FLOW ENERGY HARVESTING FROM PIEZOELECTRIC FLAGS

Sebastien Michelin∗

LadHyX – Department of Mechanics

Ecole polytechnique

Route de Saclay

91128 Palaiseau, France

Email: sebastien.michelin@ladhyx.polytechnique.fr

Olivier DoareUnite de Mecanique (UME)

ENSTA, ParisTech

Chemin de la Huniere

91761 Palaiseau, France

Email: olivier.doare@ensta-paristech.fr

ABSTRACTThe present work investigates the possibility to pro-

duce electrical energy from the flutter oscillations of aflexible plate placed in an axial flow and covered withpiezoelectric patches that convert bending and stretchingof the plate’s surface into electric charge displacement.A fully-coupled nonlinear model for the dynamics of thefluid-solid-electric system is presented and used to deter-mine the influence of the different system parameters onthe stability, nonlinear dynamics and energy harvestingefficiency of the system. In particular, the role of the tun-ing between the fluid-solid and electrical systems is in-vestigated as well as the impact of the flow velocity andfluid-solid mass ratio.

INTRODUCTIONGlobal climate change and scarcity of traditional fos-

sil fuels motivate the development of innovative and re-newable energy sources with low environmental impact.Converting the kinetic energy associated with geophysi-cal flows such as winds, rivers, oceanic and tidal currentsis particularly attractive, given the wide availability of thisresource. Classical fluid-solid instabilities such as Vortex-Induced Vibrations, galloping and flutter in axial flowseffectively extract energy from the flow and transformit into solid mechanical energy that can then be used toproduce electricity using either displacement-based (e.g.electromagnetic converters) or deformation-based energyconverters (e.g. piezoelectric materials) [1–5]

In the present work, we are interested in the feasi-bility and performance of flow energy harvesting usingpiezoelectric flexible plates, or flags. Thin flexible platesplaced in a steady axial flow are known to become un-stable to flutter at a critical velocity [6–8], above which

∗Address all correspondence to this author.

a large amplitude flapping regime can develop, character-ized by the propagation of deformation waves along theflag [9]. When a piezoelectric patch is attached on thedeforming flag surface, its periodic bending can be con-verted into electric charge transfers [10]. Use of piezo-electric materials is not adapted to large scale harvesters,but such materials show competitive and promising per-formances in the domain of small-power devices [11].

Classically, energy harvesting is represented in thefluid-solid system as an additional damping [12], but thisapproach overlooks the possible coupling between theharvesting circuit and the fluid-solid system powering it.In this study, an explicit description of the harvestingsystem is introduced in terms of piezoelectric patches,and following Refs. [13, 14], the equations for the fully-coupled fluid-solid-electrical nonlinear dynamics are de-rived. This approach provides insight on the double im-pact of energy harvesting, namely producing electricalenergy and modifying the behavior of the fluid-solid sys-tem. For example, increasing the coupling coefficientmight lead to a better energy transfer to the output cir-cuit but could also eventually result in the restabilizationof the system due to its damping effect. Considering themodel problem of a slender flexible plate in axial flow, weanalyze the energy harvesting efficiency of the system interms of the different fluid-solid-electric parameters.

The present paper is organized as follows: the fluid-solid-electric model is first presented in Section 1. Theeffect of the piezoelectric coupling on the stability of thesystem is then analyzed using linear analysis (Section 2).Section 3 finally focuses on the nonlinear dynamics of thesystem to determine the amplitude and frequency of thesaturated flapping motion and assess the energy harvest-ing efficiency.

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1 MODEL1.1 Presentation of the piezoelectric flag

In this work, we consider a rectangular inextensibleflexible plate of length L and span l, and negligible thick-ness h. The plate is clamped at its leading edge and placedin a steady incoming inviscid flow of velocity U∞ anddensity ρ . The surface of the plate is covered by pairsof piezoelectric patches (one patch on each side) whosenegative electrodes have been shunted through the plateand whose positive electrodes are connected to an ouputresistive circuit (Figure 1). We neglect here any spanwiseplate deformation and charge transfer so that the mechan-ical and electrical state variables are only functions of thestreamwise curvilinear coordinate s and time t. In the fol-lowing, for any function a(s, t), its derivatives with re-spect to t and s are noted a and as, respectively.

