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Harvesting wind energy with bi-stable snap-through excited by vortex-induced vibration and galloping
Journal Pre-proof Harvesting wind energy with bi-stable snap-through excited by vortex-induced vibration and galloping Weiyang Qin, Wangzheng Deng, Jianan Pan, Zhiyong Zhou, Wenfeng Du, Pei Zhu PII:
S0360-5442(19)31932-2
DOI:
https://doi.org/10.1016/j.energy.2019.116237
Reference:
EGY 116237
To appear in:
Energy
Received Date: 7 February 2019 Revised Date:
16 September 2019
Accepted Date: 27 September 2019
Please cite this article as: Qin W, Deng W, Pan J, Zhou Z, Du W, Zhu P, Harvesting wind energy with bi-stable snap-through excited by vortex-induced vibration and galloping, Energy (2019), doi: https:// doi.org/10.1016/j.energy.2019.116237. 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. © 2019 Published by Elsevier Ltd.
Harvesting wind energy with bi-stable snap-through excited by vortex-induced vibration and galloping Weiyang Qina , Wangzheng Denga, Jianan Pana, Zhiyong Zhoua, b∗∗, Wenfeng Dub, Pei Zhua, b a
Department of Engineering Mechanics, Northwestern Polytechnical University, Xi’an 710072, People’s Republic of China b School of Civil Engineering and Architecture, Henan University, Kaifeng 475004, People’s Republic of China Abstract Scavenging wind energy has been considered to be a promising solution for providing reliable power for sensors and transducers in smart structures. In this study, a novel wind harvesting design is presented, which is based on vortex-induced vibration and galloping. This design consists of a cruciform cantilever beam with three bluff bodies including two square cylinders and a circular cylinder. A tip magnet and two fixed magnets are added to introduce multi-stability. As the air flow passes through, the proposed harvester could begin executing snap-through at low wind speed and sustain this motion over a wide range of wind speed, thus generating a large output. Especially, the coherence resonance could be excited and sustained over a wide range of wind speed, by which the high output voltage can be generated and kept. For validation, the experiments were conducted. The results show that the proposed configuration could give a large output within the wind speed between 2.0 m/s to 7.0 m/s. This proposed structure may provide a new scenario to improve the performance of wind energy harvesting by combining the advantages of multi-stability and flow-induced instability. Keywords: energy harvesting; snap-through; air flow excitation; vortex-induced vibration; galloping 1. Introduction Harvesting ambient environmental energy has drawn increasing attention since it could realize the self-powered sensors and transducers with low consumption [1, 2]. In this field, many research works are devoted to generating inexhaustible electricity by utilizing the natural energy sources, such as solar energy [3], wind energy [4], noise energy [5], water flow energy [6], mechanical vibration [7], thermal reservoir [8] and ocean wave [9]. Of these various energy sources, wind energy is a practical and vital natural renewable energy source due to its eco-friendly characteristics and wide ∗
Corresponding author.
