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S2352-4316(20)30008-0 https://doi.org/10.1016/j.eml.2020.100633 EML 100633

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Extreme Mechanics Letters

Received date : 5 November 2019 Revised date : 7 January 2020 Accepted date : 11 January 2020 Please cite this article as: Z. Yan, L. Wang, M.R. Hajj et al., Energy harvesting from iced-conductor inspired wake galloping, Extreme Mechanics Letters (2020), doi: https://doi.org/10.1016/j.eml.2020.100633. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Energy harvesting from iced-conductor inspired wake galloping Zhimiao Yana , Lingzhi Wangb , Muhammad R Hajjc , Zhitao Yanb , Yi Sunb , Ting Tand,∗ a

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State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean & Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China b Key Laboratory of New Technology for Construction of Cities in Mountain Area (Ministry of Education), School of Civil Engineering, Chongqing University, Chongqing 400030, China c Department of Civil, Environmental and Ocean Engineering, Stevens Institute of Technology, Hoboken, NJ, 07030, United States d State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

Abstract

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Energy harvesting from wake galloping of crescent-shape and D-shape cross-section bluff bodies inspired by galloping of iced conductors is investigated. Analytical models are derived in the framework of electromechanical extension of the Hamilton’s principle, Euler-Bernoulli beam assumption and quasi-steady hypothesis. Experiments are performed in an opencircuit boundary layer wind tunnel to investigate the wake effects of the upstream fixed interference cylinder on the performance of the downstream energy harvester and validate the analytical model. The observations are explained with the wake characteristics and lift force as determined from Large Eddy Simulation. The results show that the wake effects of the upstream fixed interference cylinders can positively enhance the performance of the downstream energy harvester. The wake galloping energy harvesters with the iced D-shape and crescent-shape bluff bodies display remarkable enhancement in terms of the harvested power density for wide interference spacing range when compared to energy harvesters placed in the wake of a non-iced circular bluff body. The power density of the wake galloping energy harvester with the iced-conductor inspired bluff bodies at the optimal interference spacing is about 63 times that of the power density of the wake galloping energy harvester with the classic circular bluff body. Keywords: piezoelectric energy harvester, wake galloping, iced conductors, Large Eddy Simulation

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Ambient energy harvesting is considered as an essential technology for developing selfpowered wireless sensor networks [1] and monitoring systems [2, 3] and devices [4, 5]. Energy harvesting from fluid-induced vibrations [6], such as vortex-induced vibration (VIV) [7–9], galloping [10, 11], flutter [12, 13] and stall-induced oscillations [14, 15], has been investigated in the past few years. Of particular interest is the wake galloping phenomenon, which is induced by placing a leeward cylinder in the wake of the windward cylinders. This aerodynamic instability phenomenon might be more appropriate for energy harvesting than VIV, ∗

Corresponding author Email address: [email protected] (Ting Tan)

Preprint submitted to Elsevier

January 8, 2020

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galloping and flutter due to larger oscillations and higher power output [16]. Jung and Lee used wake galloping for electromagnetic energy harvesting and experimentally verified it to be an efficient alternative energy source [17]. Abdelkefi et al. investigated the wake effects of an upstream circular cylinder on the performance of a downstream piezoelectric energy harvester consisting of square section cylinder [18]. The optimal distance between the upstream interference cylinder and the galloping cylinder was reported by Zhang and Wang [19]. To improve the frequency bandwidth of the lock-in region, Alhadidi and Daqaq supplemented a linear wake-galloping system with a bi-stable restoring force [20]. The energy harvesting of two bluff bodies in tandem arrangement was also investigated using two-dimensional numerical simulations. The circular cylinder was found to be much more easily influenced by the wake [21]. Usman et al. proposed an arrangement of upstream and downstream cylinders for energy harvesting from a broad wind spectrum [22]. Zhang et al. investigated enhancing the ability to harvest energy from galloping systems at relatively low speeds by addressing the effects of various cross-sections of the downstream interference cylinder on the upstream energy harvester [23]. Petrini and Gkoumas evaluated the implementation of piezoelectric energy harvesting from vortex shedding and galloping induced vibrations inside heating, ventilation and air conditioning systems [24]. A double-beam piezo-magneto-elastic wind energy harvester with bistable nonlinearity was studied by Yang et al. [25]. The nonlinear characteristic decreased the critical wind speed up to 41.9%.

