Nonlinear DynDOI 10.1007/s11071-014-1355-8

ORIGINAL PAPER

Piezoelectric energy harvesting from concurrentvortex-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 apiezoelectric energy harvester to concurrently harnessenergy from base excitations and vortex-induced vibra-tions. The harvester consists of a multilayered piezo-electric cantilever beam with a circular cylinder tipmass 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 avan der Pol wake oscillator. The Euler–Lagrange prin-ciple and the Galerkin procedure are used to derive anonlinear distributed-parameter model for a harvesterunder a combination of vibratory base excitations andvortex-induced vibrations. Linear and nonlinear analy-ses are performed to investigate the effects of the elec-trical load resistance, wind speed, and base accelerationon 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 Scienceand Technology, Wuhan 430074, Chinae-mail: wanglindds@mail.hust.edu.cn

H. L. Dai · L. WangHubei Key Laboratory for Engineering Structural Analysisand Safety Assessment, Wuhan 430074, China

A. AbdelkefiDepartment of Engineering Science and Mechanics,MC 0219, Virginia Tech, Blacksburg, VA 24061, USA

harvested power is obtained. When the wind speed isin the lock-in or synchronization region, the resultsshow that there is a significant improvement in thelevel of the harvested power which can attain 150 %compared to using two separate harvesters. The resultsalso show that an increase of the base accelerationresults in a reduction in the vortex-induced vibrationseffects, an increase of the difference between the res-onant excitation frequency and the pull-out frequency,and a significant effects associated with the quenchingphenomenon.

Keywords Energy harvesting · Piezoelectricmaterial · Vortex-induced vibrations · Combinedloadings · Nonlinear analysis · Quenching phenomenon

1 Introduction

Energy harvesting through conversion of wastedmechanical energy to electrical power has received sig-nificant attention over the past decade [1–5]. Theseenergy harvesting systems have been proposed to oper-ate self-powered electronics, such as wireless andhealth monitoring sensors, data transmitters, medicalimplants, and cameras [6–9]. These harvesters are alsodesigned to replace small batteries that have a finite lifespan and would require expensive or time consumingreplacement. Of the different options of transductionmechanisms which include electrostatic [3], electro-magnetic [3], and piezoelectric [1,3], the piezoelectric

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H. L. Dai et al.

one has received the most attention because of its easeof application and it can effectively be placed in smallvolumes. In addition, the piezoelectric option does notrequire an external input voltage.

Several concepts have been introduced in order toharvest energy from unused mechanical energy avail-able in the environment. The most common conceptconsists of a piezoelectric cantilever beam subjectedto external excitations applied at its base. When theexcitation frequency matches the natural frequency ofthe harvester, resonant motions are obtained and thenthe piezoelectric material strains and produces an elec-trical charge and hence an output voltage is obtainedthrough 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 havebeen 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 conceptof harvesting energy from aeroelastic or flow-inducedvibrations, such as flutter of airfoil sections [20–24],vortex-induced vibrations (VIVs) of circular cylinders[25–28], galloping of prismatic structures [29–33], andwake galloping of parallel cylinders [34,35]. In thisconcept, the piezoelectric energy harvester is placed ina flow field and excited to undergo large limit-cycleoscillation amplitudes that can be converted to usableelectrical power. Depending on the operating windspeed, these piezoaeroelastic systems can be designedand 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 theidea of energy harvesting from a combination of bothsources of excitation. Bibo and Daqaq [36,37] inves-tigated theoretically and experimentally the responsebehavior of a flutter-based piezoaeroelastic energyharvester when subjected to a combination of vibra-tory base excitations and aerodynamic loadings. Theydemonstrated that the efficiency as well as the powerdensity of the harvester are significantly improvedwhen the wind speed is either below or above the linearflutter speed. Furthermore, there is only one theoreti-cal study investigating the influence of both vibratory

base and aeroelastic excitations on a galloping-basedpiezoaeroelastic energy harvester [38]. Yan et al. [38]investigated the effects of the base acceleration, windspeed, and electrical load resistance on the performanceof the harvester and the associated nonlinear phenom-ena that take place.

