Small-scale wind energy harvesting has emerged as a promising approach for realizing self-powered wireless sensor networks or Internet of Things (IoT).1–31. J. Wang, D. Yurchenko, G. Hu, L. Zhao, L. Tang, and Y. Yang, “ Perspectives in flow-induced vibration energy harvesting,” Appl. Phys. Lett. 119, 100502 (2021). https://doi.org/10.1063/5.00634882. D. Li, Y. Wu, A. Da Ronch, and J. Xiang, “ Energy harvesting by means of flow-induced vibrations on aerospace vehicles,” Prog. Aerosp. Sci. 86, 28–62 (2016). https://doi.org/10.1016/j.paerosci.2016.08.0013. A. Abdelkefi, “ Aeroelastic energy harvesting: A review,” Int. J. Eng. Sci. 100, 112–135 (2016). https://doi.org/10.1016/j.ijengsci.2015.10.006 Unlike a large-scale wind energy conversion system, e.g., a wind turbine, which is generally applicable when strong wind is available, this technology can be utilized to scavenge the untapped energy from a gentle or light breeze, which is ubiquitous in environment. Therefore, designing efficient small-scale wind energy harvesters has been an ongoing research topic in recent decades.4–84. F.-R. Liu, H.-X. Zou, W.-M. Zhang, Z.-K. Peng, and G. Meng, “ Y-type three-blade bluff body for wind energy harvesting,” Appl. Phys. Lett. 112, 233903 (2018). https://doi.org/10.1063/1.50294155. A. H. Alhadidi and M. Daqaq, “ A broadband bi-stable flow energy harvester based on the wake-galloping phenomenon,” Appl. Phys. Lett. 109, 033904 (2016). https://doi.org/10.1063/1.49591816. J. Dias, C. De Marqui, Jr., and A. Erturk, “ Hybrid piezoelectric-inductive flow energy harvesting and dimensionless electroaeroelastic analysis for scaling,” Appl. Phys. Lett. 102, 044101 (2013). https://doi.org/10.1063/1.47894337. M. Rezaei and R. Talebitooti, “ Wideband PZT energy harvesting from the wake of a bluff body in varying flow speeds,” Int. J. Mech. Sci. 163, 105135 (2019). https://doi.org/10.1016/j.ijmecsci.2019.1051358. K. Yang, J. Wang, and D. Yurchenko, “ A double-beam piezo-magneto-elastic wind energy harvester for improving the galloping-based energy harvesting,” Appl. Phys. Lett. 115, 193901 (2019). https://doi.org/10.1063/1.5126476Flow-induced vibrations, as typical carriers of wind kinetic energy, can be converted to electricity using piezoelectric, electromagnetic, electrostatic, and triboelectric mechanisms.9–119. J. Xing, S. Fang, X. Fu, and W.-H. Liao, “ A rotational hybrid energy harvester utilizing bistability for low-frequency applications: Modelling and experimental validation,” Int. J. Mech. Sci. 222, 107235 (2022). https://doi.org/10.1016/j.ijmecsci.2022.10723510. G. Xu, J. Fu, C. Li, J. Xing, C. Chen, W.-H. Liao, Z. Wang, and Y. Zi, “ A nonlinear triboelectric nanogenerator with a broadened bandwidth for effective harvesting of vibration energy,” iEnergy 1, 236–242 (2022). https://doi.org/10.23919/IEN.2022.002811. A. Bibo, A. Abdelkefi, and M. F. Daqaq, “ Modeling and characterization of a piezoelectric energy harvester under combined aerodynamic and base excitations,” J. Vib. Acoust. 137, 031017 (2015). https://doi.org/10.1115/1.4029611 Among these, piezoelectric transducers (PZTs) are given special attention because of their higher energy density, ease of implementation, and independency of external voltage input or magnetic field. Flow-induced vibrations can be classified as vortex-induced vibrations (VIVs),12,1312. L. Zhang, H. Dai, A. Abdelkefi, and L. Wang, “ Improving the performance of aeroelastic energy harvesters by an interference cylinder,” Appl. Phys. Lett. 111, 073904 (2017). https://doi.org/10.1063/1.499976513. L. Zhang, H. Dai, A. Abdelkefi, and L. Wang, “ Experimental investigation of aerodynamic energy harvester with different interference cylinder cross-sections,” Energy 167, 970–981 (2019). https://doi.org/10.1016/j.energy.2018.11.059 galloping,14,1514. M. Rezaei and R. Talebitooti, “ Effects of higher-order terms in aerodynamic force on the nonlinear response of a galloping PZT energy harvester,” J. Theor. Appl. Vib. Acoust. 6, 271–280 (2020). https://doi.org/10.22064/tava.2021.530769.118215. A. H. Alhadidi, H. Alhussein, and M. F. Daqaq, “ Improving the sensitivity of galloping energy harvesters to flow fluctuations,” Appl. Phys. Lett. 116, 263902 (2020). https://doi.org/10.1063/5.0011118 flutter,4,164. F.-R. Liu, H.-X. Zou, W.-M. Zhang, Z.