Figure 1: (a) Side view of the piezoelectric wake galloping energy harvesting, (b) cross-section view of the bluff bodies of the energy harvester and interference cylinder

Inspired by large amplitude motions associated with wake galloping of iced-bundled conductors of overhead transmission lines, we investigate the levels from wake galloping of 2

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crescent and D-shaped cross-section bluff bodies with the objective of evaluating the impact of interference spacing of the wake galloping energy harvester. Conversion from ambient mechanical energy to electric energy via piezoelectric materials has been investigated in the last two decades [26, 27]. The proposed energy harvester is composed of a piezoelectric beam and a bluff body attached to the beam tip and subjected to the wake of an upstream interference cylinder in a tandem configuration as shown in Fig. 1(a). The upstream interference cylinder induces the uneven pressure distribution abound the tip mass and thus causes the larger oscillations and high power output of the harvester. The performance of the piezoelectric energy using iced D-shape and crescent-shape bluff bodies is compared to the performance of the non-iced cylinder according to the configurations presented in Fig. 1(b). The areas filled with the oblique lines in the shapes presented in this figure denote the iced areas on the galloping cylinders. For consistency with the terminology in galloping investigations of iced transmission lines, we refer to the two iced shapes as D shape and crescent shape. To establish the electromechanical coupled distributed parameter model for the proposed energy harvester, the electromechanical extension of the Hamilton’s principle [28] is employed which yields R t2 P P [δ(T − V + We∗ − Wm ) + fi δui (x, t) + ij δλj (x, t)]dt = 0 (1) t1 i

where T , V , We∗ , Wm ,

P

fi δui (x, t) and

i

P

j

ij δλj (t) are respectively the kinetic energy, po-

j

and

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∂u(x, t) ∂ 5 u(x, t) ∂u2 (x, t) dδ(x − L1 ) dδ(x − L2 ) ∂ 4 u(x, t) + c I + c + m +( − )ϑp V (t) s a 4 4 2 ∂x ∂x ∂t ∂t ∂t dx dx Z ρU 2 btip Ltip u(L ˙ s , t) + η u˙ 0 (Ls , t) u(L ˙ s , t) + η u˙ 0 (Ls , t) 3 = [ a1 ( ) + a3 ( ) dηδ(x − Ls ) 2 U U 0 Z Ltip u(L ˙ s , t) + η u˙ 0 (Ls , t) u(L ˙ s , t) + η u˙ 0 (Ls , t) 3 δ(x − Ls ) − η(a1 ( ) + a3 ( ) )dη ] U U dx 0 (2)

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tential energy, electrical coenergy, magnetic energy, virtual work due to the nonconservative forces and virtual work done by the nonconservative electric element, which are denoted by the current ij and the flux linkage λj . Using the Euler-Bernoulli beam assumption [29] and quasi-steady hypothesis [30], the electromechanical coupled distributed parameter model is expressed as:

ϑp

Z

L2

L1

∂ 3 u(s, t) s33 bp Lp dV (t) V (t) ds − = ∂t∂s2 hp dt R

(3)

where u(x, t) is the beam displacement in the y direction, V (t) is the output voltage, cs and ca are respectively the equivalent viscous strain and air damping coefficient of the cantilever beam, ρ is the air density, U is the incoming wind speed, a1 and a3 are the empirical 3

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aerodynamic coefficients, and s33 is the permittivity of the piezoelectric material, m and EI are respectively the mass and bending stiffness of the cantilever beam and given by m1 = ρs bs hs 0 ≤ x < L1 m2 = ρs bs hs + ρp bp hp L1 ≤ x ≤ L2 m= (4) m3 = ρs bs hs L2 < x ≤ Ls EI =

EI2 =

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and

1 = 12 Es bs h3s 0 ≤ x < L1 3 − y0 ) + 31 Ep bp (y23 − y13 ) L1 ≤ x ≤ L2 1 EI3 = 12 L2 < x ≤ Ls Es bs h3s

EI1 1 E b (y 3 3 s s 1

(5)

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where ρs and ρp are respectively the density of the substructure and piezoelectric layers, bs and bp are respectively the width of the substructure and piezoelectric layers, hs and hp are respectively the thickness of the substructure and piezoelectric layers, Es and Ep are respectively the Young’s modulus of the substructure and piezoelectric materials, y0 = −¯ y, y1 = hs − y¯ and y2 = hs + hp − y¯ where y¯ is the position of the neutral axis and expressed as E b h2p +2Ep bp hp hs +Es bs h2s y¯ = p p 2(E , L2 = L1 +Lp as shown in Fig. 1(a), ϑp is the piezoelectric coupling s bs hs +Ep bp hp ) 2 term and given by ϑp = −Ep d31 bp y1 +y where d31 is the strain coefficient of piezoelectric layer. 2 To predict the response and energy harvesting levels, the Galerkin procedure is implemented to separate the displacement of the beam u(x, t) into spatial and time variables as X u(x, t) = φi (x)qi (t) (6) where i represents the ith model, φi (x) and qi (t) are respective mode shape and mode coordinate of the cantilever beam. The mode shapes are given by [29]