Based on the authors’ knowledge, no research studyhas considered the potential of integrating both vibra-tory base and aeroelastic excitations on a VIV-basedpiezoaeroelastic energy harvester. In this work, we aimto investigate the transduction of piezoelectric energyharvesting under a combination of vibratory base exci-tations and vortex-induced vibrations. To this end, amultilayered 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 thederived distributed-parameter model are presented inSect. 2. In Sect. 3, the derived model is validatedwith previous experimental measurements when onlyvortex-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 accelerationon 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 cantileverbeam 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 subjectedto an external harmonic base motion, zb = z0cos(Ωt),where z0 is the amplitude of the base displacement andΩ is the base excitation frequency. In addition to thebase motion, when the shedding frequency is close tothe coupled frequency of the harvester, the harvestercan undergo steady-state limit-cycle oscillations in thecross flow direction. The piezoelectric sheets are placedon both sides of the cantilever beam. The two piezo-electric sheets are bounded by two in-plane electrodesof negligible thickness connected in series to an elec-trical load resistance, R. The thickness of the beam isassumed to be small compared to its length so that the

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Piezoelectric energy harvesting from vibrations and excitations

L b

Lp

m0

U0

D y

zx

o

( )w t

x1zb

Fig. 1 Schematic of the piezoelectric energy harvester underconcurrent loadings

shear deformation and rotary inertia can be neglected.The geometric and material properties of the systemare presented in Table 1.

To develop a reduced–order model of the consideredenergy harvester, we apply the Galerkin discretizationto the Lagrangian (L = T − U ) and nonconservativework (Wnc). To this end, we first calculate the totalpotential 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 ε

bx dVb+

∫

Vp

σp

x εpx dVp−

∫

Vp

E3 D3dVp

⎤⎥⎦ ,

(1)

where Vb and Vp are, respectively, the volumes of thesubstrate and the piezoelectric sheets. ρb and ρp are thedensities 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 relationsof piezoelectricity, the strains and stresses in the alu-

minum and piezoelectric layers are, respectively, givenby:

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 thebeam. 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 piezoelectricsheets, the electric field is related to the voltage, V (t),by E3 = −V (t)/2tp, where tp is the thickness of thepiezoelectric sheet [39].

In addition, D3 is the electric displacement given bythe following piezoelectric constitutive relation:

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

where ε33 is the permittivity component at constantstrain.

The kinetic energy of the harvester is the sum of thekinetic energies of the beam and tip mass. Taking intoaccount for the moment of inertia of the cylindrical tipmass, 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 andproperties of the harvestercomponents

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 – –

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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 endof the beam and It mass moment of inertia defined asIt = 1/3 Mt (D/2)2, which is considered here becausethe 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 powerdelivered to an electrical load resistance R, the workdone by the aerodynamic forces, F(t), and the workdone by the damping force. Hence, the variation of thenonconservative 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 respectto x and c represents the viscous damping coefficient.Q R denotes the extracted charge and is related to thegenerated 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 verychallenging [40] and several researchers have focusedon this topic [41–48]. The widely employed approachis the so-called wake oscillator model, which was firstput forward by Birkhoff and Zarantanello [41]. Bishopand Hassan [42] were the first to suggest modeling thelift over a circular cylinder due to vortex shedding by anonlinear self-excited oscillator. A Rayleigh oscillatorwas used by Hartlen and Currie [43] in order to modelthe lift force and they considered a velocity term tocouple it with the motion of the cylinder. Skop andGriffin [44,45] used a modified van der Pol oscillatorto 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 workingsof wake oscillator model were contributed by Skop andBalasubramanian [46] and Facchinetti et al. [47]. Skopand Balasubramanian [46] recommended a stall termin the description of the cross-flow force. However,Facchinetti et al. [47] systematically investigated thepattern of the force induced by the cylinder’s motionon the wake oscillator. Furthermore, they found that anacceleration coupling term is better than a velocity termto model the lock-in region. The representation of thevortex-induced force and its wake oscillator were thenconfirmed by Violette et al. [48] through comparisonwith numerical simulation and experiments.