-K. Peng, and G. Meng, “ Y-type three-blade bluff body for wind energy harvesting,” Appl. Phys. Lett. 112, 233903 (2018). https://doi.org/10.1063/1.502941516. M. Z. Abdehvand, S. A. S. Roknizadeh, and H. Mohammad-Sedighi, “ Modeling and analysis of novel coupled magneto-electro-aeroelastic continuous system for flutter-based energy harvesting system,” Energy 230, 120742 (2021). https://doi.org/10.1016/j.energy.2021.120742 and buffeting.5,75. A. H. Alhadidi and M. Daqaq, “ A broadband bi-stable flow energy harvester based on the wake-galloping phenomenon,” Appl. Phys. Lett. 109, 033904 (2016). https://doi.org/10.1063/1.49591817. M. Rezaei and R. Talebitooti, “ Wideband PZT energy harvesting from the wake of a bluff body in varying flow speeds,” Int. J. Mech. Sci. 163, 105135 (2019). https://doi.org/10.1016/j.ijmecsci.2019.105135 In particular, galloping oscillations have a large vibration amplitude and possess the ability of oscillating in a wide range of wind speeds. Therefore, galloping piezoelectric energy harvesting (GPEH) technology has received a great attention and is used in some cutting-edge designs. Yang et al.1717. Y. Yang, L. Zhao, and L. Tang, “ Comparative study of tip cross-sections for efficient galloping energy harvesting,” Appl. Phys. Lett. 102, 064105 (2013). https://doi.org/10.1063/1.4792737 conducted a comparative study of different cross sections (square, rectangles with various aspect ratios, equilateral triangle, and D-section) of bluff bodies for GPEH. The results revealed that the square-sectioned bluff body outperforms the others with a low cut-in speed of 2.5 m/s and a high peak output power of 8.4 mW. Thereafter, researchers proposed modified designs to enhance energy harvesting performance. Hu et al.1818. G. Hu, K.-T. Tse, and K. C. Kwok, “ Enhanced performance of wind energy harvester by aerodynamic treatment of a square prism,” Appl. Phys. Lett. 108, 123901 (2016). https://doi.org/10.1063/1.4944555 verified that fitting fins to the leading edge of a square prism can improve the efficiency of GPEH up to 150%. Alhadidi et al.1515. A. H. Alhadidi, H. Alhussein, and M. F. Daqaq, “ Improving the sensitivity of galloping energy harvesters to flow fluctuations,” Appl. Phys. Lett. 116, 263902 (2020). https://doi.org/10.1063/5.0011118 proposed a method to improve the sensitivity of GPEH by adding Y-shaped attachments of various lengths and fork angles on the rear face of a square prism. Results manifested that the galloping rise time can be reduced by 75%, compared to the finless square prism.Integrating metasurfaces into flow-induced energy harvesters is a promising method of improving flow-induced energy harvesting performance. However, a batch of pertinent research exists in the field of VIV. Wang et al.1919. J. Wang, S. Sun, L. Tang, G. Hu, and J. Liang, “ On the use of metasurface for vortex-induced vibration suppression or energy harvesting,” Energy Convers. Manage. 235, 113991 (2021). https://doi.org/10.1016/j.enconman.2021.113991 for the first time explored the use of metasurfaces for VIV suppression or energy harvesting. It was shown that the existence of metasurfaces can influence the flow field around cylindrical bluff bodies and, hence, alter the generated aerodynamic force. Tang et al.2020. B. Tang, X. Fan, J. Wang, and W. Tan, “ Energy harvesting from flow-induced vibrations enhanced by meta-surface structure under elastic interference,” Int. J. Mech. Sci. 236, 107749 (2022). https://doi.org/10.1016/j.ijmecsci.2022.107749 further investigated the effect of a metasurface structure on the enhancement of VIV energy harvesting at the downstream of an interference long cylinder. It was indicated that the larger size of metasurface structures is more conducive to energy harvesting performance improvement at lower wind speeds, and ‘V’-shaped metasurface performs the best. Nevertheless, in the field of GPEH, there has been little research on this topic. Only one systematic study conducted by Wang et al.2121. J. Wang, S. Sun, G. Hu, Y. Yang, L. Tang, P. Li, and G. Zhang, “ Exploring the potential benefits of using metasurface for galloping energy harvesting,” Energy Convers. Manage. 