E-mail address: [email protected] (Z. Zhou)
1
distribution in environment [10, 11]. Wind energy harvesters are generally designed to convert the air flow energy into the electrical energy through flow-induced vibration by electromagnetic, electrostatic, or piezoelectric transduction mechanisms [12-14]. Aeroelastic effect has been widely exploited in flow energy harvesting through vortex-induced vibration (VIV) [15], galloping [16, 17], flutter [18] and buffeting [19]. Among these flow-induced vibrations, VIV and galloping are of great interest to the researchers due to their excellent performance in harvesting wind energy, which can be realized by adopting simple bluff bodies. Up till now, extensive studies have focused on converting flow energy to electrical power based on VIV and galloping [20]. VIV scenario generally uses the dynamic instability of a bluff body to generate Kármán vortices shedding from either side of the bluff body. Bernitsas et al [21] designed a VIV-based energy harvester to successfully convert ocean/river current hydro-kinetic energy to electric energy. Abdelkefi et al. [22] numerically investigated the energy harvesting from VIV oscillations of rigid circular cylinder and proposed a model with a modified van der Pol equation and the Gauss law. Akaydin et al. [23] proposed to utilize a flexible piezoelectric cantilever beams in the wake of a circular cylinder to harvest unsteady, turbulent fluid flow energy. Mehmood et al. [24] numerically investigated the energy harvesting from VIV of a circular cylinder for different Reynolds covering pre-synchronization, synchronization, and post-synchronization regimes. Dai et al. [25] experimentally compared the four distinct VIV-based piezoelectric energy harvesters in different orientations. Hu et al. [26] conducted a modeling and experimental study of piezoelectric energy to predict the optimum position for vortex shedding-induced vibration. The sensitivity of such energy harvester to the properties of cylinders such as cross sections [27], mass ratio [28], bending stiffness [29] and Reynolds number [30] has been analyzed to further improve the efficiency of energy extraction. However, typical VIV-based energy harvesters are designed to possess a linear restoring force. For such energy harvester, the efficiency of VIV-based generators will dramatically decrease when the vortex shedding frequency mismatches the natural frequency of system [31]. Therefore, the linear VIV-based scenarios have a very narrow frequency response bandwidth and are not desirable when operating in a wide flow speed range. On the other hand, some scholars have tried using galloping to scavenge the wind energy. Galloping is a kind of dynamic instability created by prismatic bluff bodies in flowing fluid, which might lead to a large-amplitude oscillation when the speed of incident flow exceeds the threshold one. Barrero-Gil et al. [32] exploited transverse galloping phenomena to extract flow energy and studied the effects of mass, mechanical properties, cross-section geometry and flow speed on the energy conversion efficiency. Abdelkefi et al. [33] investigated energy harvesting from transverse galloping of a bluff body with different cross-section geometries and explored the influence of parameters on the performance of output power. Sirohi and Mahadik [34, 35] compared the different galloping-based energy harvesters including D-shaped and triangular-cross-section geometries of bluff bodies. Zhao et al. [36] compared the modeling methods of galloping piezoelectric energy harvester including the lumped-parameter single-degree-of-freedom model and the Euler–Bernoulli 2
distributed parameter model. And the effects of wind exposure area and mass of the bluff body, load resistance, and length of piezoelectric sheets on output power performance have also been investigated. Bibo and Daqaq [37] established the universal design curves depending on the aerodynamic properties of the bluff bodies in order to optimize the performance of piezoelectric galloping-based energy harvesters. Abdelmoula and Abdelkefi [38] considered the potential of electrical impedance on the performance of galloping energy harvester with a square cross-section bluff body at its free end subjected to a uniform cross-flow. Hu et al. [39] investigated the effects of the inclining angles of the slender square-section cylinder on the galloping behavior. Based on Hu’s work, Javed and Abdelkefi [40] Further gave insights to the effects of damping-ratio, mass-ratio [41], the cross-sectional aspect ratio [42, 43], cross section [44] of a cylinder on the efficiency of galloping energy harvesters. Galloping featured sustained vibrations without self-limited amplitudes often occur at a speed higher than the flow speed of VIV at resonance. As a result, the piezoelectric galloping-based energy harvesters should work at a relatively high speed so as to get the high harvesting efficiency. Generally, many galloping configurations adopt the design of a bluff body connecting to a piezoelectric cantilever beam. Such similar designs have also appeared in many VIV designs [45]. However, for this kind of design, an unavoidable drawback exists that the energy efficiency