31)

and ψ1 = ϕ1(Lb) + D2ϕ′

1(Lb).

The matrix B has a set of five eigenvalues λi ,

i = 1, 2, 3, 4, 5. The first two eigenvalues are complex

conjugates and they are associated with the transverse

displacement motion of the cantilever beam. In fact,

the real part of these two eigenvalues represents the

electromechanical damping coefficient and their pos-

itive imaginary part represents the coupled frequency

of the harvester. These two eigenvalues depend on the

electrical load resistance and the wind speed when the

rest of the parameters are determined based on the given

values in Table 1. The third eigenvalue λ3 is a result of

the piezoelectric coupling and it is always real and neg-

ative. As for λ4 and λ5, they are complex conjugates

and they are associated with the response of the fluctu-

ating lift coefficient.

The plotted curves in Fig. 4a, b, respectively, depict

the variations of the coupled frequency and the electro-

mechanical damping as a function of the electrical load

resistance when the wind speed is set equal to zero.

Inspecting Fig. 4a, there is a steep increase over the

electrical load resistance values between 4×105 and

107. In addition, when the load resistance is smaller

than 2 × 105 or larger than 2 × 107, the cou-

pled frequency is almost constant with different values.

When the load resistance is near 102; the coupled fre-

quency is equal to 22.7 rad/s. We refer to this frequency

as the short coupled frequency. On the other hand,

when the load resistance is near 108; the coupled fre-

quency of the harvester is equal to 24.85 rad/s. This

frequency is referred as the open coupled frequency.

Clearly, an increase of the electrical load resistance

results in an increase of the onset of synchronization.

Figure 4b shows that the electromechanical damping

is relatively high when the electrical load resistance

is between 4 × 105 and 107. This region of load

123

Piezoelectric energy harvesting from vibrations and excitations

0 0.5 1 1.5 2

22

23

24

25

26

R=102Ω

R=106Ω

R=109Ω

0 0.5 1 1.5 20

2

4

6

8

10

12

R=102Ω

R=106Ω

R=109Ω

( )0 m sU ( )0 m sU

(b)(a)(

)F

req

uen

cy

rad

s

()

2C

ouple

d d

ampin

g r

atio

10

−

Fig. 5 Variations of the a coupled frequency and b electromechanical damping as a function of the wind speed for different load

resistances

resistances over which the electromechanical damp-

ing is high coincides with the region over which the

steep increase of the coupled frequency occurs. It is also

noted that small values of the electromechanical damp-

ing are obtained when the load resistance is smaller

than 104 or larger than 108. These two regions

coincide with the constant values of the coupled fre-

quency (short- and open-coupled frequencies, respec-

tively). Clearly, these effects of the electrical load resis-

tance on the coupled frequency and electromechanical

damping have a significant impact on the displacement

and harvested power.

To determine the influence of the wind speed on the

coupled frequency and electromechanical damping of

the harvester, we follow the same performed strategy in

Fig. 4. To this end, we plot, in Fig. 5a, b, the variations

of the coupled frequency and electromechanical damp-

ing as a function of the wind speed for three distinct

values of the electrical load resistance. It follows from

Fig. 5a that the variation of coupled frequency with the

wind speed has the same tendency for different values

of the electrical load resistance. In fact, the coupled fre-

quency decreases when the wind speed varies from 0

to 1 m/s. Then, it increases significantly when the wind

speed is between 1 and 1.7 m/s. Between 1.7 and 2 m/s,

the variation of the coupled frequency is almost negligi-

ble. It is also noted that the coupled frequency increases

when the electrical load resistance is increased for all

values of wind speed between 0 and 2 m/s. Inspecting

Fig. 5b, we note that there is a maximum value of the

electromechanical damping for a specified value of the

wind speed. However, this specific value of the wind

speed changes with the considered value of the electri-

cal load resistance. In fact, an increase of the electrical

load resistance value is accompanied by an increase

of the wind speed value that gives maximum electro-

mechanical damping. This result can be explained by

the presence of the drag force and the influence of the

load resistance on the coupled frequency. Furthermore,

maximum values of the electromechanical damping are

always obtained for an electrical load resistance around

106.

This performed linear analysis gives a clear idea

about the effects of the electrical load resistance and

wind speed on the onset of synchronization or coupled

frequency and electromechanical damping variations.

These effects have an important impact in determining

the synchronization region and the level of the har-

vested power.