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φij (x) = Aij sin βij x + Bij cos βij x + Cij sinh βij x +Dij cos βij x, j = 1, 2, 3

(7)

where j = 1 denotes 0 ≤ x < L1 , j = 2 denotes L1 ≤ x < L1 + Lp and j = 3 denotes L1 + Lp ≤ x ≤ Ls . The coefficients of Aij , Bij , Cij , Dij and βij are determined from the boundary conditions:

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φi1 (0) = 0, φi1 0 (0) = 0, φi1 (L1 ) = φi2 (L1 ), φi1 0 (L1 ) = φi2 0 (L1 ), EI1 φi1 00 (L1 ) = EI2 φi2 00 (L1 ), EI1 φi1 000 (L1 ) = EI2 φi2 000 (L1 ), φi2 (L2 ) = φi3 (L2 ), φi2 0 (L2 ) = φi3 0 (L2 ), EI2 φi2 00 (L2 ) = EI3 φi3 00 (L2 ), EI2 φi2 000 (L2 ) = EI3 φi3 000 (L2 ), EI3 φi3 00 (Ls ) − ωi2 Mt Lc φi3 (Ls ) − ωi2 It φi3 0 (Ls ) = 0, EI3 φi3 000 (Ls ) + ωi2 Mt φi3 (Ls ) + ωi2 Mt Lc φi3 0 (Ls ) = 0

(8)

To make the equivalent mass of the governing equation to be equal to 1, the following normalized orthogonality conditions are chosen as

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RL1 0

φs1 (x)m1 φr1 (x)dx +

RL2

φs2 (x)m2 φr2 (x)dx +

RLs

φs3 (x)m3 φr3 (x)dx + φs3 (Ls )Mt φr3 (Ls )

L2

L1

+φs3 0 (Ls )It φr3 0 (Ls ) + φs3 (Ls )Mt Lc φr3 0 (Ls ) + φs3 0 (Ls )Mt Lc φr3 (Ls ) = δrs , RLs RL2 RL1 00 φs1 (x)EI1 φr1 00 (x)dx + φs2 00 (x)EI2 φr2 00 (x)dx + φs3 00 (x)EI3 φr3 00 (x)dx = δrs ωr2 0

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(9) where the 0 denotes the derivation with respective to x, ω is the i natural frequency of the th qi EI harvester and has the relationship with βij as ωi = βij2 mjj , Mt is the tip mass, It is the rotary inertia of the tip mass relative to the end of the beam structure, Lc = 1/2Ltip , and δrs is Kronecker delta. With the experimental parameters shown in Table 1, the first three natural frequencies of the harvester are determined as 2.37 Hz, 30.27 Hz and 305.04 Hz, respectively. The vortex shedding frequency of the interference cylinder at U = 7.6 m/s (the experimental data) is calculated as fv = SDt U =50.67 Hz, where the Strouhal number St is 0.2 [31]. The vortex shedding frequency leaves far away from the natural frequencies of the harvester and thus the vortex induced vibration is not considered in this work. For the wake induced galloping, it is a self-excitation motion and the onset galloping speed is proportional to the natural frequency [29]. Because the natural frequency of the second or high mode is much larger than that of the first mode, the second or high mode starts galloping at much higher wind speed than the first mode. The purpose of investigating energy harvester from icedconductor inspired wake galloping is to harvest the energy from the low-speed environmental wind. Therefore, the second or high mode does not contribute to energy harvesting from low-speed wind. To simplify our simulation, only the first mode is considered in the following analysis. Substituting Eq. (6) into Eqs. (2) and (3), the reduced electromechanical governing equations considering the first mode shape are calculated as

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q¨(t) + 2ξω q(t) ˙ + ω 2 q(t) + θp V (t) = k1 q(t) ˙ + k3 q(t) ˙ 3 ˙ =0 Cp V˙ (t) + VR(t) − θp q(t)

(10)