could not sustain desirable for the flow with a variable speed, while in practice the fluctuation in wind speed is inevitable. Moreover, there have been many interests in exploiting nonlinear forces, e.g. magnetic forces or external axial loading forces, to excite the large-amplitude motion so as to obtain a high efficiency in energy harvesting. Nonlinear forces could produce mono-stable, bi-stable, tri-stable, even quad-stable motions, resulting in snap-through motion and thus large energy output. As a result some researchers have tried to employ such scenario in harvesting the fluid energy. Abdelkefi et al. [46] investigated the influences of nonlinear parameters of the piezoaeroelastic energy harvesters on the output powers. Bibo et al. [47] designed a galloping energy harvester with a quartic-potential-energy function to improve the output power and investigate the influence the nonlinearity on the energy harvesting performance. Huynh and Tjahjowidodo [48] simulated a nonlinear vortex induced vibration system and found the existence of chaotic vibrations in such structure. An experimental validation of bi-stable VIV system was carried out to investigate the effect of structural parameters including bi-stable gap and damping factor [49]. Zhang et al. [50] used nonlinear magnetic forces to enhance the output power form vortex-induced vibration. Naseer et al. [51] designed a mono-stable VIV-based piezoelectric energy harvester to widen the synchronization region. Alhadidi and Daqaq [52] proposed a wake-galloping flow energy harvester with bi-stable characteristics for broadband energy harvesting when flow speed varies. Usman et al. [53] designed an energy harvester employing the wake galloping phenomenon. The results of experimental validation indicates that there exists an optimal spacing distance between two cylinders. In the classical configurations, most flow-induced energy harvesters are designed by employing one of some types of instability phenomena, e,g., VIV, galloping or flutter [54]. 3
In this paper, the strategy for enhancing the performance of wind energy harvesting is proposed and realized by introducing nonlinear forces and using interactions among different instability phenomena. A novel bi-stable piezoelectric energy harvester (BPEH) is proposed to improve the efficiency of scavenging wind energy by introducing the dynamic multi-stability and incorporating the merits of VIV and galloping. For the purpose of working efficiently for variable speed winds, the configuration is designed to consist of a cruciform piezoelectric substrate with a circular, two square cylinders and three magnets. Different from the general scheme introduced in the literatures, this configuration owns two or three stable equilibria, between or among which the snap-through motion could be activated to occur. As a result, the proposed BPEH could generate a very large output voltage over a wide range of wind speed. 2. Proposed system
Fig. 1. The schematic diagram of the BPEH based on vortex-induced vibration and galloping.
The configuration of proposed BPEH is shown in Fig. 1. It consists of a cruciform cantilever substrate (including a horizontal part and a vertical part) covered with a piezoelectric patch. A circular cylinder, two square cylinders and a tip magnet are attached to the cruciform substrate, while two external magnets are fixed opposite to the tip magnet to introduce attractive forces. The square and circular cylinder bluff bodies subjected to the flow can induce two different types of vibrations: galloping and VIV, respectively. The cruciform cantilever substrate and the attached cylinders are designed such as to induce interactions between VIV and galloping. Owing to the existence of magnetic attractive force, when the wind speed is low, the harvester will stabilize on one stable position near a fixed magnet. Then with the wind speed increasing, the flow passing through the circular cylinder will create vortex and cause the cantilever beam to oscillate; furthermore, the two square cylinders will introduce the galloping phenomenon. These two kinds of vibration could both facilitate the beam to leave the original stable positions, then the beam will start to oscillate with the release of bending potential energy. This process could 4
repeatedly happen and perform an inter-well vibration that will induce high energy output. In practice, the onset speed of galloping is relatively high. Therefore, for the low speed wind, the snap-through will only be triggered by VIV; while the performance of VIV often degrades out of its best speed. To overcome this defect, the square cylinders were designed so that the increasing wind speed could exceed a critical value to enable the galloping to be excited. As a consequence the beam could maintain the snap-through motion, or inter-well vibration and produce a high energy output. Using the interaction between VIV and galloping and the resulted bi-stability, the proposed system can maintain a nearly periodic snap-through over a wide range of wind speed and thus generate high output voltages. It should be noted that in our configuration, the nonlinear forces could turn the system to be a bi-stable one, which plays an important role in improving the harvesting efficiency. Nomenclature Ls