The deformation and position of the cantileveredplate are characterized by the local orientation θ(s, t) ofthe unit tangent vector τ(s, t) with the flow direction ex.The difference of potential between the free electrodeson each side of the plate and the charge transfer betweenthose electrodes are respectively noted v(s, t) and q(s, t)(Figure 1). In the limit of continuous coverage by patchesof infinitesimal length in the streamwise direction [14],the piezoelectric effect results in an additional torque in-side the plate imposed by the voltage between the elec-trodes, namely

Mpiezo =−χv, (1)

with χ the piezoelectric coupling, and in a charge transferq resulting from the plate’s deformation:

q = cv+χθs (2)

with c the characteristic capacity of the piezoelectricpatch pair. Noting g the lineic conductivity of the outputcircuit, q and v are related through Ohm’s law

q+gv = 0. (3)

1.2 Non-linear solid dynamicsThe system {plate + piezo} is represented as an

Euler–Bernoulli beam, and large amplitude displace-ments of the beam are considered. The beam is inexten-sible and, taking into account the piezoelectric torque in

g

q

v χθs v

q

cg

(a)

(b)

(c)

U∞

ey

ex

θ(s, t)A

ex

ey

ez

U∞

l

L

τ (s, t)n(s, t)

FIGURE 1: (a) Two-dimensional motion of a cantileveredslender flexible plate in axial flow. (b) Piezoelectric flag:the flexible plate (black) is covered on each side by piezo-electric patches (grey). (c) each piezoelectric patch pair isconnected to a dissipative circuit and the electrical equiv-alent is given on the right.

Eq. (1), its position x(s, t) satisfies

µ x =�T τ − (Bθss −χv)n

�

s+ ffluid, (4)

xs = τ, (5)

where B and µ are respectively the uniform rigidity andlineic mass of the piezoelectric flag, T (s, t) is the plate’stension and acts as a Lagrangian multiplier to enforce theinextensibility of the plate, and ffluid is the fluid force perunit length on the flag. Clamped-free boundary condi-tions are used so that

x = 0, θ = 0 at s = 0, (6)Bθs −χv = Bθss −χvs = 0 at s = L. (7)

1.3 Fluid modelWe focus here on the slender body limit (l � L),

when the fluid force applied to the plate can be decom-posed into two distinct parts: a reactive force [15], result-ing from the added inertia of the fluid put in motion by

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each cross-section of the plate, and a resistive force [16]accounting for inertial drag and dissipative effects:

ffluid =−maρl2�

un − (uτun)s +12

u2nθs

�n− 1

2ρlcd |un|unn,

(8)where ma and cd are the non-dimensional added mass anddrag coefficient associated with the plate’s cross-section,respectively, and uττ + unn = x−U∞ex is the local rela-tive velocity of the solid to the fluid at rest. The validity ofthis approach, initially developed to evaluate flow forcescreated by the motion of a fish body in large amplitude de-formations, was recently confronted to and found in goodagreement with Reynolds-Averaged Numerical Simula-tions (RANS) at high Reynolds number [17]. Althoughlimited, in theory, to asymptotically small aspect ratiosl/L, it was also shown recently that predictions using thisapproach provide satisfying results even up to O(1) as-pect ratio [18]. In the following, the aspect ratio is takenas H∗ = l/L = 0.5, and we focus on thin flat plates so thatma = π/4 and cd = 1.8.

1.4 Harvesting efficiencyThe total power P(t) harvested by the system is

equal to the energy actually transferred to the output cir-cuit and dissipated in the useful load g:

P(t) =� L

0gv(s, t)2ds. (9)

The efficiency of the system is defined as the ratioof the harvested energy to the fluid kinetic energy fluxthrough the section occupied by the system:

η =�P�

12 ρU3

∞A l, (10)

with A the peak-to-peak flapping amplitude, and �P�the time-average of the output power.

1.5 Non-dimensional parametersEquations (1)–(9) are non-dimensionalized using L,

L/U∞ and ρL3 as reference length, time and mass, respec-tively. v and q are non-dimensionalized by U∞

�µ/c and

U∞√µc, respectively. The system is then characterized

by five non-dimensional coefficients:

M∗ =ρlLµ

, U∗ =U∞L�

µB, H∗ =

lL

(11)

α =χ√Bc

, β =cU∞gL

, (12)

corresponding to the fluid-solid inertia ratio, the non-dimensional flow velocity, the aspect ratio, the piezoelec-tric coupling and the tuning of the output circuit, respec-tively. β is indeed the ratio of the non-dimensional time-scale associated with the output RC-loop. In the follow-ing, all quantities are non-dimensionalized, unless speci-fied otherwise.