Following Facchinetti et al. [47], the vortex-inducedforce is expressed as:

F(t) = CLρ0 DU 20 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 ofthe cylinder, and D is the diameter of the cylinder. Thefluctuating lift coefficient is denoted by CL = CL0

2 q(t).CL0 and CD are the steady lift and mean sectionaldrag coefficients which can be, respectively, consid-ered equal to 0.3 and 1.2 in the region of well-developedwakes [47].

The wake oscillator employed in the present studyis based on the model of Facchinetti et al. [47] whichstates:

q + λωs

(q2 − 1

)q + ω2

s 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 andstands for the lift acting on the cylinder. w(t) is thedisplacement of the cylinder. The values of λ and A areconstants and are identified from the experimental mea-surements. ωs is the vortex-shedding frequency, whichis defined as ωs = 2π StU0/D, St being the Strouhalnumber which is determined by the Reynolds numbersrange [49] and U0 represents the wind speed.

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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, weexpress 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 cantileverbeam with a tip mass. Taking into account for the iner-tial force at the offset D/2 from the free end of thebeam and the rotational moment of inertia of the cylin-der around the end of the beam, the mode shapes of thissystem 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 )] − κ1κ2β2i

[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 thecylinder radius and beam length, and the frequencyparameters βi are the roots of the following frequencyequation

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

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

− 4/3κ1κ22β

3 [cos(β) sinh(β)+ cosh(β) sin(β)]

+ 1/3κ21κ

22β

4 [1 − cos(β) cosh(β)] = 0. (14)

To determine the modal amplitude constant, C1, weapply 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)E

bϕ′′j (x)dVb

+∫

Vp

z2ϕ′′i (x)E

pϕ′′j (x)dVp = ω2

i δi j ,

(15)

where δi j is the Kronecker delta, defined as being equalto unity for i = j and equal to zero for i �= j . And ωi

is the undamped natural frequency of the i th vibrationmode.

The discretization is performed by substitutingEq. (11) into Eqs. (1), (6), and (7). After that, we usethe Euler–Lagrange equations which are based on thefollowing 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 thisenergy 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 + ω2

i ri − θi V

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

C pV + 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 themaximum generated voltageas a function of the windspeed for different modes inthe Galerkin discretizationwhen 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 mode2nd mode3rd mode4th mode5th mode

()

Vp

V

( )0 m sU

where a = zb�2 is the base acceleration, m1 =

wbρbtb,m2 = wbρbtb + 2wpρptp, and the rest of thecoefficients 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 baseexcitations and vortex-induced vibrations has beendeveloped. To determine the accuracy of the deriveddistributed-parameter model, we first compare ournumerical predictions with the experimental measure-ments of Akaydin et al. [26]. In their experiments, onlyvortex-induced vibrations were present.Therefore, thebase acceleration is considered to be zero in the derivedreduced-order model. To make an accurate comparison,we plot, in Fig. 2, the predicted numerical values of thegenerated voltage as a function of the wind speed fordifferent number of modes in the Galerkin procedurewhen the load resistance is set equal to 2.46×106�. Wehave to mention that this plot is performed for the same

geometric and physical properties as used by Akaydinet al. [26]. It follows from the plotted curves in Fig. 2that 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 numericalpredictions are performed when using five modes inthe Galerkin discretization.

