Fluid-solid-electric lock-in of energy-harvesting piezoelectric flags Yifan Xia∗ and S´ebastien Michelin LadHyX–D´epartement de M´ecanique, E´cole Polytechnique, Route de Saclay, 91128 Palaiseau, France Olivier Doar´e ENSTA Paristech, Unit´e de M´ecanique (UME), 828 boulevard des Mar´echaux, 91762, Palaiseau, France (Dated: January 12, 2015) The spontaneous flapping of a flag in a steady flow can be used to power an output circuit using piezoelectric elements positioned at its surface. Here, we study numerically the effect of inductive circuits on the dynamics of this fluid-solid-electric system and on its energy harvesting efficiency. Inparticular,adestabilizationofthesystemisidentifiedleadingtoenergyharvestingatlowerflow 5 velocities. Also,afrequencylock-inbetweentheflagandthecircuitisshowntosignificantlyenhance 1 the system’s harvesting efficiency. These results suggest promising efficiency enhancements of such 0 flow energy harvesters through the output circuit optimization. 2 n I. INTRODUCTION researchers have contributed to this field in order to im- a prove the efficiency of such energy-harvesting systems J [24–27]. Somestudiesshowthatsimpleresonantcircuits, 9 Flowinducedinstabilitiesandvibrationshaverecently i.e. resistive-inductive circuits combined with the piezo- received a renewed attention as potential mechanisms to ] produce electrical energy from geophysical flows (wind, electricmaterial’sintrinsiccapacitance[20],offerpromis- n ing opportunities to achieve high efficiency [28, 29]. tidal currents, river flows, etc.). They indeed enable a y Flow energy harvesting can be achieved by exploiting d spontaneous self-sustained motion of a solid body which the unsteady forcing of the vortex wake generated by an - can be used as a generator, effectively converting this u upstream bluff body to force the deformation of a piezo- mechanical energy into electrical form [1–5]. l electric membrane [16, 17, 30]. Fluid-solid instabilities f A canonical example of such instability is the flapping . offer a promising alternative as they are able to generate s of a flexible plate in an axial flow (e.g. a flag), which c spontaneousandself-sustainedstructuraldeformationof has been extensively investigated for its rich and com- i s plex dynamics [6]. The origin of this instability lies in the piezoelectric structure, e.g. cross-flow instabilities y [29, 31, 32]. In their work, De Marquis et al. [29] used a a competition between the destabilizing fluid force and h resistive circuit and a resistive-inductive one, and found the stabilizingstructural stiffness. Beyond a critical flow p inadditiontothebeneficialeffectoftheresonancetothe [ velocity Uc, the flag becomes unstable, leading to large energy harvesting, that a resistive-inductive circuit may amplitude self-sustained flapping [7–14]. 1 also affect the stability of the vibration source. How- Energyharvestingbasedonflappingplatesmayfollow v ever, the resonant circuit’s influence on the structure’s two routes: producing energy either from the displace- 1 dynamics was not reported in this work. 9 ment [15] or from the deformation of the plate [16–18]. Fluid-solid instabilities in axial flows, including the 1 The latter has recently been the focus of several studies aforementioned flapping flag instability, are also stud- 2 based on active materials [19]. 0 ied in the context of piezoelectric energy-harvesting [33– Piezoelectric materials, considered in this article, pro- . 36]. Inparticular,Michelin&Doar´e[34,36]considereda 1 duce electric charge displacement when strained [20], a piezoelectric flag coupled with a purely resistive output. 