Length of horizontal substrate
bs
Width of horizontal substrate
hs
Thickness of horizontal substrate
ρs
Density of horizontal substrate
Es
Young’s modulus of horizontal substrate
Lsv
Length of vertical substrate
bsv
Width of vertical substrate
hsv
Thickness of vertical substrate
ρsv
Density of vertical substrate
Esv
Young’s modulus of vertical substrate
Lp
Length of piezoelectric layer
bp
Width of piezoelectric layer
hp
Thickness of piezoelectric layer
Ep
Young’s modulus of piezoelectric layer
ρp
Density of piezoelectric layer
Ip
Piezoelectric moment of inertia
ezx
Young’s modulus of piezoelectric layer
V
Voltage of generated by piezoelectric layer
Ls
Length of square cylinder
Ws
Width of square cylinder
Hs
Height of square cylinder
ρs
Density of square cylinder
dc
Diameter of circular cylinder
Lc
Length of circular cylinder
ρc
Density of circular cylinder
Is
Substrate moment of inertia
a1
Effective magnetic moments of tip magnet
a2 (a3)
Effective magnetic moments of fixed magnet
µ0
Permeability constant
w(x,t)
The transverse displacement of cantilever beam at distance x and instant t 5
d
Separation distance
dg
Gap distance
σε
Variance of strain
Since the potential energy and its shape could reflect the multi-stable characteristic of system, we will calculate the potential energy and plot its shape first. The total potential energy could be subdivided into two parts: the magnetic potential energy and the elastic potential energy. The elastic potential energy can be formulated by the Euler-Bernoulli theory. Euler-Bernoulli theory states that the linear inextensible strain through the substrate is a result of bending motion only and proportional to the second spatial derivative of the bending coordinate. Therefore, the elastic potential energy can be given as [55]
= ′′, + ′′,
(1)
where the prime notation ′′ represents ⁄ . In this study, the permanent magnets are modeled as the dipoles in calculating magnetic potential energy, which can be expressed as [56, 57] =−
"# $%
%$&
-
'() , − * + + ,
. /
−
"# $% $.
&
'() , + * + + ,
-
The definition of parameters and their values are listed in Table 1. Table 1 Model parameters used for numerical and experimental studies. Symbol
Parameter
Value
Substrate properties Ls (Lsv)
Length
0.500 m
bs (bsv)
Width
0.01 m
hs (hsv)
Thickness
0.001 m
ρs (ρsv)
Density
7800 kg/m3
Es (Esv)
Young’s modulus
205 Gpa
Piezoelectric laminate properties Lp
Length
0.005 m
bp
Width
0.005 m
hp
Thickness
0.00015 m
Ep
Young’s modulus
2 Gpa
dc
Diameter
0.05 m
Lc
Length
0.25 m
ρc
Density
750 kg/m3
Circular cylinder
Square cylinder 6
. /
(2)
Ls
Length
0.25 m
Ws
Width
0.25 m
Hs
Height
0.05 m
ρs
Density
750 kg/m3
0 (0 , 01 )
Effective magnetic moments
0.213 Am2
µ0
Permeability constant
4π×10-7 NA-2
Potential energy (mJ)
Magnet properties
Fig. 2. Potential energy functions.
The geometric, material and magnetic parameters used in simulations and experiments are listed in Table 1. Fig. 2 shows the results of potential energy versus the tip deflection for different separate distance d. At d=90 mm, i.e., the distance is quite large, the magnetic field is weak and has little influence on the potential energy, thus the system behaves like a linear one. Namely, they own a narrow working bandwidth near the natural frequency and thus can’t work efficiently for the variable wind speed. Then, the distance is reduced to d=26 mm, at which two symmetric potential energy wells begin to emerge, with a potential barrier between them. In this case, if the bluff bodies undergo VIV and galloping oscillations, the system response could take snap-through between the two potential wells, thereby generating high energy output. With a further decrease in d (d=22 mm), the distance between the two potential well’s bottoms will increase, along with a rise of the potential barrier. Thus, the distance d is a key parameter for the system and has a large influence on the potential well and the barrier. The large distance could produce a low barrier and a small separation distance. With this merit, the system could execute snap-through easily, but the resulting amplitude will be small, thus the output is small; whereas the small distance leads to a high barrier and large separation distance, implying that the system needs more excitation energy to execute snap-through. Thus the distance d should be optimized so as to obtain the best harvesting performance. 3. Experimental verification To prove the advantages of the proposed BPEH, corresponding validation experiments were performed for different air flow speeds. The prototype of the proposed BPEH was fabricated, and the experimental setup was built as shown in Fig. 7
3. The dynamic strain response and the dynamic voltage across the piezoelectric patch are recorded by a data acquisition device (DH5922D, DONG HUA). A digital anemometer (AS8336, XI MA) is used to measure the air flow speed. Fig. 4 illustrates the static stable positions of BPEH. If the two fixed magnets are removed, the system becomes a linear piezoelectric energy harvester (LPEH). For comparison with BPEH, the corresponding static stable position of LPEH are shown as well in Fig. 5.