A comparison between the numerical predictionsof our developed distributed-parameter model and theexperimental measurements of Akaydin et al. [26] isperformed. The plotted curves in Fig. 3a show the vari-ation of the average harvested power as a function ofthe wind speed when the load resistance is set equal to2.46×106�. The average harvested power is calculatedas follows:

Pe = V 2rms

R, (25)

where Vrms denotes the root mean square (RMS) valueof the generated voltage.

It follows from the plotted curves in Fig. 3a that thereis 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 modeland particularly its starting and its peak. In fact, themaximum value of the average power is the same andequal to 0.1 mW when the wind speed is U0 = 1.20 m/s.However, in the post-synchronization region, there isan 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

0.12SimulationExperimentAkaydin 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 thederived 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 speedvalues larger than 1.4 m/s. This result is probably dueto 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 whenconsidering 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 isset equal to zero. It follows from the plotted curves inFig. 3(b) that the synchronization region takes placewhen the wind speed is near U0 = 1.4 m/s.

4 Linear analysis: effects of the load resistanceand wind speed on the coupled frequencyand electromechanical damping

To investigate the effects of the electrical load resis-tance and wind speed on the coupled frequency andelectromechanical damping, a linear analysis is per-formed in which the base excitation terms as wellas the nonlinear terms in Eqs. (18) and (20) aredropped. Because piezoaeroelastic systems are gener-ally designed to harvest energy at low wind speeds andhence 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 discretizationis considered. Hence, the equations of motion can beexpressed as:

r1+[

2ς1ω1+ CDρ0 DU0 L0

2

(ϕ1(Lb)+ D

2ϕ′

1(Lb)

)2]

r1

+ω21r1 − θ1V − α1q = 0 (26)

C pV + 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

qq

⎤⎥⎥⎥⎥⎥⎥⎦. (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

()

Freq

uenc

y r

ad s

102

103

104

105

106

107

108

109

1010

1

2

3

4

5

( )R Ω

()

2C

oupl

ed d

ampi

ng 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 whenU0 = 0 m/s

where

B =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 1 0 0 0

−ω21 −

(2ς1ω1 + CDρ0 DU0 L0

2 ψ21

)θ1 α1 0

0 − θ1C p

− 1RC p

0 0

0 0 0 0 1

− ADψ1ω

21 − A

Dψ1

(2ς1ω1 + CDρ0 DU0 L0

2 ψ21

)ADψ1θ1

ADψ1α1 − ω2

s λωs

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(31)

and ψ1 = ϕ1(Lb)+ D2 ϕ

′1(Lb).

The matrix B has a set of five eigenvalues λi ,i = 1, 2, 3, 4, 5. The first two eigenvalues are complexconjugates and they are associated with the transversedisplacement motion of the cantilever beam. In fact,the real part of these two eigenvalues represents theelectromechanical damping coefficient and their pos-itive imaginary part represents the coupled frequencyof the harvester. These two eigenvalues depend on theelectrical load resistance and the wind speed when therest of the parameters are determined based on the givenvalues in Table 1. The third eigenvalue λ3 is a result ofthe piezoelectric coupling and it is always real and neg-ative. As for λ4 and λ5, they are complex conjugatesand they are associated with the response of the fluctu-ating lift coefficient.

The plotted curves in Fig. 4a, b, respectively, depictthe variations of the coupled frequency and the electro-

mechanical damping as a function of the electrical loadresistance when the wind speed is set equal to zero.Inspecting Fig. 4a, there is a steep increase over theelectrical load resistance values between 4×105� and107�. In addition, when the load resistance is smallerthan 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 frequencyas 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. Thisfrequency is referred as the open coupled frequency.Clearly, an increase of the electrical load resistanceresults in an increase of the onset of synchronization.Figure 4b shows that the electromechanical dampingis relatively high when the electrical load resistanceis between 4 × 105� and 107�. This region of load

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Piezoelectric energy harvesting from vibrations and excitations