5 Nonlinear analysis: effects of the load resistance,

wind speed and base acceleration

on the harvester’s performance

In the presence of the base and vortex-induced exci-

tations, the frequency-response curves as well as the

response behavior of the harvester are expected to be

strongly dependent on the value of wind speed which

can change the response of the harvester from periodic

123

H. L. Dai et al.

10 15 20 25 30 35

0

0.05

0.1

0.15

0.2

0.25

0.3U

0=0

U0=0.6m/s

U0=1.2m/s

U0=1.4m/s

U0=1.6m/s

U0=1.8m/s

( )rad sΩ

()

mW

eP

()

mW

eP(

)m

WeP(

)m

WeP

10 15 20 25 30 35

0

0.1

0.2

0.3

0.4

0.5U

0=0

U0=0.6m/s

U0=1.2m/s

U0=1.4m/s

U0=1.6m/s

U0=1.8m/s

(a)

a=0.005g

(b)

a=0.01g

10 15 20 25 30 350

0.2

0.4

0.6

0.8

1

1.2U

0=0

U0=0.6m/s

U0=1.2m/s

U0=1.4m/s

U0=1.6m/s

U0=1.8m/s

10 15 20 25 30 35

0

0.5

1

1.5

2U

0=0

U0=0.6m/s

U0=1.2m/s

U0=1.4m/s

U0=1.6m/s

U0=1.8m/s

(c)

a=0.03g

(d)

a=0.05g

( )rad sΩ

( )rad sΩ( )rad sΩ

Fig. 6 Frequency-response curves of the harvested power when the load resistance is set equal to 106 for different base accelerations

and wind speeds

to period-n or quasi-periodic. This dependence is due

to the presence of two frequencies which are the excita-

tion frequency and the shedding frequency. For a con-

stant wind speed value, the shedding frequency is fixed

through this relation ωs = 2π StU0/D. At this wind

speed, if the shedding frequency matches the coupled

frequency of the harvester, resonant transverse vibra-

tions occur and then lock-in or synchronization takes

place. Furthermore, if the excitation frequency matches

the coupled frequency of the harvester, resonant trans-

verse vibrations take place. Consequently, a significant

improvement in the level of the harvested power can

be observed if both the excitation frequency and the

shedding frequency match the coupled frequency of

the harvester. Away from the synchronization region,

there are no oscillations from the aerodynamic loading

and one expects that the harvester response is always

periodic and only contains the excitation frequency. On

the other hand, when the shedding frequency is close

to the coupled frequency, the response of the harvester

under the concurrent dynamic loads becomes complex.

This complexity depends on the values of the excitation

frequency and the coupled frequency of the harvester.

It is expected that the response of the harvester is com-

posed of a free-oscillation one due to vortex-induced

vibrations effect and a forced-oscillation one due to

base excitation. As such, the harvester mainly oscillates

with two harmonic frequencies which are the excitation

frequency and the shedding frequency. The existence

of these two frequencies is right only when these fre-

quencies are not commensurable (off-resonance). In

this case, the response of the harvester is generally

aperiodic. Away from the resonance region, the oscil-

lations due to vortex-induced vibrations are expected

to dominate. This process of unlocking between the

two frequencies is called pulling-out. When the exci-

123

Piezoelectric energy harvesting from vibrations and excitations

tation frequency is close to the coupled frequency of

the harvester (resonance region), its response changes

significantly and the oscillations due to the vortex-

induced vibrations entrained or locked onto the forced

oscillations. Therefore, a synchronization response at

the excitation frequency takes place. In addition, the

free-oscillation contribution decays with time and the

steady-state motion becomes periodic. The process

of decaying the free-oscillation contribution is called

quenching and the frequency at which the system

changes from aperiodic response to periodic one is

called pull-out frequency [51]. To demonstrate these

expectations, we investigate the effects of the electrical

load resistance, wind speed, and base acceleration on

the response of the harvester and its performance.