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where q(t) and ω are respectively the first mode coordinate and natural frequency of the cantilever beam, ξ is the mechanical damping ratio, Cp is the capacitance of the harvester and εs b (L −L ) given by Cp = 33 p hp2 1 , θp is piezoelectric coupling term and expressed as θp = (φ0 (L2 ) − φ0 (L1 ))ϑp , k1 and k3 are aerodynamic coefficients and given by k1 = 12 a1 ρU 2 btip [φ2 (Ls )Ltip + RL φ(Ls )φ0 (Ls )L2tip + 13 φ02 (Ls )L3tip ] and k3 = 12 a3 ρU 2 btip [φ(Ls ) 0 tip (φ(Ls ) + sφ0 (Ls ))3 ds + φ0 (Ls ) R Ltip s(φ(Ls ) + sφ0 (Ls ))3 ds]. 0 To validate the analytically predicted response, experiments were conducted in an opencircuit blow-out boundary layer wind tunnel at Tianjin Institute of Water Transportation Engineering as shown in Fig 2. The wind tunnel is composed of a diffuser, test section, contraction, stable section, transition and power section. The total length is 40.75 m. The test section is 15 m long, 4.4 m wide and 2.5 m high. In order to reduce the streamwise static pressure gradient in the test section, a diffusion angle of 0.195 degree is set on both sides of the test section. The maximum wind speed is 30 m/s generated by a direct-current motor with 400 kW power. The wind speed is regulated by a direct-current speed regulating 5

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Figure 2: Schematic of the open-circuit blow-out boundary layer wind tunnel

Figure 3: Experimental setup for the tested downstream energy harvester placed in the wake of an upstream fixed cylinder in the open-circuit wind tunnel

device (6RA70 series governor, Siemens) and measured by a three-dimensional fluctuating wind speed meter named Cobra probe (Cobra 307). Pressure and temperature sensors are 6

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Table 1: The geometric and physical parameters of the substructure layer, MFC layer and bluff body

90 mm 20 mm 0.6 mm 8.9 g/cm3 127 GPa 28 mm 14 mm 0.4 mm 5.44 g/cm3 30.336, 15.857 GPa -170 pC/N 14.04 nF/m 260 mm 30 mm 75 g

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Length of substructure layer Width of substructure layer Thickness of substructure layer Density of substructure layer Young’s modulus of substructure layer Active length of MFC layer Active width of MFC layer Thickness of MFC layer Density of MFC layer Young’s modulus of MFC layer Piezoelectric strain coefficient Permittivity component of MFC layer Height of bluff body Windward width of the bluff body Mass of the bluff body

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Beam Displacement (mm)

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Iced D shape Iced crescent shape Uniced circuilar shape VIV

4 2 0

1

2

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5

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Figure 4: Variations of the beam displacement of the measured point with the non-dimensional spacing L/d

placed at the entrance of the test section and used to control the speed of the motor to achieve the closed-loop control of wind speed. As shown in Fig. 3, the tested downstream energy harvester and upstream interference cylinder are fixed on the transverse sliding chutes of a frame in the wind tunnel. The bluff bodies with different cross sections were fabricated by a 3D printer (HORI Z500) using the renewable biodegradable material polylactic acid (PLA). The piezoelectric layer (M-2814-P2, Smart Materials Corp.) was glued on the surface

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Iced D shape Iced crescent shape Uniced circuilar shape VIV

0.3 0.25 0.2 0.15 0.1 0.05 0

1

2

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Harvested Power Density (W/dm3)

0.35

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Figure 5: Variations of the harvested power density from wake galloping with the non-dimensional spacing L/d

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of the substructure layer and connected to an external load resistance of R = 105 Ω. The geometric and physical properties of the substructure layer, MFC layer and bluff bodies are presented in Table 1. The beam displacement of the tested downstream energy harvester was measured using the laser displacement meter (HL-C235BE, Panasonic Company). The beam displacement and voltage output were recorded by a data acquisition system (DASP v10.0). The root mean square of the generated voltage across the load resistance was used to compute the harvested power. Variations of the beam displacement and harvested power density with the non-dimensional spacing L/d where L is the distance between the downstream energy harvester and the upstream interference cylinder and d is the width of the bluff body are respectively presented in fig. 4 and fig. 5. At U = 7.6 m/s, the harvested power of energy harvester without upstream interference is close to zero. That is to say, vortex-induced-vibrations of a single circular cylinder does not occur at such a wind speed for the particular prototype. This can be explained by the fact that the vortex shedding frequency (fv = SDt U =50.67 Hz, where the Strouhal number St is 0.2 [31]) at U = 7.6 m/s is much larger than the natural frequency (fn = 2.37 Hz) of the energy harvester. In contrast, the aerodynamic characteristics of the cylinder placed in the wake of the upstream interference cylinder cause significant displacement and enhanced levels of harvested energy. In the tandem arrangement, the lift force of the downstream cylinder is increased and its drag force is decreased [32]. The experimental plots in figs. 4 and 5 show that optimal spacing of the wake-galloping energy harvesters is 3 irrelevant of the shape of the harvester. This is expected as the fluctuating lift force of the downstream cylinder first increases and then decreases with integer multiples of the spacing [33]. Besides, the largest fluctuating force of the down stream cylinder is experimentally found in the range of spacing L/D between 2.5 and 3 for the tandem arrangement of two cylinders [33–35]. In the range near the optimal spacing, the shear layer of the upstream 8