Fig. 3. Experimental setup of the BPEH.
Fig. 4. Static stable positions of BPEH.
8
RMS Voltage [V]
Variance of strain
Fig. 5. Static stable position of LPEH
Fig. 6. (a) Variance of strain (23 ) and (b) RMS voltage (45 ) versus wind speed (LPEH and BPEH). The bar graphs are presented as mean ± standard deviation (n=5).
The variance of dynamic strain (26 ) and open-circuit RMS voltage (45 ) are chosen as the significant quantities to describe the dynamic behaviors of LPEH and BPEH. Fig. 6 shows the results of 23 and 45 versus wind speed v, which varies from 1.0 m/s to 7.0m/s. In order to show potential error and uncertainty of the experimental results, the standard deviation of measured data (5 sets of 200-second responses) has been added in the form of error bars. In Fig. 6(a), we can see that there appears a sharp increase in 26 at v=1.5 m/s, which corresponds to the occurrence of snap-through. The generated 45 is shown in Fig. 5(b), we can see that the 45 is directly proportional to 23 , and the BPEH outperforms the LPEH significantly when the speed exceeds the threshold of snap-through. To show the dynamics of small-amplitude and large-amplitude motions clearly, the dynamic strain and resulting voltage, in terms of time histories, are illustrated in the insets of Figs. 7-19, as the wind speed increased from 1.0 to 7.0 m/s. Moreover, the spectra of dynamic strain are illustrated to show the frequency components.
9
10
Strain amplitude
Voltage [V] Strain
Voltage [V]
Strain
Fig. 7. (a) Strain of LPEH, (b) voltage of LPEH, (c) strain of BPEH, (d) voltage of BPEH and (e) strain FFT (v=1.0 m/s).
Voltage [V]
Strain
Voltage [V]
Strain
First, at a relatively low wind speed (v=1.0 m/s), the dynamic responses of LPEH and BPEH are shown in Fig. 7. Both of them undertake small-amplitude motions confined in a single potential well. From the corresponding spectra (Fig. 7 (e)), it can be seen that the LPEH owns a dominant nearly periodic component near f=1.16Hz, while the BPEH has a dominant component between f=1.18 Hz and f=1.53 Hz due to involvement of nonlinear restoring forces.
11
Strain amplitude
Fig. 8. (a) Strain of LPEH, (b) voltage of LPEH, (c) strain of BPEH, (d) voltage of BPEH and (e) strain FFT (v=1.5 m/s).
Strain
Voltage [V]
Strain
Then adding a small increment to the wind speed, v=1.5 m/s, the BPEH could be triggered to undertake snap-through between the stable equilibrium positions and generate a quiet large output voltage. It can be found that there exists a large amplitude in the low frequency area other than at the first resonance frequency, implying that the snap-through could transfer the excitation energy to the low frequency component.
12
Voltage [V] Strain amplitude Voltage [V]
Strain
Fig. 9. (a) Strain of LPEH, (b) voltage of LPEH, (c) strain of BPEH, (d) voltage of BPEH and (e) strain FFT (v=2.0 m/s).
13
Strain Voltage [V] Strain amplitude Strain
Fig. 10. (a) Strain of LPEH, (b) voltage of LPEH, (c) strain of BPEH, (d) voltage of BPEH and (e) strain FFT (v=2.5 m/s).
14
Voltage [V] Strain Voltage [V] Strain amplitude
Fig. 11. (a) Strain of LPEH, (b) voltage of LPEH, (c) strain of BPEH, (d) voltage of BPEH and (e) strain FFT (v=3.0 m/s).
15
16
Strain amplitude
Voltage [V]
Strain
Voltage [V]
Strain
Voltage [V]
Strain
Voltage [V]
Strain
Fig. 12. (a) Strain of LPEH, (b) voltage of LPEH, (c) strain of BPEH, (d) voltage of BPEH and (e) strain FFT (v=3.5 m/s).