0 0.5 1 1.5 2

22

23

24

25

26R=102Ω

R=106ΩR=109Ω

0 0.5 1 1.5 20

2

4

6

8

10

12R=102Ω

R=106ΩR=109Ω

( )0 m sU ( )0 m sU

(b)(a)(

)Fr

eque

ncy

rad

s

()

2C

oupl

ed d

ampi

ng r

atio

10−

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

resistances over which the electromechanical damp-ing is high coincides with the region over which thesteep increase of the coupled frequency occurs. It is alsonoted that small values of the electromechanical damp-ing are obtained when the load resistance is smallerthan 104� or larger than 108�. These two regionscoincide 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 electromechanicaldamping have a significant impact on the displacementand harvested power.

To determine the influence of the wind speed on thecoupled frequency and electromechanical damping ofthe harvester, we follow the same performed strategy inFig. 4. To this end, we plot, in Fig. 5a, b, the variationsof the coupled frequency and electromechanical damp-ing as a function of the wind speed for three distinctvalues of the electrical load resistance. It follows fromFig. 5a that the variation of coupled frequency with thewind speed has the same tendency for different valuesof the electrical load resistance. In fact, the coupled fre-quency decreases when the wind speed varies from 0to 1 m/s. Then, it increases significantly when the windspeed 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 increaseswhen the electrical load resistance is increased for allvalues of wind speed between 0 and 2 m/s. InspectingFig. 5b, we note that there is a maximum value of the

electromechanical damping for a specified value of thewind speed. However, this specific value of the windspeed changes with the considered value of the electri-cal load resistance. In fact, an increase of the electricalload resistance value is accompanied by an increaseof the wind speed value that gives maximum electro-mechanical damping. This result can be explained bythe presence of the drag force and the influence of theload resistance on the coupled frequency. Furthermore,maximum values of the electromechanical damping arealways obtained for an electrical load resistance around106�.

This performed linear analysis gives a clear ideaabout the effects of the electrical load resistance andwind speed on the onset of synchronization or coupledfrequency and electromechanical damping variations.These effects have an important impact in determiningthe synchronization region and the level of the har-vested power.

5 Nonlinear analysis: effects of the load resistance,wind speed and base accelerationon the harvester’s performance

In the presence of the base and vortex-induced exci-tations, the frequency-response curves as well as theresponse behavior of the harvester are expected to bestrongly dependent on the value of wind speed whichcan change the response of the harvester from periodic

123

H. L. Dai et al.

10 15 20 25 30 350

0.05

0.1

0.15

0.2

0.25

0.3 U0=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 30 350

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 30 350

0.5

1

1.5

2 U0=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 accelerationsand wind speeds

to period-n or quasi-periodic. This dependence is dueto 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 fixedthrough this relation ωs = 2π StU0/D. At this windspeed, if the shedding frequency matches the coupledfrequency of the harvester, resonant transverse vibra-tions occur and then lock-in or synchronization takesplace. Furthermore, if the excitation frequency matchesthe coupled frequency of the harvester, resonant trans-verse vibrations take place. Consequently, a significantimprovement in the level of the harvested power canbe observed if both the excitation frequency and theshedding frequency match the coupled frequency ofthe harvester. Away from the synchronization region,there are no oscillations from the aerodynamic loadingand one expects that the harvester response is alwaysperiodic and only contains the excitation frequency. On

the other hand, when the shedding frequency is closeto the coupled frequency, the response of the harvesterunder the concurrent dynamic loads becomes complex.This complexity depends on the values of the excitationfrequency 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-inducedvibrations effect and a forced-oscillation one due tobase excitation. As such, the harvester mainly oscillateswith two harmonic frequencies which are the excitationfrequency and the shedding frequency. The existenceof these two frequencies is right only when these fre-quencies are not commensurable (off-resonance). Inthis case, the response of the harvester is generallyaperiodic. Away from the resonance region, the oscil-lations due to vortex-induced vibrations are expectedto dominate. This process of unlocking between thetwo frequencies is called pulling-out. When the exci-

