Resonance-induced enhancement of the energy harvesting performance of piezoelectric flags
aa r X i v : . [ phy s i c s . f l u - dyn ] D ec Resonance-induced enhancement of the energy harvesting performance ofpiezoelectric flags
Yifan Xia ∗ and S´ebastien Michelin LadHyX–D´epartement de M´ecanique, ´Ecole Polytechnique – CNRS, Route de Saclay, 91128 Palaiseau, France
Olivier Doar´e
IMSIA, ENSTA ParisTech, CNRS, CEA, EDF, Universit´e Paris-Saclay,828 bd des Mar´echaux, 91762 Palaiseau cedex France (Dated: October 11, 2018)The spontaneous flapping of a flag can be used to produce electrical energy from a fluid flowwhen coupled to a generator. In this paper, the energy harvesting performance of a flag coveredby a single pair of PVDF piezoelectric electrodes is studied both experimentally and numerically.The electrodes are connected to a resistive-inductive circuit that forms a resonant circuit with thepiezoelectric’s intrinsic capacitance. Compared with purely resistive circuits, the resonance betweenthe circuit and the flag’s flapping motion leads to a significant increase in the harvested energy. Ourexperimental study also validates our fluid-solid-electric nonlinear numerical model.
Flow-induced instabilities and vibrations have recentlyreceived a renewed attention as potential mechanisms toproduce electrical energy from ambient flows. Such insta-bilities enable a spontaneous and self-sustained motion ofa solid body which can be used to convert this mechanicalenergy into electrical form.[1–3]The flapping of a flexible plate in an axial flow (there-after referred to as a “flag”) is a canonical example ofsuch instabilities. Due to its rich and complex dynam-ics, this instability has been extensively studied duringthe last century.[4] The origin of this instability lies in acompetition between the destabilizing fluid force and thestabilizing structural elasticity. The flag becomes unsta-ble when the flow velocity exceeds a critical value, leadingto a large amplitude self-sustained flapping.[5–10]Piezoelectric materials produce electric charge dis-placements when attached to a deformable structure,[11]showing a “direct piezoelectric effect” that effectivelyqualifies them as electric generators. An output circuitconnected to the electrodes of the piezoelectric elementscan then exploit the generated electric current, as in vi-bration control applications.[12, 13] In the meantime, afeedback coupling is introduced by the inverse piezoelec-tric effect: any voltage between the electrodes creates anadditional structural stress that would potentially influ-ence the dynamics of the structure.Piezoelectric energy generators received an increasingattention over the last two decades,[14, 15] and a bur-geoning research effort has been invested on this topicever since.[16–19] The work of Allen & Smits using apiezoelectric membrane [20] inspired several recent stud-ies on flapping piezoelectric flags as energy-harvestingsystems.[21–25] Xia et al. [25] investigated numericallythe coupling between a piezoelectric flag and a resonantcircuit, and identified an electro-mechanical frequencylock-in phenomenon where the output circuit dictates ∗ [email protected] to the flag its flapping frequency and that significantlyincreases the energy-harvesting performance, comparedwith a simple resistive circuit.[24] This frequency lock-in phenomenon was however obtained for an idealizedconfiguration, where (i) the flag is continuously coveredwith infinitesimally-small piezoelectric patches and (ii)the piezoelectric coupling is strong. The present workfocuses on a configuration that is easier to obtain experi-mentally: the flag is covered by a single pair of piezoelec-tric patches made of Polyvinylidene Difluoride (PVDF), amaterial characterized by a relatively weak coupling. Theenergy-harvesting performance of this system is investi-gated both experimentally and numerically when con-nected to a resonant circuit.Experiments are performed using a PVDF piezoelec-tric flag. The PVDF film is cut into two patches of iden-tical size: 9.5 cm × × − kg. This flag is clampedat one end in a wind tunnel with a 50 cm ×
50 cm testsection (Fig. 1 a ). Using this single flag, the flag’s “ef-fective length”, denoted by L , is varied by adjusting theposition at which the flag is clamped. The remainingupstream section of the flag is attached to a rigid plateparallel to the flow so that its influence on the flow isminimized and the system effectively behaves as a flagof length L (Fig. 1 b ). The electrodes of the flag are con-nected to a data acquisition board (DAQ) recording thevoltage V , as well as an output circuit consisting of avariable resistor R ranging from 5 Ω to 10 Ω, and an in-ductor of inductance L in parallel connection. From anelectrical point of view, the piezoelectric flag is equivalentto a current source connected in parallel to an internalcapacitance C and the equivalent circuit of the experi-mental setup is shown in Fig. 1 c , where R L and R d arerespectively the internal resistance of the inductor andthe DAQ. The mean harvested power P , defined as theenergy dissipated in the output resistance R is obtained FIG. 1. ( a ) PVDF piezoelectric flag placed in the wind tunnelwith opaque walls (gray). ( c ) Photo of the clamped PVDFflag. ( c ) Equivalence of the harvesting circuit and data acqui-sition board (DAQ). as: P = h V iR − , (1)where h·i is the time-averaging operator. The average istaken over 5 seconds, sufficiently long compared with theflag’s flapping period (cf. Table I)The numerical model presented in previous works[24,25] is adapted to the present configuration. Its maincharacteristics are briefly presented here and the readeris referred to Michelin & Doar´e[24] for more details. Theflag of length L and height H is modeled as a clamped-free Euler-Bernoulli beam, whose mass per unit length isdenoted by µ , placed in a fluid of density ρ flowing uni-formly at velocity U ∞ . The fluid’s forcing on the flag iscomputed using a local force model based on the relativemotion between the flag and the flow.[26, 27] Contrary tothe previous studies referenced above, the flag is coveredby one single pair of piezoelectric patches; consequently,the electric charge displacement Q between the electrodesdepends only on the flag’s orientation at the trailing edge,Θ T , and on the voltage V , while the inverse piezoelectriceffect introduces an added torque M piezo applied locallyat the flag’s trailing edge. The relation between Q and V is prescribed by considering the electric circuit shownin Fig. 1 c .Using L , L/U ∞ , ρHL , U ∞ p µL/ C , U ∞ √ µ C L respec-tively as characteristic length, time, mass, voltage andelectric charge, the system’s electric state and the piezo-electric added torque M piezo are given, in dimensionlessforms by:¨ v + (cid:18) β L ω + 1 β e (cid:19) ˙ v + (cid:18) β L β e (cid:19) ω v + αU ∗ ¨ θ T + αU ∗ β L ω ˙ θ T = 0 , (2) M piezo = − αU ∗ v. (3) Also, the problem is characterized by the following di-mensionless parameters M ∗ = ρHLµ , U ∗ = U ∞ r µL B , H ∗ = HL ,α = χ r LB C , β = R U ∞ C L , ω = LU ∞ √LC , (4)with M ∗ the fluid-solid inertia ratio, U ∗ the reduced flowvelocity, and H ∗ the aspect ratio. The piezoelectric cou-pling coefficient, α , characterizes the mutual forcing be-tween the piezoelectric pair and the flag. Finally β and ω characterize the resistive and inductive properties of thecircuit. In the following, β , β d , and β L correspond to theharvesting resistor R , the DAQ’s internal resistance R d ,and the inductor’s internal resistance R L , respectively.In Eq. (2), β e corresponds to the equivalent resistance to R and R d connected in parallel: β e = ββ d β + β d . (5)The coupling coefficient α defined in Eq. (4) is deter-mined from the conversion factor χ , the bending rigid-ity B , and the capacitance C = 15 nF, which is mea-sured using a multimeter. Note that because the wholePVDF flag (i.e. upstream and downstream of the clamp,Fig. 1 b ), is covered by two patches which are entirelyconnected in the circuit through their electrode, the to-tal intrinsic capacitance C is independent of the effectivelength L . The bending rigidity B of the flag is deter-mined by measuring the flag’s free vibration frequency f , which, at the first vibration mode, is given by: [28] f = 3 . πL s Bµ . (6)The conversion factor χ is measured by connecting theflag uniquely to the DAQ, whose input impedance is R d = 10 Ω. The voltage V and deflection Θ T are re-lated through the direct piezoelectric effect,[24, 25] whichtogether with Ohm’s law leads to: R d χ ˙Θ T + R d C ˙ V + V = 0 . (7)Assuming purely harmonic signals, we write V = V e πft and Θ T = Θ e i(2 πft + φ ) , and using the previous equation: χ = V πf Θ s R d + 4 π f C . (8)The measurements of the flapping frequency f , the am-plitude of trailing edge orientation Θ , and the amplitudeof the voltage V are performed in a second wind tunnelwhose walls are transparent and allow for direct videorecording of the flag’s motion. This second wind tunnelis however not suitable for the rest of the present workdue to its strong confinement (test section of 10 cm × FIG. 2. Examples of measurement of Θ T for L = 8 cm and U = 12 . χ sinceEq. (8) is valid regardless of the fluid forcing applied onthe plate in the limit of linear piezoelectric coupling as-sumed here. Moreover, χ is an intrinsic property of thepiezoelectric material that does not depend on the flowcondition.The voltage is recorded using the DAQ (Figure 3), and V is obtained by averaging the signal’s peak values overthe duration of the recording ( > f is equal to the voltage signal’s frequency, whichis obtained using the Fourier transform of the recordedsignal. In order to measure the trailing edge orientationΘ T , the flag’s motion is recorded using a high-speed cam-era (Phantom R (cid:13) v9) at 960 frames per second. In eachframe, Θ T is measured using ImageJ R (cid:13) (Fig. 2) and Θ is obtained from these measurements (Fig. 3 b ). Usingthis procedure, we obtained B = 2 . × − N · m and χ = 1 . × − C. As a consequence, for the PVDF flagused here, α ∼ .