17
Strain amplitude
Fig. 13. (a) Strain of LPEH, (b) voltage of LPEH, (c) strain of BPEH, (d) voltage of BPEH and (e) strain FFT (v=4.0 m/s).
Strain
Voltage [V]
Strain
As the wind speed increases further and attains v=2.0-4.0 m/s, there appears a frequent snap-through motion, resulting in a frequent peak in the output voltage. As for the spectrum, as shown in Figs.9-13, the extremely low frequency components (0-1.3Hz) are dominant.
18
Voltage [V] Strain amplitude Voltage [V]
Strain
Fig. 14. (a) Strain of LPEH, (b) voltage of LPEH, (c) strain of BPEH, (d) voltage of BPEH and (e) strain FFT (v=4.5 m/s).
19
Strain Voltage [V] Strain amplitude Strain
Fig. 15. (a) Strain of LPEH, (b) voltage of LPEH, (c) strain of BPEH, (d) voltage of BPEH and (e) strain FFT (v=5.0 m/s).
20
Voltage [V] Strain Voltage [V] Strain amplitude
Fig. 16. (a) Strain of LPEH, (b) voltage of LPEH, (c) strain of BPEH, (d) voltage of BPEH and (e) strain FFT (v=5.5 m/s).
21
22
Voltage [V]
Strain
Voltage [V]
Strain
Strain amplitude
Fig. 17. (a) Strain of LPEH, (b) voltage of LPEH, (c) strain of BPEH, (d) voltage of BPEH and (e) strain FFT (v=6.0 m/s).
Voltage [V]
Strain
Further increasing the wind speed to v=4.5-6.0 m/s, the times of snap-through continue to increase. There occurs frequently jumping in the response, as shown in Figs. 14–17, which results in a large output, e.g. at v=6.0 m/s. The frequently jumping is desired in energy harvesting, because the voltage across the piezoelectric patch is proportional to the amplitude of vibration. From the spectrum, it can be found that the energy exhibits a likely uniform distribution over a region of low frequency (0-1.1Hz). This desired vibration is beneficial for energy harvesting.
23
Strain Voltage [V] Strain amplitude Strain
Fig. 18. (a) Strain of LPEH, (b) voltage of LPEH, (c) strain of BPEH, (d) voltage of BPEH and (e) strain FFT (v=6.5 m/s).
24
Voltage [V] Strain Voltage [V] Strain amplitude
Fig. 19. (a) Strain of LPEH, (b) voltage of LPEH, (c) strain of BPEH, (d) voltage of BPEH and (e) strain FFT (v=7.0 m/s).
When the wind speed reaches a high value, e.g., v=6.5-7.0 m/s, as plotted in Figs.18 and 19, the main energy gradually concentrates at the first natural frequency, showing that the magnetic force is small compared to the aerodynamic force now and can’t trap the system. Nevertheless, the existence of two fixed magnets extend the amplitude of vibration effectively, as Figs. 19(a) and 19(c) illustrated. Thus the BPEH still could keep a high output. The maximum voltage could reach nearly 4.2 V now. 25
Strain Voltage [V]
Fig. 20. (a) Strain of LPEH and BPEH, (b) voltage of LPEH and BPEH (v=2.0 m/s).
Strain
In this system, owing to involvement of nonlinear forces and the combination of two kinds of flow-induced vibration: VIV and galloping, there appear some complicated nonlinear phenomena. In order to show the snap-through and its influence on the output voltage clearly, the time histories of response at v=2.0, 3.5, 5.5 and 7.0 m/s are magnified and shown in Figs. 20-23. Fig. 20 shows a short time period of strain and the corresponding output voltage at v=2.0 m/s. At the relatively low speed, the BPEH vibrate around one equilibrium point for a while and then jump to another, undergoing a snap-through between the two equilibrium positions.
26
Voltage [V] Voltage [V]
Strain
Fig. 21. (a) Strain of LPEH and BPEH, (b) voltage of LPEH and BPEH (v=3.5 m/s).
Fig. 22. (a) Strain of LPEH and BPEH, (b) voltage of LPEH and BPEH (v=5.5 m/s).