0 “directpiezoelectriceffect”thateffectivelyqualifiesthem 5 They observed moderate efficiency, which is maximized aselectricgenerators. Thiselectricchargecanbeusedin 1 when the characteristic timescale of the circuit is tuned an output circuit connected to their electrodes, as in vi- : to the frequency of the flag. A significant impact of the v bration control applications [21]. Piezoelectric materials circuit’s properties on the fluid-solid dynamics was also i alsointroduceafeedbackcouplingofthecircuitontothe X identified. mechanical system: any voltage between the electrodes r The present work therefore focuses on the coupling a creates an additional structural stress that modifies its of the fluid-solid system, i.e. the flapping piezoelectric dynamics (inverse piezoelectric effect). flag, to a basic resonant circuit (resistive-inductive loop The concept of piezoelectric energy generator has re- with the piezoelectric material’s intrinsic capacitance). ceived an increasing amount of interest in the last 20 Resonance is expected when the flapping frequency ω of years [22, 23]. Its basic idea is to convert ambient vibra- theflag, forcingthecircuit, matchesthecircuit’snatural tion energy to useful electric energy through piezoelec- frequency. Using linear stability analysis and nonlinear tric materials implemented on vibration sources. Many numerical simulations of the fluid-solid-electric coupled system, we investigate the impact of such resonance on the dynamics and on the amount of energy that can be extracted from the device (i.e. the energy dissipated in ∗ [email protected] the resistive elements). This explicit description of both 2 the fluid-solid and the electric systems’ dynamics pro- vides a deeperand more accurate insight into the energy harvesting process than its classical modeling as a pure damping [4, 15, 18]. II. FLUID-SOLID-ELECTRIC MODEL A. Fluid-solid coupling The coupled system considered here is a cantilevered plate of length L and span H, placed in an axial flow of density ρ and velocity U . The flag’s surface is cov- f f ered by pairs of piezoelectric patches (Fig. 1a). Within each pair, two patches of reversed polarities are con- nected through the flag, the remaining electrodes being connected to the output circuit [34, 37]. The resulting three-layer sandwich plate is of lineic mass µ and bend- ing rigidity B. We restrict here to purely planar de- formations (bending in the z-direction and twisting are neglected). The flag’s dynamics are described using an inextensible Euler–Bernoulli beam model forced by the piezoelectric pair fluid: equivalence µX¨ =(Tτ −M(cid:48)n)(cid:48)+F, (1) f FIG. 1. Flapping piezoelectric flag in a uniform axial flow. X(cid:48) =τ (2) (a): 3-dimension view; (b): zoom of the circled area in (x,y) plane;(c): equivalentcircuitofapiezoelectricpairconnected with clamped-free boundary conditions: with a parallel RL circuit. at s=0: X=X˙ =0, (3) at s=L: T =M=M(cid:48) =0. (4) where C is the drag coefficient for a rectangular plate d in transverse flow. It is important to mention that the HereT andMarerespectivelythetensionandthebend- flapping flag dynamics considered here implies a large ing moment. Throughout this article, ˙ and (cid:48) denote Reynolds number. A 10 cm long/wide flag in a wind derivativeswithrespecttotands,respectively. Thefluid flowing at around 5 m/s leads to Re∼104, while in wa- loadingFfiscomputedusingalocalforcemodelfromthe ter we would have Re ∼ 105 for a flow velocity around relative velocity of the flag to the incoming flow: 1 m/s. These values of Re are sufficiently large to jus- tify a constant value of C [41]. A Re-dependence could U n+U τ =X˙ −U e . (5) d n τ f x howeverbeintroducedtoextendtheapplicabilityofthis model to intermediate Re. The present fluid model includes two different contribu- The fluid forcing is the sum of these two terms: tions. The first one results from the advection of the fluid added momentum by the flow, an inviscid effect, F =F +F . (8) f react resist and can be obtained analytically in the slender body limit through the Large Amplitude Elongated Body The- The applicability of this result to flapping flag was con- ory [38]: firmed experimentally, at least up to an aspect ratio H/L=0.5, the value considered in our work [42]. (cid:20) (cid:21) 1 F =−m ρH2 U˙ −(U U )(cid:48)+ U2θ(cid:48) n. (6) react a n n τ 2 n B. Piezoelectric effects Candelier et al [39] recently proposed an analytic proof ofthisresult,andsuccessfullycomparedittoRANSsim- Piezoelectric patch pairs are positioned on either side ulationsforfishlocomotionproblems. Theseauthorsalso oftheplate,withoppositepolarity. Thisguaranteesthat, stated that in the case of spontaneous flapping, it is nec- duringtheplate’sbending,thetwopatchesreinforceeach essarytoaccountfortheeffectoflateralflowseparation, other, rather than cancel out. Each piezoelectric pair is which is empirically modeled by the following term [40]: connected to a resistance and an inductance in parallel connection (Fig. 1c). 