243, 114414 (2021). https://doi.org/10.1016/j.enconman.2021.114414 manifests that when convex cylinder ornaments (each with a diameter of 6 mm and a length of 9 mm) are attached to a bluff body, the maximum output voltage can be increased by 26.14%. However, it is worth mentioning that all the aforementioned studies are based on full-surface metasurface structures, i.e., the surface modifications are distributed throughout the whole bluff body. No relevant study has been conducted to investigate the potential effect of single-sided metasurfaces on GPEH, which may differ when the surface protrusions are considered on different sides.In this Letter, with the aim of investigating the potential effect of different surface protrusions on the GPEH performance, three types of protruded bluff bodies (with rectangular/triangular/elliptical metasurfaces) are proposed as shown in Fig. 1(a). For each of the shapes, four kinds of surface treatments, including all-sided, frontal, side, and backward, which are, respectively, depicted from left to right in Fig. 1(a), are considered. The original bluff body, which is a 30 × 30 × 70 mm3 square prism, serves as a reference for the purpose of comparison in further experimental and analytical investigations. Figures 1(b) and 1(c), respectively, present the overall experimental setup and the details of the GPEH system. The proposed energy harvester is composed of a cantilever beam (PLA, 82 × 20 × 1 mm3), a Macro Fiber Composite (MFC) patch (M2814-P2, 28 × 14 × 0.33 mm3, Smart Material Corporation), and a tip bluff body. The whole experimental setup was placed inside the open-circuit wind tunnel, and the output signals were recorded by an oscilloscope (STO1104C, Micsig). The anemometer (testo 405i) was used to measure the wind speed, and the laser sensor (HG-C1100, Panasonic) was mounted parallel to the beam structure to detect its real-time tip deflection. A series of experiments were conducted in the speed range of 0–5.1 m/s, as a simulation of the natural gentle breeze.2222. T. Tan, L. Zuo, and Z. Yan, “ Environment coupled piezoelectric galloping wind energy harvesting,” Sens. Actuators, A 323, 112641 (2021). https://doi.org/10.1016/j.sna.2021.112641Under open-circuit conditions, three types of protruded bluff bodies with four different surface treatments of a fixed protruded length of 5 mm are deployed for an initial forward wind sweep experiment. The beam tip deflection and RMS of generated voltage vs wind speed are plotted in Fig. 2. Based on this figure, the original bluff body experiences galloping oscillations as expected. On the contrary, adding the protrusions affects the mechanical and electrical responses of the GPEH considerably. Specifically, for the backward protrusions, all three protruded shapes (rectangle/triangle/ellipse) can enhance the GPEH performance through improving its response and voltage amplitudes; for the frontal protrusions, the triangular and elliptical shapes can also give rise to galloping, but attenuated, while the rectangular shape cannot. Furthermore, for the side and all-sided protrusions, none of the shapes can lead to galloping, but vortex-induced vibrations, so as to reduce the energy harvesting performance in almost the whole range of the considered wind speeds. Therefore, based on the results of Fig. 2, the backward protrusions enhance the performance of the proposed GPEH, in comparison with the ordinary GPEH, through increasing the amplitudes of vibrations and harvested voltage. Specifically, under the test speed of 5.1 m/s, the GPEH carrying backward protruded bluff bodies with rectangular/triangular/elliptical metasurfaces outperform the original one by 12.31%, 10.61%, and 15.06%, respectively, in terms of the harvested voltage. Therefore, the finding of this study discloses that backward metasurfaces can enhance the GPEH performance, and this motivates a deeper investigation regarding the influence of different backward protruded lengths. As shown in Fig. 1(a), except the length of 5 mm, two other protruded lengths of 10 and 15 mm are also considered as a comparative study. The experimental results of considering different lengths for backward protrusions are depicted in Figs. 3(a)–3(c). It can be observed that, for each shape of protrusions, the galloping cut-in speed increases slightly as the protruded length increases. At the same time, the galloping oscillations amplitude get amplified at higher wind speeds. Under the highest test speed of 5.1 m/s, the associated harvested voltage of the rectangular/triangular/elliptical backward length of 15 mm outperforms the reference system by 33.03%, 35.97%, and 53.76%, respectively. This implies that elliptical backward protrusions perform the best in terms of the efficiency enhancement of GPEH.In order to account for the discovered experimental phenomenon, mathematical modeling can be established based on the extended Hamilton's principle11,2311. A. Bibo, A. Abdelkefi, and M. F. Daqaq, “ Modeling and characterization of a piezoelectric energy harvester under combined aerodynamic and base excitations,” J. Vib. Acoust. 