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Piezoelectric energy harvesting from vibrations and excitations

tation frequency is close to the coupled frequency ofthe harvester (resonance region), its response changessignificantly and the oscillations due to the vortex-induced vibrations entrained or locked onto the forcedoscillations. Therefore, a synchronization response atthe excitation frequency takes place. In addition, thefree-oscillation contribution decays with time and thesteady-state motion becomes periodic. The processof decaying the free-oscillation contribution is calledquenching and the frequency at which the systemchanges from aperiodic response to periodic one iscalled pull-out frequency [51]. To demonstrate theseexpectations, we investigate the effects of the electricalload resistance, wind speed, and base acceleration onthe 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 harvesterfor different wind speed values when the electrical loadresistance is set equal to 106�. It follows from theseplots that the value of the resonant excitation frequencychanges when the wind speed is changed. In fact, whenthe wind speed is increased from 0 to 1.2 m/s, the res-onant excitation frequency decreases. When the windspeed is increased from 1.2 to 1.6 m/s, the resonantexcitation frequency is increased. This result is pre-dicted because the coupled frequency of the harvestervaries with the wind speed, as shown in Fig. 5a. Further-more, inspecting the off-resonance regions, it is clearthat the effects of the vortex-induced vibrations areimportant for wind speed values equal to 1.2, 1.4, and1.6 m/s. This result is also expected because the syn-chronization region takes place for wind speed valuesbetween 1.1 and 1.7 m/s, as shown in Fig. 3b. For windspeed values equal to 0.6 and 1.8 m/s, the tendency ofthe frequency-response curves are similar to the caseof base excitation without vortex-induced vibrationseffect. This is explained due to the fact that the vortex-induced vibrations effect is vanished in the pre- andpost-synchronization regions, as shown in Fig. 3b. Inaddition, 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 counterpartwhen U0 = 0.6 and 1.8 m/s. This result is due to highervalues of the electromechanical damping at U0 = 0.6and 1.8 m/s compared to its counterpart at U0 = 0 m/s,as shown in Fig. 5b. Consequently, in the absence ofthe vortex-induced vibrations effect which correspondsto the cases when the wind speed is not between 1.1and 1.7 m/s, there is a reduction in the harvested power

which due to the electromechanical damping effect (seeFig. 5b). It follows from Fig. 6c, d that for high val-ues of the base acceleration, the maximum harvestedpower when U0 = 1.6 m/s is larger than its counterpartwhen U0 = 1.2 and 1.4 m/s although higher valuesof the harvested power are obtained at these speedsin the off-resonance regions. This result is also due tothe electromechanical damping effect which is muchhigher 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 harvesterchanges between aperiodic and periodic motions.

The plotted curves in Fig. 8a–d depict the frequency-response curves of the harvested power for variousvalues of the electrical load resistance and four val-ues of the wind speed. The base acceleration is setequal to 0.01 g. It follows from these plots that theharvested power is strongly dependent on the electri-cal load resistance. In the case without vortex-inducedvibrations effect (U0 = 0 m/s), Fig. 8a shows that max-imum values of the harvested power are obtained whenR = 105� and 107�, respectively. Furthermore, thevalue of the resonant excitation frequency depends onthe considered value of the electrical load resistance.In fact, an increase of the electrical load resistanceis followed by an increase of the resonant excitationfrequency. This is expected because the coupled fre-quency of the harvester has the same variations withthe 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 optimumelectrical load resistance when varying the wind speedfrom 0 to 0.6 m/s can be explained due to the fact thatincreasing the electrical load resistance and the windspeed from 0 to 0.6 m/s is accompanied by a significantchange in the electromechanical damping. InspectingFig. 8a, b, it is noted that there is a reduction in the levelof harvested power when the wind speed is increasedfrom 0 to 0.6 m/s for all considered values of electri-cal load resistance. This result is predicted because theelectromechanical damping increases when the windspeed is increased from 0 to 0.6 m/s for all consideredvalues of electrical load resistance, as shown in Fig. 5b.