2 LINEAR STABILITY ANALYSISWe first consider the impact of the piezoelectric cou-

pling on the linear stability of the flexible plate. In the ab-sence of any coupling (α = 0), the plate becomes unstableto flutter for a flow velocity U∗ greater than a thresholdvalue U0

c (M∗) [8, 9, 19].

2.1 Linearized equationsIn the limit of small vertical displacement y(s, t) of

the plate, the linearized equations for y and v are obtainedfrom Eqs. (2)–(4) as

(1+maM∗H∗) y+2maM∗H∗ys +maM∗H∗yss

+1

U∗2 yssss −α

U∗ vss = 0, (13)

β v+ v+αβU∗ yss = 0. (14)

Assuming [y,v] = Re�[Y (s),V (s)]eiωt� with ω complex,

Eqs (13)–(14), together with the boundary conditions

Y (0) = Ys(0) = 0, (15)

Yss(1)−α

U∗V (1) = Ysss(1)−α

U∗Vs(1) = 0, (16)

become an eigenvalue problem for [Y,V ] and ω , solvednumerically using a collocation method on N Chebyshev-Gauss-Lobatto points (typically N ≈ 60–120).

2.2 Impact of the piezoelectric coupling on the in-stability threshold

For a given output circuit (fixed β ), an increase inthe piezoelectric coupling α impacts the dynamics of thesystem in two ways: (i) it increases the effective rigid-ity of the system and (ii) it increases the energy transferto the output circuit where it is dissipated. One there-fore expects an increase of the critical velocity (i.e. a sta-bilization) due to the piezoelectric coupling. In Fig. 2,the critical velocity is represented for increasing valuesof α . A stabilization is indeed observed for low M∗ (lightfluid or short flag), while a destabilization of the higher

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10 1 100 101 1020

5

10

15

20

25

M*

Uc

FIGURE 2: Critical velocity threshold Uc(M∗) for β =0.1 and α = 0 (uncoupled, thick light grey), α = 0.3(dashed), α = 0.5 (dash-dotted) and α = 0.9 (dotted).

order dominant modes is observed for higher M∗ (typi-cally heavier fluid or long flag). This effect was observedrecently for a purely two-dimensional model (flag of in-finite span), and can be related to the destabilization bydamping of negative energy waves in the local stabilityanalysis [14]. One should emphasize that the influence ofthe piezoelectric coupling is highly dependent on the tun-ing β of the fluid-solid and electric systems. The resultspresented in Fig. 2 are obtained when both systems havesimilar fundamental time-scales.

For a fixed piezoelectric coupling α , the frequencytuning of the fluid-solid and electrical systems, β , is ameasure of the resistance in the output circuit. For β � 1,the piezoelectric patches’ electrodes are short-circuited:no electric potential is applied to the piezoelectric ele-ment and the piezoelectric torque in Eq. (1) vanishes. Theflag dynamics is not modified by the piezoelectric, and thecritical velocity threshold is that of a flag with no piezo-electric coupling, U0

c (Figure 3). On the other hand, whenβ � 1, the conductivity of the output circuit is negligi-ble and no charge transfer can occur between the elec-trodes. Equations (1)–(2) impose that the potential v, andtherefore Mpiezo are proportional to curvature and effec-tively, the piezoelectric coupling acts as an increase in theflag’s rigidity. As a result, the critical velocity threshold inthis open-circuit limit is obtained directly as U0

c√

1+α2

(Figure 3). Between these two limit cases, the impactdescribed above is recovered: the fluid-solid-electric cou-pling induces a destabilization of the system at large M∗

10 1 100 101 1020

5

10

15

20

25

M*

Uc

FIGURE 3: Critical velocity threshold Uc(M∗) for α =0.5 and β = 0 (short circuit, thick light grey), β = 0.1(dashed), β = 0.3 (dash-dotted), β = 1 (dotted), β = 3(solid) and β = ∞ (open circuit, thick dark grey)

and a stabilization for small M∗.This destabilization by damping is particularly inter-

esting from the point of view of harvesting energy from aflow. In general, the optimal operating regime of the de-vice should be a trade-off between adding enough damp-ing on the system to dissipate energy in the output circuitwhile maintaining sufficiently high flapping amplitude.When the system is destabilized by damping, initially atleast, the addition of damping to the system satisfies bothconstraints: it increases the energy transfer while main-taining, or possibly enhancing, the flapping dynamics.