137, 031017 (2015). https://doi.org/10.1115/1.402961123. M. Rezaei, R. Talebitooti, W.-H. Liao, and M. I. Friswell, “ Integrating PZT layer with tuned mass damper for simultaneous vibration suppression and energy harvesting considering exciter dynamics: An analytical and experimental study,” J. Sound Vib. 546, 117413 (2023). https://doi.org/10.1016/j.jsv.2022.117413 and Euler–Bernoulli beam theory.2424. M. Rezaei, S. E. Khadem, and P. Firoozy, “ Broadband and tunable PZT energy harvesting utilizing local nonlinearity and tip mass effects,” Int. J. Eng. Sci. 118, 1–15 (2017). https://doi.org/10.1016/j.ijengsci.2017.04.001 The derived governing equations of the proposed GPEH can be described as ′+″v=Fa(δ(x−Lb)−Db2δ′(x−Lb)),Cpv̇+1Rv=∂∂t,(1)where w, m, YI, and Lb are the transverse displacement, mass, bending stiffness, and length of the cantilever beam, respectively. Furthermore, c is the dimensional damping of the system, θ is the electromechanical coupling coefficient, Cp is the capacitance, R is the load resistance, and v denotes the output voltage of the PZT circuit. In addition, Fa is the galloping aeroelastic force, which can be expressed as follows: where ρ is the air density and Uf denotes the wind speed. Moreover, L and Db, respectively, stand for the frontal length and width of the bluff body and CFy denotes the transverse force coefficient, which can be derived by lift coefficient CL, drag coefficient CD, and the angle of attack α (AoA). Using a third order polynomial approximation,1414. M. Rezaei and R. Talebitooti, “ Effects of higher-order terms in aerodynamic force on the nonlinear response of a galloping PZT energy harvester,” J. Theor. Appl. Vib. Acoust. 6, 271–280 (2020). https://doi.org/10.22064/tava.2021.530769.1182 CFy can be explicitly expressed as a function of the AoA as25,2625. J. P. Den Hartog, Mechanical Vibrations ( Courier Corporation, 1985).26. M. P. Païdoussis, S. J. Price, and E. De Langre, Fluid-Structure Interactions: Cross-Flow-Induced Instabilities ( Cambridge University Press, 2010). CFy=−(CL+CD tan α)sec α≈A1α−A3α3.(3)Furthermore, according to the relationship between AoA and beam deflection,1111. A. Bibo, A. Abdelkefi, and M. F. Daqaq, “ Modeling and characterization of a piezoelectric energy harvester under combined aerodynamic and base excitations,” J. Vib. Acoust. 137, 031017 (2015). https://doi.org/10.1115/1.4029611 α=(ẇLb+Db2ẇ′Lb)Uf,(4)CFy is also a function of the beam tip deflection.According to the Den Hartog criterion,2525. J. P. Den Hartog, Mechanical Vibrations ( Courier Corporation, 1985). which is expressed in Eq. (5), for a system to undergo galloping oscillations, the initial slope of CFy vs α curve should be positive. In other words, a positive linear aerodynamic coefficient (A1) is an essential prerequisite for the occurrence of galloping instability, otherwise the GPEH cannot oscillate. Therefore, investigating CFy behavior can also be beneficial for analyzing the experimental results. Moreover, using the curve fitting method, it can be further expressed explicitly with respect to the AoA in the expression of galloping aeroelastic force. Thus, the governing equations [Eq. (1)] can be solved numerically using MATLAB by the Runge–Kutta method, Based on the theory of quasi-steady assumption,2626. M. P. Païdoussis, S. J. Price, and E. De Langre, Fluid-Structure Interactions: Cross-Flow-Induced Instabilities ( Cambridge University Press, 2010). the values of CL and CD at each α in the course of oscillations are the same as the values measured at the same value of α in the static wind tunnel experiments or computational fluid dynamics (CFD) simulations.2121. J. Wang, S. Sun, G. Hu, Y. Yang, L. Tang, P. Li, and G. Zhang, “ Exploring the potential benefits of using metasurface for galloping energy harvesting,” Energy Convers. Manage. 