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interference cylinder reattaches to the surface of the downstream bluff body. This leads to enhancing the fluctuating force of the harvester’s tip mass and thus increasing the vibrating amplitude and harvested power. Beyond the optimal spacing, the fluctuating lift force of the downstream cylinder decreases as the spacing increases since the vortices induced by the upstream interference cylinder has weak effect on the tip mass. The performance of the tested energy harvester is significantly enhanced as the cross-section of the bluff body varied from uniced circular shape to iced D and crescent shapes. The effective interference spacing range is broadened in the iced cases in comparison to the case of non-iced bluff body.

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Figure 6: The vortex contour of the wake galloping energy harvester with the iced crescent-shape bluff body at L/d = 3

Figure 7: The time histories of the lift coefficients for the wake galloping energy harvesters with the uniced circular shape, iced D and crescent shape bluff bodies

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Figure 8: The time histories of the voltage and displacement for the wake galloping energy harvesters with the uniced circular shape, iced D and crescent shape bluff bodies

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To explain the enhanced performance of the iced shapes, we assess the aerodynamic forces determined from Large Eddy Simulation (LES) of the flow around the two cylinders. Vorticity contours for the iced crescent case at L/d = 3 are shown in Fig. 6. The incoming uniform flow becomes uneven with vortex after the upstream interference cylinder and thus leads to the relatively large variation of the lift with the time as shown in Fig. 7. A comparison of the lift coefficients for the uniced and iced cases presented in Fig. 7 shows that the lift coefficients of the two iced shape bluff bodies are more than 40 times that of the noniced shape bluff body. The larger lift forces result in larger oscillations and, subsequently, in higher voltage outputs, which is consistent with the experimental data and theoretical predictions via Eq. (10) that are compared in Fig. 8. Inspecting the time series of the measured displacements and voltages, we note that the oscillations amplitude and frequency are accurately predicted with the model presented in Eq. (10). The harvested energy from the low-speed wind is mainly contributed by the first mode. Based on this validated analytical mode, the effect of the resistance on energy harvesting from wake galloping is investigated for these three different cross sections in Fig. 9. The optimum resistances are marked and located in the range between 3 × 106 Ω and 6 × 106 Ω. The optimum harvested power decreases as the tip mass increases. However, the optimum resistance is increased with the tip mass. In conclusion, iced-conductor inspired design for energy harvesting from wake galloping has been shown to be more effective than other shapes. Theoretical modeling that accounts for electromechanical dynamics and aerodynamics for the system was performed. Wind tunnel tests were conducted and the optimal interference spacing was determined. Large Eddy Simulations of the vorticity field and generated lift force were performed to examine

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(b) Iced crescent shape

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(c) Uniced circular shape

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Figure 9: Variations of the harvested power from wake galloping with the resistance for different tip masses when L/d = 3 and U = 7.6 m/s: (a) iced D shape, (b) iced crescent shape and (c) uniced circular shape.

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the reasons for observed differences. Compared to the wake galloping energy harvester with the non-iced circular-shape bluff body, the wake galloping energy harvesters with the iced D-shape and crescent-shape bluff bodies showed remarkable enhancement of the levels of the harvested power density over a broad interference spacing range in wind tunnel tests. The energy harvester with crescent shape bluff body has the best output since the generated vortices caused by the upstream interference cylinder have more significant impact on aerodynamic instability of the crescent bluff body. The analytical model predictions almost reproduce the experimental data of the oscillations amplitude and frequency of the voltage and displacement.

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Acknowledgment

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This study is supported by the Natural Science Foundation of China (Grant Nos. 11902193, 11802071, 51778097), Natural Science Foundation of Shanghai (Grant No. 19ZR1424300), Shanghai Research Project (Grant No. 19JC1412900) and Key Laboratory of New Technology for Construction of Cities in Mountain Area Chongqing University. Reference

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The authors declare that they have no conflicts of interest to this work.