1 We focus on the limit of continuous coverage by in- Fresist =−2ρHCd|Un|Unn, (7) finitesimal piezoelectric pairs [34, 36]. Within this limit, 3 theelectricstateofthepiezoelectricpairsischaracterized energy transferred to the circuit, and as such critically by the local voltage v and lineic charge transfer q, which impacts the energy harvesting performance. Finally β are continuous functions of the streamwise Lagrangian and ω characterize respectively the resistive and induc- 0 coordinate s. The electrical circuits are characterized by tive properties of the circuit. alineicconductanceg, andalineicinductiveadmittance The effect of the mechanical parameters, M∗, U∗ and 1/l. The electrical charge displacement across a piezo- H∗, on the flapping flag dynamics have been extensively electric pair resulting from the direct piezoelectric effect studied in the literature [34, 36]. In the following, we is given by: focus specifically on the dynamical properties of the cir- cuit and maintain M∗ = 1 and H∗ = 0.5 throughout q =χθ(cid:48)+cv, (9) this study. Unless stated otherwise, we will also con- sider α = 0.3, a value consistent with existing material where χ is a mechanical/piezoelectric conversion coeffi- properties [34]. The effect of varying the piezoelectric cientandcisthelineicintrinsiccapacityofapiezoelectric coupling will also be briefly discussed. The full nonlin- pair[43]. Equation(9)showsthattheeffectonthecircuit eardynamicsofthecoupledsystemarenowdescribedin of the piezoelectric components is that of a current gen- non-dimensional form by: eratorwithaninternalcapacitance(Fig.1c). Thecharge conservation of the resulting RLC circuit leads to: x¨=M∗(Tτ)(cid:48)−(θ(cid:48)(cid:48)n)(cid:48)+α(v(cid:48)n)(cid:48)+M∗fn, (16) f v+glv˙ +lq¨=0. (10) x(cid:48) =τ, (17) βv¨+v˙ +βω2v+αβθ¨(cid:48) =0, (18) The inverse piezoelectric effect manifests as an added 0 bending moment, so that the total bending moment in and the nondimensional boundary conditions are: the structure is given by [37]: at s=0: x=x˙ =0, (19) M=Bθ(cid:48)−χv. (11) at s=1: T =θ(cid:48)−αv =θ(cid:48)(cid:48)−αv(cid:48) =0. (20) Finally we define the harvested energy as the time av- erage of the total rate of dissipation in the resistive ele- The non-dimensional tension T is computed using the ments, which is formally given as: inextensibility of the beam [44]. Finally the nondimensional fluid loading is obtained (cid:42) (cid:43) (cid:90) L as: P = gv2ds , (12) (cid:20) (cid:21) 0 1 1 f =−m H∗ u˙ −(u u )(cid:48)+ u2θ(cid:48) − C |u |u , f a n n τ 2 n 2 d n n and the efficiency is defined as: (21) P with m = π/4 and C = 1.8, the added mass and drag a d η = P . (13) coefficients for a rectangular plate in transverse flow [45, ref 46]. P is the kinetic energy flux of fluid passing through ref the cross section occupied by the flag: III. CRITICAL VELOCITY 1 P = ρU3 ×AH, (14) ref 2 ∞ ThecriticalvelocityU∗ isdefinedastheminimumflow c where A is the peak-to-peak amplitude of the flapping velocity above which self-sustained flapping can develop flag. and energy can be harvested. In this part, the influence In the following, the problem is nondimensionalized of the RL loop on U∗ is studied using linear stability c using the elastic wave velocity U =(cid:112)B/L2µ as charac- analysis, in the limit of small vertical displacement, i.e. s teristicvelocity. L,L/U ,ρHL2,U (cid:112)µ/c,U √µcarere- y (cid:28) 1, allowing linearization of