We plot, in Fig. 6a–d, the effects of the base acceler-

ation on the frequency-response curves of the harvester

for different wind speed values when the electrical load

resistance is set equal to 106. It follows from these

plots that the value of the resonant excitation frequency

changes when the wind speed is changed. In fact, when

the wind speed is increased from 0 to 1.2 m/s, the res-

onant excitation frequency decreases. When the wind

speed is increased from 1.2 to 1.6 m/s, the resonant

excitation frequency is increased. This result is pre-

dicted because the coupled frequency of the harvester

varies with the wind speed, as shown in Fig. 5a. Further-

more, inspecting the off-resonance regions, it is clear

that the effects of the vortex-induced vibrations are

important for wind speed values equal to 1.2, 1.4, and

1.6 m/s. This result is also expected because the syn-

chronization region takes place for wind speed values

between 1.1 and 1.7 m/s, as shown in Fig. 3b. For wind

speed values equal to 0.6 and 1.8 m/s, the tendency of

the frequency-response curves are similar to the case

of base excitation without vortex-induced vibrations

effect. This is explained due to the fact that the vortex-

induced vibrations effect is vanished in the pre- and

post-synchronization regions, as shown in Fig. 3b. In

addition, in the absence of the vortex-induced vibra-

tions effect, it is noted that more power can be har-

vested when U0 = 0 m/s compared to its counterpart

when U0 = 0.6 and 1.8 m/s. This result is due to higher

values of the electromechanical damping at U0 = 0.6

and 1.8 m/s compared to its counterpart at U0 = 0 m/s,

as shown in Fig. 5b. Consequently, in the absence of

the vortex-induced vibrations effect which corresponds

to the cases when the wind speed is not between 1.1

and 1.7 m/s, there is a reduction in the harvested power

which due to the electromechanical damping effect (see

Fig. 5b). It follows from Fig. 6c, d that for high val-

ues of the base acceleration, the maximum harvested

power when U0 = 1.6 m/s is larger than its counterpart

when U0 = 1.2 and 1.4 m/s although higher values

of the harvested power are obtained at these speeds

in the off-resonance regions. This result is also due to

the electromechanical damping effect which is much

higher at U0 = 1.2 and 1.4 m/s, as shown in Fig. 5b.

The typical time history curves of the voltage are plot-

ted in Fig. 7, which give a clear idea that the harvester

changes between aperiodic and periodic motions.

The plotted curves in Fig. 8a–d depict the frequency-

response curves of the harvested power for various

values of the electrical load resistance and four val-

ues of the wind speed. The base acceleration is set

equal to 0.01 g. It follows from these plots that the

harvested power is strongly dependent on the electri-

cal load resistance. In the case without vortex-induced

vibrations effect (U0 = 0 m/s), Fig. 8a shows that max-

imum values of the harvested power are obtained when

R = 105 and 107, respectively. Furthermore, the

value of the resonant excitation frequency depends on

the considered value of the electrical load resistance.

In fact, an increase of the electrical load resistance

is followed by an increase of the resonant excitation

frequency. This is expected because the coupled fre-

quency of the harvester has the same variations with

the electrical load resistance, as presented in Fig. 4b.

At U0 = 0.6 m/s, Fig. 8b shows that maximum val-

ues of the harvested are obtained when R = 107

and 105, respectively. This change in the optimum

electrical load resistance when varying the wind speed

from 0 to 0.6 m/s can be explained due to the fact that

increasing the electrical load resistance and the wind

speed from 0 to 0.6 m/s is accompanied by a significant

change in the electromechanical damping. Inspecting

Fig. 8a, b, it is noted that there is a reduction in the level

of harvested power when the wind speed is increased

from 0 to 0.6 m/s for all considered values of electri-

cal load resistance. This result is predicted because the

electromechanical damping increases when the wind

speed is increased from 0 to 0.6 m/s for all considered

values of electrical load resistance, as shown in Fig. 5b.

When U0 = 1.2 and 1.6 m/s, it is clear that the

vortex-induced vibrations effect takes place. In the off-

resonance regions, we note that higher levels of the har-

vested power are obtained when R = 107 and 105,

respectively. Furthermore, it is noted that the harvested

123

H. L. Dai et al.

0 10 20 30 40-30

-20

-10

0

10

20

30

t (s)

V (

V)

0 10 20 30 40-30

-20

-10

0

10

20

30

t (s)

V (

V)

(b)(a)

Fig. 7 Time history of the voltage for different excitation frequencies a = 20 rad/s, b = 23 rad/s with a = 0.01 g, U0 = 1.2 m/s

when the load resistance is set equal to 106

10 15 20 25 30 35

0

0.01

0.02

0.03

0.04

0.05

0.06R=104

Ω

R=105Ω

R=106Ω

R=107Ω

R=108Ω

10 15 20 25 30 35

0

0.01

0.02

0.03R=104

Ω

R=105Ω

R=106Ω

R=107Ω

R=108Ω

10 15 20 25 30 350

0.05

0.1

0.15

0.2

0.25

0.3

0.35R=104

Ω

R=105Ω

R=106Ω

R=107Ω

R=108Ω

10 15 20 25 30 350

0.2

0.4

0.6

0.8R=104

Ω

R=105Ω

R=106Ω

R=107Ω

R=108Ω

()

mW

eP

()

mW

eP(

)m

WeP

()

mW

eP

( )rad sΩ ( )rad sΩ

( )rad sΩ ( )rad sΩ

U0=0.6m/sU0=0

U0 U 0=1.6m/s=1.2m/s

(a) (b)

(c) (d)