243, 114414 (2021). https://doi.org/10.1016/j.enconman.2021.114414 Hence, in this study, a series of wind tunnel simulations are performed by ANSYS Workbench 18 and the obtained results are shown in Fig. 4. It can be observed that for all backward cases, whatever their protruded shape and length are, the initial slope is positive, which implies the occurrence of galloping. However, for all the side and all-sided cases, no matter what the protruded shape is, the initial slope is negative or nearly zero, which means that the system cannot vibrate in a galloping mode. It is interesting to note that for the frontal cases, the triangular and elliptical protruded shapes have positive slopes, but the rectangular shape does not. This indicates that the rectangular frontal protrusions cannot render galloping oscillations, but the other two shapes are able to render. Moreover, according to the previous research conducted by Hu et al.,27,2827. G. Hu, K. T. Tse, M. Wei, R. Naseer, A. Abdelkefi, and K. C. Kwok, “ Experimental investigation on the efficiency of circular cylinder-based wind energy harvester with different rod-shaped attachments,” Appl. Energy 226, 682–689 (2018). https://doi.org/10.1016/j.apenergy.2018.06.05628. G. Hu, K.-T. Tse, and K. C. Kwok, “ Galloping of forward and backward inclined slender square cylinders,” J. Wind Eng. Ind. Aerodyn. 142, 232–245 (2015). https://doi.org/10.1016/j.jweia.2015.04.010 the larger the peak value of the CFy and the α corresponding to CFy = 0, the greater the galloping response. Therefore, the trends in Figs. 4(d)–4(f) suggest a greater response for the bluff bodies with a larger backward protruded length. Most importantly, the above analysis from the wind tunnel CFD simulations is in agreement with the experimental findings of Fig. 2, which in turn proves the validity of the quasi-steady assumption applied in this study.Using the third-order polynomial curve fitting method, the linear (A1) and cubic (A3) aerodynamic coefficients for the galloping cases are obtained and listed in Table I. It can be observed that both A1 and A3 increase by increasing the backward protruded length. According to the theory of Parkinson,2929. G. Parkinson, “ Wind-induced instability of structures,” Philos. Trans. R. Soc. London., Ser. A 269, 395–413 (1971). https://doi.org/10.1098/rsta.1971.0040 this is due to the fact that the shear layer, which has separated from one of the upstream corners of the bluff body, reattaches to one of the downstream corners at a larger angle of attack α. However, by comparing them, there seems to be an increasing convergence of both A1 and A3 in the rectangular case as the protruded length is increased from 10 to 15 mm. This also implies a convergence of the dynamic response as shown in Figs. 3(a) and 3(d). Corresponding numerical simulations can be performed by substituting each pair of the aerodynamic coefficients into the governing equations. Using this, the simulated results of the RMS voltage are plotted in Figs. 3(d)–3(f). It can be seen that the established modeling can well predict the galloping response, especially the cut-in wind speed and its variance in different cases. However, after the cut-in wind speed, there is an over-prediction of the output voltage at lower speeds (2–4 m/s), and an under-prediction at higher speeds (4–5.1 m/s). This discrepancy is probably due to the fact that the wind tunnel CFD simulations normally cannot be absolutely accurate, which leads to some errors in predicting the actual galloping force. Next, the underlying reason for the slight increase in the cut-in wind speed with the increase in the backward protrusion length, which is observed in Fig. 3, is discussed. According to Eq. (6), which is obtained by assuming the equivalent linear damping equal to be zero,25,2625. J. P. Den Hartog, Mechanical Vibrations ( Courier Corporation, 1985).26. M. P. Païdoussis, S. J. Price, and E. De Langre, Fluid-Structure Interactions: Cross-Flow-Induced Instabilities ( Cambridge University Press, 2010). the cut-in speed is proportional to the dimensional viscous damping (c) and inversely proportional to the linear aerodynamic coefficient. Therefore, although increasing the backward protruded length leads to an increased A1, the ratio of c/A1 determines the variation of cut-in wind speed with protruded length. After conducting vibration tests on GPEHs with different protruded lengths, it was observed that the ratio of c/A1 increases by increasing the protruded length. These observations justify the slight increase in cut-in wind speed with increasing protruded length, Ucut−in∝2cρLDbA1.(6)

TABLE I. Linear and cubic aerodynamic coefficients under third-order polynomial curve fitting approximation.