Eqs. (16)–(18). The s s s resulting linear equations are: spectivelyusedascharacteristiclength,time,mass,volt- age and lineic charge. As a result, six non-dimensional (1+M )y¨+2M U∗y˙(cid:48)+M U∗2y(cid:48)(cid:48)+y(cid:48)(cid:48)(cid:48)(cid:48)−αv(cid:48)(cid:48) =0, parameters characterize the coupled system: a a a (22) M∗ = ρfHL, U∗ = Uf, H∗ = H, βv¨+v˙ +βω02v+αβy¨(cid:48)(cid:48) =0, (23) µ U L s (15) α= √χ , β = cUs, ω = L√ , where Ma =πM∗H∗/4. Bc gL 0 U lc Eqs. (22) and (23) are then projected onto the funda- s mentalbeammodesandtheirsecondderivatives,respec- withM∗ thefluid-solidinertiaratio,U∗ thereducedflow tively, andrecastasaneigenvalueproblem. Thecoupled velocity, and H∗ the aspect ratio. The piezoelectric cou- system is unstable if one of its eigenfrequencies has a plingcoefficient,α,characterizesthefractionofthestrain positive imaginary part. 4 12 ← RC Short circuit→ 1 (a) 11 Uc∗= Uc0 0 ℑ(ω) 10 −1 U∗ c 9 −2 8 3 10 (b) 7 ℜ(ω) 1 10 6 0 1 2 10 10 10 ω 0 −1 10 −1 0 1 2 3 FIG. 2. Evolution of the critical velocity with ω at α=0.3 10 10 10 10 10 0 ω andβ =0.05(solid),β =1(dash-dot),β =4(dashed),β =8 0 (dotted). U∗ =U0 is plotted (dash-dot, gray) as a reference. c c FIG.3. (coloronline)Evolutionwithω of(a)theimaginary 0 part (growth rate) and (b) the real part (frequency) of the second and third pairs of eigenvalues for α = 0.3, U∗ = 10 The evolution of U∗ with ω is computed using lin- c 0 and β = 4. In both (a) and (b), solid lines represent me- ear stability analysis and is shown on Fig. 2. For inter- chanical modes and dashed lines represent electrical modes. mediate values of ω (3 < ω < 10), we observe a sig- 0 0 The destabilized pair is plotted in red and the other plotted nificant destabilizing effect of inductance that increases in black. In (b), ω is plotted with thin gray solid line as a 0 with β, as the circuit becomes dominated by inductive reference. effects. For small β, however, no such destabilization is observed, as the inductance plays little role in this re- sistive limit. These results highlight a major benefit of IV. NONLINEAR DYNAMICS AND ENERGY the circuit’s inductive behavior: the instability thresh- HARVESTING old may be lowered, resulting in energy harvesting with slower flow velocity. For ω0 (cid:29) 1, the inductance acts Above the critical velocity, the unstable coupled sys- as a short-circuit, and U∗ converges, regardless of β, to tem experiences an exponential growth in its amplitude, c U0, the critical flow velocity without coupling (α = 0). which eventually saturates due to nonlinear effects. A c For ω0 (cid:28)1, the effects of inductance are negligible, and direct integration of the fluid-solid-electric system’s non- U∗ (cid:62) U0, illustrating the stabilizing effect of the resis- linear equations Eqs. (16)–(18) is performed using an c c tance. Note that a destabilizing effect of the resistance implicit second order time-stepping scheme [44]. The can be observed at higher values of M∗[34], and in more flag is meshed using Chebyshev-Lobatto nodes, and a general cases of damping [47]. Chebyshevcollocationmethodisusedtocomputespatial To determine the origin of this destabilization, Fig. 3 derivatives and integrals. At each time step, the result- shows the evolution of the two most unstable pairs of ing nonlinear system is solved using Broyden’s method eigenvalues with ω at U∗ = 10, which is lower than U0 [48]. The simulation is started with a perturbation in 0 c but higher than the minimum critical velocity (Fig. 2). the flag’s orientation (θ(s,t = 0) (cid:54)= 0), and is carried Starting from ω (cid:29) 1 and decreasing ω , the electrical out over a sufficiently long time frame so as to reach a 0 0 circuit evolves successively from short circuit, to RLC permanent regime. loop, and finally to a purely resistive circuit. Instability ThereducedflowvelocityischosenatU∗ =13,avalue occurs when the imaginary part of any mode becomes sufficientlyhigherthanthecriticalvelocityU∗. Theflag’s c positive. behavior is observed to drastically differ with varying ω 0 In the absence of coupling (ω0 (cid:29) 1), both pairs con- (Fig. 4). When ω0 is within the range of destabiliza- sist of two eigenvalues: (i) an electrical mode with a tion, the flag undergoes a remarkably larger deformation frequency equal to ω0; (ii) a mechanical mode, with a (Fig. 4b) than with other values of ω0 (Fig. 4a). frequency independent of ω . Decreasing ω leads to in- Figure5ashowstheevolutionoftheefficiencywithω , 0 0 0 teractions between the electrical and mechanical modes, and demonstrates that this increased flapping amplitude successively within each pair. This interaction destabi- indeed leads to a significant efficiency improvement: a lizes the mechanical mode, leading to the flag’s instabil- maximum efficiency of 6% is obtained here, significantly ityatintermediateω (Fig.3). Notethatthisinteraction higher than the optimized efficiency obtained at M∗ =1 0 within other pairs also leads to an increase of (cid:61)(ω) for and U∗ =13 without inductance (∼0.1%) [36]. themechanicalmode,butdoesnotleadtoinstability(at Within the high efficiency range, the flapping fre- least for M∗ =1). quency is deviated and locks onto the natural frequency 5 0.1 (a) (a) 0.06 y 0 −0.1 η0.04 0.4 (b) 0.02 0.2 0 2 10 (b) y 0 ω 1 10 −0.2 0 10 −0.4 0 1 2 0 0.2 0.4 0.6 0.8 1 10 10 10 x ω 0 FIG. 4. Flapping motion of flags at α=0.3, β =4, U∗ =13 FIG.5. (a)Harvestingefficiencyηand(b)flappingfrequency and (a) ω0 =3.25 (no lock-in), (b): ω0 =4.12 (lock-in). ω as a function of ω0 for α = 0.3, U∗ = 13, and β = 0.05 (solid), β =1 (dash-dot), β =4 (dashed), β =8 (dotted). of the circuit, ω (Figure 5b). A frequency lock-in is 0 12 therefore observed here, similar to the classical lock-in observed in Vortex-Induced Vibrations (VIV) [49, 50]: 11 thefrequencyofanactiveoscillator(theflag)isdictated by the natural frequency of a coupled passive oscillator (the circuit). The lock-in range is extended by a re- 10 U∗ duction of the circuit’s damping (1/β), consistently with c what is observed in VIV for varying structural damping 9 [51]. The lock-in range leading to high efficiency coin- cides with the range of ω associated with the desta- 0 8 bilization by inductance (Fig. 2). This suggests that a coupled piezoelectric flag, once destabilized by inductive 7 effects, may flap at a frequency close to the natural fre- 0 1 2 10 10 10 quencyofthecircuit. Asaresult,apermanentresonance ω 0 takes place between the flag and the circuit, leading to increased flapping amplitude and harvesting efficiency. FIG. 6. Evolution of the critical velocity with ω and β =4, By varying ω within the lock-in range, Fig. 5 shows 0 0 M∗ =1andα=0.1(dash-dot),α=0.2(dashed)andα=0.3 that when the frequency of the output circuit matches (solid). the short-circuit natural frequency of the flapping flag (ω ∼ 17.5), the maximal efficiency is obtained for every valueofβ. Thisobservationhighlightsagaintheinterest cuitviapiezoelectriceffects. Inpractice,thiscouplingco- of exciting piezoelectric structures at their natural fre- efficient α, defined in Eq. (15), depends primarily on the quencies for energy harvesting, as suggested by previous materialsusedforthepiezoelectricflag. Theimportance studies, where maximal efficiency is observed when the ofthecouplingfactorhasbeenreportedbymanystudies external forcing resonates with the piezoelectric system onenergyharvestingbypiezoelectricsystems[25,27]. In [17, 52, 53]. The existence of a lock-in extends this ef- Refs. [34, 36], a dependence of the harvesting efficiency fect to a larger range of parameters, by maintaining the to α2 was identified. system at resonance, hence guaranteeing efficient energy Inthepresentwork,theinfluenceofαisobservedboth transfers from the flag to the circuit. intermsofcriticalvelocity(Fig.6)andthelock-inrange (Fig. 7). In Fig. 6, lower critical velocities and larger destabilization range in terms of ω are observed with 0 V. IMPACT OF PIEZOELECTRIC COUPLING increasing α. Fig. 7 shows that the range of frequency lock-in also increases with increasing