When U0 = 1.2 and 1.6 m/s, it is clear that thevortex-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/swhen the load resistance is set equal to 106�

10 15 20 25 30 350

0.01

0.02

0.03

0.04

0.05

0.06R=104Ω

R=105Ω

R=106Ω

R=107Ω

R=108Ω

10 15 20 25 30 350

0.01

0.02

0.03R=104Ω

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=104Ω

R=105Ω

R=106Ω

R=107Ω

R=108Ω

10 15 20 25 30 350

0.2

0.4

0.6

0.8R=104Ω

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 isset equal to 0.01 g

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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

0.8 a=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 isset 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 theelectrical load resistance except R = 106�. Clearly,the wide of the synchronization or lock-in region of thevortex-induced vibrations is strongly dependent on theelectrical 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 effectsof the wind speed on the coupled frequency, as pre-sented in Fig. 5a. When the wind speed is increasedto 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 curvesof the harvested power for different base accelerationvalues and four distinct values of the wind speed whenthe electrical load resistance is set equal to 106�. It

follows from these plotted curves that the harvestedpower increases when the base acceleration is increasedfor all considered wind speed values. Inspecting theseplots, we note that the harvester is affected by thevortex-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 aremore important when U0 = 0 m/s than its counterpartwhen 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 isincreased from 0 to 0.6 m/s. On the other hand, the pres-ence of the vortex-induced vibrations (when U0 = 1.2and 1.6 m/s) results in a significant improvement in thelevel of the harvested power for all considered valuesof the base acceleration. In fact, there is an enhance-ment by 100 % and 150 % in the level of the harvestedpower at U0 = 1.2 and 1.6 m/s, respectively, compared

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H. L. Dai et al.

to their counterparts at U0 = 0 and 0.6 m/s when thevortex-induced vibrations effect is absent. InspectingFig. 9a–d, it is clear that the difference between the res-onant excitation frequency and the pull-out frequency isincreased when the base acceleration is increased. Fur-thermore, an increase of the base acceleration results insignificant effects associated with the quenching phe-nomenon.

6 Conclusions

The transduction of a piezoelectric energy harvesterunder the combination of vibratory base excitations andvortex-induced vibrations has been investigated. Theharvester consists of a multilayered piezoelectric can-tilever beam with a circular cylinder-tip mass attachedto its free end which is placed in a uniform air flowand subjected to direct harmonic excitations. A non-linear distributed-parameter model was derived basedon the Euler–Lagrange principle and the Galerkin dis-cretization. The fluctuating lift coefficient was mod-eled by a van der Pol oscillator. The results showedthat at least four modes have to be considered in theGalerkin discretization to accurately predict the per-formance of the harvester. The numerical predictionsbased on the derived model were in a very good agree-ment with experimental measurements when there isno base excitations effect. A linear analysis was per-formed to investigate the effects of the electrical loadresistance and wind speed on the coupled frequencyand electromechanical damping. The effects of the baseacceleration, wind speed, and electrical load resistanceon the performance of the harvester and the quenchingphenomenon were determined. When the wind speed isin the pre- or post-synchronization regions, the resultsshowed that the presence of this wind speed results in anincrease of the electromechanical damping and hence areduction in the level of the harvested power. When thewind speed is in the lock-in or synchronization region,it was demonstrated that using a single piezoelectricenergy harvester under a combination of vibratory baseexcitations and vortex-induced vibrations significantlyimproves the level of the harvested power compared tousing two separate harvesters. It was also demonstratedthat increasing the base acceleration results in a reduc-tion in the vortex-induced vibrations effect, an increaseof the difference between the resonant excitation fre-quency and the pull-out frequency, and a significanteffects associated with the quenching phenomenon.

Acknowledgments The authors gratefully acknowledge thesupport 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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