Fig. 8 Frequency-response curves of the harvested power for different load resistances and wind speeds when the base acceleration is

set equal to 0.01 g

123

Piezoelectric energy harvesting from vibrations and excitations

10 15 20 25 30 350

0.2

0.4

0.6

0.8

1a=0.005g

a=0.01g

a=0.03g

a=0.05g

10 15 20 25 30 350

0.2

0.4

0.6

0.8a=0.005g

a=0.01g

a=0.03g

a=0.05g

U0=0.6m/sU0=0

10 15 20 25 30 350

0.4

0.8

1.2

1.6a=0.005g

a=0.01g

a=0.03g

a=0.05g

10 15 20 25 30 350

0.5

1

1.5

2

2.5a=0.005g

a=0.01g

a=0.03g

a=0.05g

U0 U 0=1.6m/s=1.2m/s

()

mW

eP

()

mW

eP(

)m

WeP

()

mW

eP

( )rad sΩ ( )rad sΩ

( )rad sΩ ( )rad sΩ

(a) (b)

(c) (d)

Fig. 9 Frequency-response curves of the harvested power for different base accelerations and wind speeds when the load resistance is

set equal to 106

power is higher when U0 = 1.6 m/s than its counter-

part when U0 = 1.2 m/s for all considered values of the

electrical load resistance except R = 106. Clearly,

the wide of the synchronization or lock-in region of the

vortex-induced vibrations is strongly dependent on the

electrical load resistance. When U0 = 1.2 m/s, the res-

onant excitation frequencies when R = 104, 105,

and 106 are almost the same which is due to the effects

of the wind speed on the coupled frequency, as pre-

sented in Fig. 5a. When the wind speed is increased

to 1.6 m/s, the difference between the resonant exci-

tation frequencies when R = 104, 105, and 106

becomes more important and is similar to its counter-

part when U0 = 0 m/s and 0.6 m/s.

In Fig. 9a–d, we plot the frequency-response curves

of the harvested power for different base acceleration

values and four distinct values of the wind speed when

the electrical load resistance is set equal to 106. It

follows from these plotted curves that the harvested

power increases when the base acceleration is increased

for all considered wind speed values. Inspecting these

plots, we note that the harvester is affected by the

vortex-induced vibrations for different forcing exci-

tations when U0 = 1.2 and 1.6 m/s. In the reso-

nance region, the levels of the harvested power are

more important when U0 = 0 m/s than its counterpart

when U0 = 0.6 m/s for all base acceleration values.

This result is due to higher electromechanical damp-

ing values which are obtained when the wind speed is

increased from 0 to 0.6 m/s. On the other hand, the pres-

ence of the vortex-induced vibrations (when U0 = 1.2

and 1.6 m/s) results in a significant improvement in the

level of the harvested power for all considered values

of the base acceleration. In fact, there is an enhance-

ment by 100 % and 150 % in the level of the harvested

power at U0 = 1.2 and 1.6 m/s, respectively, compared

123

H. L. Dai et al.

to their counterparts at U0 = 0 and 0.6 m/s when the

vortex-induced vibrations effect is absent. Inspecting

Fig. 9a–d, it is clear that the difference between the res-

onant excitation frequency and the pull-out frequency is

increased when the base acceleration is increased. Fur-

thermore, an increase of the base acceleration results in

significant effects associated with the quenching phe-

nomenon.

6 Conclusions

The transduction of a piezoelectric energy harvester

under the combination of vibratory base excitations and

vortex-induced vibrations has been investigated. The

harvester consists of a multilayered piezoelectric can-

tilever beam with a circular cylinder-tip mass attached

to its free end which is placed in a uniform air flow

and subjected to direct harmonic excitations. A non-

linear distributed-parameter model was derived based

on the Euler–Lagrange principle and the Galerkin dis-

cretization. The fluctuating lift coefficient was mod-

eled by a van der Pol oscillator. The results showed

that at least four modes have to be considered in the

Galerkin discretization to accurately predict the per-

formance of the harvester. The numerical predictions

based on the derived model were in a very good agree-

ment with experimental measurements when there is

no base excitations effect. A linear analysis was per-

formed to investigate the effects of the electrical load

resistance and wind speed on the coupled frequency

and electromechanical damping. The effects of the base

acceleration, wind speed, and electrical load resistance

on the performance of the harvester and the quenching

phenomenon were determined. When the wind speed is

in the pre- or post-synchronization regions, the results

showed that the presence of this wind speed results in an

increase of the electromechanical damping and hence a

reduction in the level of the harvested power. When the

wind speed is in the lock-in or synchronization region,

it was demonstrated that using a single piezoelectric

energy harvester under a combination of vibratory base

excitations and vortex-induced vibrations significantly

improves the level of the harvested power compared to

using two separate harvesters. It was also demonstrated

that increasing the base acceleration results in a reduc-

tion in the vortex-induced vibrations effect, an increase

of the difference between the resonant excitation fre-

quency and the pull-out frequency, and a significant

effects associated with the quenching phenomenon.

Acknowledgments The authors gratefully acknowledge the

support provided by the Program for New Century Excel-

lent Talents in University (NCET-11-0183) and the Funda-

mental Research Funds for the Central Universities, HUST

(2013TS034).

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