α. The impact of A decisive factor is the intensity of piezoelectric cou- the coupling coefficient α on the system’s performance pling, characterized by α in this work. It quantifies the is again identified: a strong piezoelectric coupling is de- proportionofthemechanicalworktransmittedtothecir- sired so that the beneficial effects of a resonant circuit, 6 2 50 10 (a) 40 ω 30 20 ω 1 10 10 0.06 (b) 0.04 η 0 10 0 1 2 10 10 10 0.02 ω 0 0 FIG. 7. Flapping frequency ω as a function of ω for β =4, 12 14 16 18 20 22 24 0 M∗ = 1, U∗ = 13 and α = 0.1 (dash-dot), α = 0.2 (dashed) U∗ and α=0.3 (solid). FIG.8. (a)Flappingfrequencyωand(b)harvestingefficiency η as a function of U∗ for α = 0.3, M∗ = 1, β = 3.5 and namelythedestabilizationandthefrequencylock-in,can ω0 =0.1 (dotted), ω0 =29 (solid), ω0 =1000 (dashed). be obtained. Inpractice, α∼0.3couldbeexpectedwithlargescale devices [34], and in general α ∼ 0.1 is achievable, for Thecriticalimpactofthecouplingcoefficientαonthe example using two 10 cm × 10 cm MFC patches glued system’s performance is also underlined in the present by an epoxy layer with a thickness of 0.1 mm. A potent work through its strong influence on the destabilization piezoelectric material, leading to a strong coupling, is range and the lock-in range. The choice of the piezo- therefore an essential prerequisite to utilize the lock-in electric materials is therefore essential in the practical phenomenon. achievement of high efficiency. Thislock-inmechanismalsoplaysacriticalroleinthe VI. CONCLUSION robustness of the energy harvesting process with respect totheflowvelocity(Fig.8). Lock-inindeedpersistsover a wide range of U∗, effectively acting as a passive con- The results presented here provide a critical and trol of the flapping frequency in response to flow veloc- new insight on the dynamics of a piezoelectric energy- ity: while ω increases rapidly in the limit of weak fluid- harvesting flag. First and foremost, they emphasize how solid-electricinteractions, lock-inwiththeoutputcircuit the fundamental dynamics of the energy harvesting sys- maintains ω ≈ ω and high harvesting efficiency over a tem and of the output circuit may strongly impact the 0 large range of flow velocity. Such a control of the flap- motion of the structure and its energy harvesting per- pingfrequencywasshowntobeessentialforefficiencyen- formance. Also, they identify two major performance hancement and robustness (e.g. to delay mode switches enhancements associated with the resonant behavior of or frequency changes) [36]. thecircuit,namely(i)adestabilizationofthefluid-solid- electricsystem,leadingtospontaneousenergyharvesting Theseresultsnonethelessillustratethefundamentalin- at lower velocity; and (ii) a lock-in of the fluid-solid dy- sights and technological opportunities offered by the full namics on the circuit’s fundamental frequency, resulting nonlinear coupling of a passive resonant system (elec- inanextendedresonanceandasignificantincreaseofthe tric, mechanical or other) to an unstable piezoelectric harvested energy. structureforthepurposeofenergyharvesting. Thelock- Thislock-inbehaviorattheheartofbotheffectsabove in phenomenon and the enhanced performance demon- is classically observed in VIV where it is also responsible stratedbythecouplingbetweenthepiezoelectricflagand formaximumenergyharvesting[5]; itisinfactageneral asimpleresonantcircuitopentheperspectiveofapplying consequenceofthecouplingofanunstablefluid-solidsys- different kinds of resonant systems to energy-harvesting tem to another oscillator’s dynamics. We therefore ex- piezoelectric system. The choice of an inductive circuit pect that the conclusions presented in the present paper ismotivatedbyitssimplicity,whileinpractice,otherde- go beyond the simple inductive-resistive circuit consid- signs of resonant circuits may present important advan- eredhere,andshouldbeapplicabletoamuchlargerclass tages over the proposed formulation in our work. Mean- of resonant systems. 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