A1A3Original1.68522.07Rec-52.14223.14Rec-102.68723.62Rec-152.81524.13Tri-52.12822.56Tri-102.76425.17Tri-153.21925.30Ellip-52.03022.13Ellip-102.67924.69Ellip-153.24624.86Further experiments are conducted to investigate the optimal resistance for each galloping energy harvester. Since the working wind speed is not constant, its influence on the optimal resistance should be considered first. As shown in Fig. 5(a), three different wind speeds are selected for the impedance matching tests of the original bluff body. Results prove that wind speed has a small effect on the optimal resistance for an identical bluff body. Therefore, a single wind speed of 5.1 m/s is selected for the subsequent experiments. Results of Figs. 5(b)–5(d) reveal that for each different protruded bluff body, the optimal resistance is nearly the same, which is around 300 kΩ. As a result, 300 kΩ will be chosen as the load resistance for each GPEH. The GPEH output power vs wind speed for the rectangular/triangular/elliptical bluff bodies is depicted in Figs. 5(e)–5(g). It can be seen that for each backward protruded case, the energy harvesting performance outperforms the original one when the wind speed is higher than the cut-in speed; with the increase in the protruded length, the harvested energy at the highest studied wind speed also increases. Moreover, the highest output power for the rectangular/triangular/elliptical backward protruded bluff bodies is, respectively, 0.528, 0.601, and 0.757 mW. Hence, compared to 0.294 mW of the original bluff body, the power corresponding to the rectangular/triangular/elliptical backward protruded bluff bodies is enhanced by 79.59%, 104.42%, and 157.48%, respectively. Therefore, it can be concluded that the elliptical surface protrusion with the 15 mm backward protruded length can provide the highest energy harvesting enhancement for the GPEH, which is the desired optimal protrusion shape in this study. It is interesting to note that, unlike the rectangular surface protrusion, which has a clear enhancement convergence when the protruded length reaches 10 mm, there is still a big potential for further output power improvement by increasing the elliptical backward protruded length. Therefore, future work will be focused on finding the optimal solution for this case.In summary, this Letter explores the potential effect of different surface protrusions on galloping energy harvesting performance. It proves that adding protrusions on the bluff body can obviously change the amplitude and mode of oscillations, and only the backward protrusions are able to enhance the galloping vibration amplitudes, which is a remarkable finding of this study. However, the galloping cut-in speed is increased slightly as the protruded length increases. Wind tunnel CFD simulations were performed to investigate the transverse force coefficients, which can be used to justify the phenomenon observed in the experiments according to the classic Den Hartog's criterion. Both the experiments and numerical simulations show that elliptical surface protrusions have the greatest potential in enhancing galloping energy harvesting performance. With a backward protruded length of 15 mm, the optimum harvested power is experimentally measured to be 0.757 mW at the optimal resistance of 300 kΩ and a wind speed of 5.1 m/s, which outperforms the system carrying the original bluff body by 157.48%. This study demonstrates that by integrating the square prism with optimally designed backward protrusions, the modified galloping energy harvester has a great potential to serve as a sustainable energy source for low-power electronic devices. In the future, advanced optimization methods, such as topology optimizations,30,3130. O. Sigmund and K. Maute, “ Topology optimization approaches,” Struct. Multidisc. Optim. 48, 1031–1055 (2013). https://doi.org/10.1007/s00158-013-0978-631. S. Boyd, S. P. Boyd, and L. Vandenberghe, Convex Optimization ( Cambridge University Press, 2004). can be utilized to optimize the backward protrusion shapes and protruded lengths for galloping energy harvesting enhancement.See the supplementary material for the video demo when the proposed protruded GPEH undergoes the limit-cycle oscillation.

This work was supported by the Hong Kong Innovation and Technology Commission (Project No. MRP/030/21), the Chinese University of Hong Kong (Project ID: 4055178), and the National Natural Science Foundation of China (Grant No. 12202151).

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Juntong Xing: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). Masoud Rezaei: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – review & editing (equal). HuLiang Dai: Funding acquisition (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal). Wei-Hsin Liao: Funding acquisition (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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