085 for L = 6 cm, and α ∼ . L = 8cm. These values are retained for the numerical study.A first comparison between experiments and simula-tions is conducted for the case without piezoelectric cou-pling ( α = 0). Figure 4 shows the dimensionless flappingfrequency ω as a function of dimensionless velocity U ∗ ,obtained both experimentally and numerically, for L = 6cm and 8 cm. Both results show a good agreement interms of the flapping frequency, suggesting that the nu-merical model is capable of capturing the flag’s essen-tial dynamics. The fact that the numerical simulation −10010 V V o l t ag e ( V ) T i m e ( s ) Θ T ( r a d ) Θ π − π FIG. 3. Measurement of voltage V and trailing edge angleΘ T for L = 8 cm and U ∞ = 12 .
16 18 20 22 2400.511.52 ( a ) ω U ∗ ExperimentalNumerical
16 18 20 22 24 U ∗ ( b ) ExperimentalNumerical
FIG. 4. Comparison of dimensionless flapping frequency ω asa function of dimensionless velocity U ∗ obtained experimen-tally and numerically with α = 0. In ( a ) Case A, ( b ) Case B(see Table I). slightly underpredicts the frequency compared with theexperiments (Fig. 4) is likely due to the wind tunnel’sweak but existing transverse confinement.[29]The energy harvesting performance is assessed bothexperimentally and numerically for two different effec-tive lengths. The parameters corresponding to the twocases A and B are shown in Table I. Figure 5 shows thedimensionless harvested power P (Fig. 5 a, b ) and flappingfrequency ω (Fig. 5 c, d ) as functions of the harvesting re-sistance β , obtained numerically and experimentally. Weobserve that for both values of L , when the circuit is inresonance with the flapping flag, the harvested power in-creases considerably compared with the purely resistivecase for almost all values of β . For a same resistance,the resonant circuit is able to harvest twice as much en- L (cm) U ∞ (m/s) f (Hz) L (H) α M ∗ H ∗ U ∗ ω Case A 6 20.9 56.8 530 0.085 0.410 0.417 17.91 1.06Case B 8 17.8 41.0 1000 0.1 0.547 0.313 21.18 1.14TABLE I. Parameter values of two piezoelectric flags and corresponding numerical simulation
012 x 10 −5 P ( a ) L = 6 c m Numerical R PureNumerical Resonant −1 ω β ( c ) L = 6 c m
012 x 10 −5 ( b ) L = 8 c m Experimental R PureExperimental Resonant −1 β ( d ) L = 8 c m FIG. 5. ( a, b ) Dimensionless harvested power P and ( c, d )flapping frequency ω as function of β obtained both from ex-periments and nonlinear numerical simulations for cases Aand B (see Table I). ergy as the optimal purely resistive circuit. Meanwhile,the optimal resistance is larger when a resonant circuitis used, showing that resonant circuits maintain a sat-isfactory energy-harvesting performance even for largeresistances for which little energy would be harvestedwith no inductance. At the optimal resistance for theresonant circuit, the system harvests 4 to 5 times moreenergy than the optimal resistive circuit. These resultssuggest that the presence of an inductance improves theenergy harvesting performance by resonance. The factthat the circuit works in resonance produces a high volt-age, leading to an enhanced harvested power. The sameresults are found using numerical