3 NON-LINEAR DYNAMICSStudying the nonlinear dynamics of the fully cou-

pled fluid-solid-electric system is necessary in order todetermine the characteristics of the limit-cycle oscilla-tions of the system (amplitude and frequency) and assessthe amount of power that can be harvested by such a de-vice. The non-dimensional form of Eqs. (1)–(8) is solvednumerically using a Chebyshev collocation in space anda second order implicit time-stepping method [20]. Theflag is initially at rest, with the piezoelectric patches intheir reference configuration (q= v= 0). At t = 0, a smallvertical perturbation is added to the horizontal flow to ini-tiate the flag motion.

For U∗ > Uc, the small perturbation results in an ex-ponential growth of the flapping amplitude that eventu-ally saturates (Figure 4). Two different behaviors can

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(a)

44 46 48 504

5

6

7

8

9x 10 3

Har

vest

ed P

ower

(b)

142 144 146 1481.5

2

2.5

3

3.5

4

4.5

x 10 3

Har

vest

ed P

ower

(c)

28 30 32 34

5

6

7

8

9

x 10 3

Har

vest

ed P

ower

t

FIGURE 4: Flapping dynamics of the piezoelectric flagfor α = 0.5, M∗ = 10, β = 0.158 and (a) U∗ = 10.5, (b)U∗ = 11 and (c) U∗ = 14.5. (Left) Successive position ofthe piezoelectric flag and (Right) evolution in time of theinstantaneous harvested power.

then be observed: (i) the flag enters a strongly periodicregime with a limit cycle oscillation clearly identified or(ii) the flag dynamics is more complex, suggesting somenonlinear interactions between different modes, and somechaotic-looking motion.

A complete discussion of the properties of the latterbehavior is beyond the scope of the present work, but it isworth noting that such complex behavior was reported inexperimental and numerical studies on the flapping flagdynamics [9, 18, 21, 22], and is not specific to the piezo-electric coupling problem. For fixed M∗, α and β , asU∗ is increased beyond the critical velocity Uc and moremodes become unstable, the system shows different modeswitching events, one being illustrated on Figures 4 and 5.The impact of such events on the harvesting performanceis important as the amplitude, frequency and curvaturedistributions are modified.

0 5 10 15 200

0.1

0.2

0.3

0.4

U*

Amplitude

0

2

4

6

8

Frequency

FIGURE 5: Evolution with U∗ of the amplitude (solid)and frequency (stars) of the limit-cycle oscillations forM∗ = 10, α = 0.5 and β = 0.158.

4 HARVESTED ENERGY AND EFFICIENCY OF THESYSTEMFor a given set of parameters (M∗,U∗,α,β ) the dy-

namics of the piezoelectric system is obtained using thenumerical method presented above, and the harvesting ef-ficiency is computed using Eq. (9). When a limit cycleis detected, the efficiency is simply obtained by takingthe time-average of P(t) over one period, and the non-dimensional flapping amplitude A =A /L is computed as

A =�

�2ye(t)2�, (17)

with ye(t) the deflection of the trailing edge of the flag.In the case where a limit cycle can not be identified,

the computation is carried over a long enough time frameto ensure that a stationary flapping state has been reached.The efficiency is then obtained using statistical averagesover this interval in order to define A and �P�.

We are interested here in the influence of the differentparameters on the system’s performance, measured hereby the efficiency η , the ratio of the energy used in the out-put circuit (i.e. dissipated in the resistive elements) and ofthe flow kinetic energy flux through the cross-section oc-cupied by the device. In Figure 6, the evolution of η withβ and U∗ is represented for two different values of M∗.