simulations, showinga good agreement with the experiments in terms of theharvested power P .However, both experiments and numerical simulationssuggest that resonance introduces little impact on theflags’ flapping dynamics, as the flapping frequencies ω for both cases are almost identical for both types of cir-cuit, and remain unchanged when varying β . The ab-sence of a strong feedback induced by the inverse piezo-electric effect in the present work, as well as in otherstudies,[30] is mainly due to the weak coupling of thechosen piezoelectric material.[25] Another important fac- tor is also the large internal resistance of the inductor( R L ∼ [1] M. M. Bernitsas, K. Raghavan, Y. Ben-Simon, and E. M.Garcia, J. Offshore Mech. Arct. Eng. , 041101 (2008).[2] Z. Peng and Q. Zhu, Phys. Fluids. , 12362 (2009).[3] J. A. C. Dias, J. C. De Marqui, and A. Erturk, App.Phys. Lett. , 044101 (2013).[4] M. J. Shelley and J. Zhang, Annu. Rev. Fluid. Mech. ,449 (2011).[5] M. Argentina and L. Mahadevan, Proc. Natl. Acad. Sci.U.S.A. , 1829 (2005).[6] C. Eloy, C. Souilliez, and L. Schouveiler, J. Fluids Struct. , 904 (2007).[7] B. S. H. Connell and D. K. P. Yue, J. Fluid Mech. ,33 (2007).[8] S. Alben and M. J. Shelley, Phys. Rev. Lett. , 074301(2008).[9] S. Michelin, S. Llewellyn Smith, and B. Glover, J. FluidMech. , 1 (2008).[10] E. Virot, X. Amandolese, and P. H´emon, J. Fluids Struct. , 385 (2013).[11] J. Yang, An Introduction to the Theory of Piezoelectricity (Springer, 2005).[12] N. W. Hagood and A. von Flotow, J. Sound Vib. ,243 (1991).[13] O. Thomas, J. F. Deu, and J. Ducarne, Int. J. Numer.Methods. Eng. , 235 (2009).[14] C. Williams and R. B. Yates, Sens. Actuat A: Physical , 8 (1996).[15] M. Umeda, K. Nakamura, and S. Ueha, Jpn. J. of Appl.Phys. , 3267 (1996).[16] H. A. Sodano, D. J. Inman, and G. Park, Shock VibrationDig. , 197 (2004). [17] S. Anton and H.A.Sodano, Smart Mater. Struct. , R1(2007).[18] A. Erturk and D. J. Inman, Piezoelectric energy harvest-ing (John Wiley & Sons, 2011).[19] R. Cali`o, U. B. Rongala, D. Camboni, M. Milazzo, C. Ste-fanini, G. de Petris, and C. M. Oddo, Sensors , 4755(2014).[20] J. J. Allen and A. J. Smits, J. Fluids Struct. , 629(2001).[21] J. A. Dunnmon, S. C. Stanton, B. P. Mann, and E. H.Dowell, J. Fluids Struct. , 1182 (2011).[22] O. Doar´e and S. Michelin, J. Fluids Struct. , 1357(2011).[23] D. T. Akcabay and Y. L. Young, Phys. Fluids. (2012).[24] S. Michelin and O. Doar´e, J. Fluid Mech. , 489(2013).[25] Y. Xia, S. Michelin, and O. Doar´e, Phys. Rev. Applied , 014009 (2015).[26] M. J. Lighthill, Proc. R. Soc. Lond. B , 125 (1971).[27] F. Candelier, F. Boyer, and A. Leroyer, J. Fluid Mech. , 196 (2011).[28] S. Timoshenko, History of strength of materials: witha brief account of the history of theory of elasticity andtheory of structures (Courier Corporation, 1953).[29] F. Belanger, M. P. Paidoussis, and E. de Langre, AIAAjournal , 752 (1995).[30] C. De Marqui, W. G. Vieira, A. Erturk, and D. J. Inman,J. Vib. Acoust. , 011003 (2011).[31] J. D. Irwin and R. M. Nelms, Basic engineering circuitanalysis (Wiley Publishing, 2008).[32] M. Pineirua, O. Doar´e, and S. Michelin, Journal of Soundand Vibration346