One first observes that η vanishes for both β � 1and β � 1 and is only significant in the intermediaterange β ∼ 0.1–1, a direct result from the filtering prop-erties of the RC-loop: for β � 1, the circuit’s resis-

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tance is infinitesimal, resulting in negligible energy dis-sipation despite the charge transfers between the piezo-electric electrodes; for β � 1, the resistance is too largefor any charge transfer to be possible. The dissipatedpower is maximum when the frequency of the forcing onthe electrical circuit (here, the flapping flag frequency) isclose to the circuit’s fundamental frequency g/c. Thisis confirmed by comparing the results for M∗ = 1 andM∗ = 10: for larger M∗, higher order modes are knownto become dominant and have typically higher frequen-cies [9]. The maximum efficiency is then obtained fora higher characteristic frequency of the electrical circuit,namely a smaller value of β (Figure 6).

For a fixed β and increasing U∗, successive optimalregions are observed separated by low-efficiency transi-tion regions. This behavior can be interpreted from themode switching events observed on Figure 4 and 5: asU∗ is increased, the efficiency of the system increaseswhile the dynamics of the system remains locked on thesame mode, because the frequency of that mode is in-creasing with U∗. However, as more modes become un-stable the limit cycle oscillations can lock onto a differentmode with lower frequency resulting in a drop in the sys-tem’s efficiency. One observes that when the fluid inertiais increased (greater M∗), mode switching events occur atmore closely-spaced values of U∗.

The effect of β can also be observed on the stabil-ity threshold: for small M∗, the critical velocity increasesmonotonically with β between the short-circuit and open-circuit limits, while destabilization by damping is ob-served for larger M∗. Figure 6(b) also shows that the im-pact of β on the mode switching thresholds is similar tothat on the stability threshold.

The performance of the system can therefore be opti-mized by tuning the output circuit to the frequency of theflag dynamics. This frequency is however not constantbut is itself modified by the fluid-solid-electric coupling.In Fig. 7, we consider the evolution of the efficiency ofthe optimally-tuned system with M∗ and U∗: for a givenvalue of these two parameters, the maximum achievableefficiency (over all possible values of β ) is represented.One observes that the efficiency is in general an increas-ing function of U∗ (as seen previously) and M∗. Thishigher efficiency of the high-fluid loading limit (large M∗)confirms the linear analysis in Ref. [14], where the energytransfer from the fluid-solid system to the electrical circuitwas shown to be more efficient on configurations destabi-lized by damping. Maximum efficiencies above 10% areachievable, but it must be pointed out that the efficiencyis strongly dependent on U∗ at higher M∗ as critical flowvelocities leading to mode switching events get closer as

U*

10 2 10 1 100 1014

6

8

10

12

14

16

18

20

0

1

2

3

4

x 10 3

U*

10 2 10 1 100 1014

6

8

10

12

14

16

18

20

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

FIGURE 6: Evolution of the harvesting efficiency withthe tuning ratio β and flow velocity U∗ for α = 0.5 andM∗ = 1 (top) and M∗ = 10 (bottom). the white line corre-sponds to the stability threshold.

M∗ is increased (Figure 6). This results in an increasedsensitivity of the device’s performance with the flow ve-locity.

CONCLUSIONSThe present study provides some important insight

on the coupled dynamics of a classical fluid-solid system(a fluttering flag) and an output circuit producing electri-cal energy from the deformation of the structure. It wasshown that the coupling can lead to a destabilization ofthe system, corresponding to an increase in the operat-ing range of the device. The destabilized range of fluidloading M∗ also corresponds to the highest harvesting ef-ficiencies. The role of the tuning of the output circuit to

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M*

U*

100 1014

6

8

10

12

14

16

18

20

0

0.02

0.04

0.06

0.08

0.1

FIGURE 7: Optimal energy harvesting efficiency as afunction of M∗ and U∗ for α = 0.5. for each value of(M∗,U∗), the maximum efficiency obtained for the opti-mal value of β is represented. The white line correspondsto the minimum stability threshold over all β .

the fluid-solid frequency is here essential: only carefully-tuned system can harvest a significant amount of energy.

Here, we illustrated the critical role played by thenonlinear flapping mode selection on the energy harvest-ing efficiency of the system. The locking of the flag on aparticular dynamics is a strongly nonlinear phenomenon,and it is expected that it is also influenced by the outputcircuit behavior. Further investigation is therefore neces-sary to understand more completely the factors determin-ing the nonlinear dominant mode in order to ensure an in-creased robustness to the harvesting system with respectto the flow conditions.

ACKNOWLEDGMENTS. M. would like to acknowledge the support of a

Marie Curie International Reintegration Grant within the7th European Community Framework Program.

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