Strain
Figs. 21 and 22 shows the portions of strain and voltage for two large wind speeds (v=3.5 m/s and v=5.5 m/s). It can be seen that the oscillation of BPEH includes a large amplitude of low frequency vibration and some small high harmonic components. The large vibration corresponds to the bi-stable snap-through, i.e., the inter-well vibration. In this case, both the BPEH’s amplitude and output voltage outperform those of the LPEH.
27
Voltage [V]
Fig. 23. (a) Strain of LPEH and BPEH, (b) voltage of LPEH and BPEH (v=7.0 m/s).
At a quite high wind speed (e.g. v=7.0 m/s), the BPEH undertake nearly the similar oscillation as the LPEH; in contrast, compared to the LPEH, the BPEH’s amplitude is considerably large, thus it gives a quite high output, as shown in Fig. 23. As a summary, the proposed BPEH incorporates the VIV and galloping effects to make the system lose stability easily for the variable speed wind. To increase the amplitude and corresponding output, a multi-stable design is proposed and realized by magnetic attractive forces, by which the instability resulted from VIV or galloping could activate a snap-through between the equilibrium positions and generate a high output. Furthermore, this design could produce a frequent snap-through motions over a wide range of wind speed, which is desired in harvesting wind energy and could generate a large output. The experimental results proved this superior performance clearly. The proposed harvester may be more suitable for application in practical environment, owing to its ability to execute snap-through easily and keep this motion for the wind speed varying in a wide range. 4. Conclusions We proposed a novel harvester (BPEH) to scavenge wind energy, which could increase the harvesting efficiency by incorporating the merits of multi-stability and flow-induced instability, i.e., the vortex-induced vibration and galloping. By introducing the multi-stability and combining the advantages of VIV and galloping, the system could maintain a large amplitude vibration over a wide range of wind speed and generate a high electric output. For low wind speed, VIV takes effect and makes the system undergo snap-through; for high wind speed, galloping happens and drive the system to execute snap-through. Thus the BPEH could generate a high output for variable speed wind. Furthermore, it is proved that over a relatively wide range of wind speed, the BPEH could reach coherence resonance and thus generate a considerably large electric output. Compared to the classical wind energy harvester, our design could keep a stable output as the wind speed fluctuates in a wide range. Thus the proposed harvester is preferred for wind energy harvesting in practical environment. Acknowledgments The authors gratefully acknowledge the support by the National Science Foundation of China (Grant No. 11672237). 28
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32
Highlights
A design including bluff bodies and magnets is proposed to harvest wind energy. Flow-induced instability and bi-stability are incorporated in the design. Snap-through could be triggered by vortex-induced vibration and galloping. Coherence resonance could be excited to happen over a range of wind speed. The harvester can keep a large output in the environment with variable wind speed.
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
S0360-5442(19)31932-2
DOI:
https://doi.org/10.1016/j.energy.2019.116237
Reference:
EGY 116237
To appear in:
Energy
Received Date: 7 February 2019 Revised Date:
16 September 2019
Accepted Date: 27 September 2019
Please cite this article as: Qin W, Deng W, Pan J, Zhou Z, Du W, Zhu P, Harvesting wind energy with bi-stable snap-through excited by vortex-induced vibration and galloping, Energy (2019), doi: https:// doi.org/10.1016/j.energy.2019.116237. 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. © 2019 Published by Elsevier Ltd.
Harvesting wind energy with bi-stable snap-through excited by vortex-induced vibration and galloping Weiyang Qina , Wangzheng Denga, Jianan Pana, Zhiyong Zhoua, b∗∗, Wenfeng Dub, Pei Zhua, b a
Department of Engineering Mechanics, Northwestern Polytechnical University, Xi’an 710072, People’s Republic of China b School of Civil Engineering and Architecture, Henan University, Kaifeng 475004, People’s Republic of China Abstract Scavenging wind energy has been considered to be a promising solution for providing reliable power for sensors and transducers in smart structures. In this study, a novel wind harvesting design is presented, which is based on vortex-induced vibration and galloping. This design consists of a cruciform cantilever beam with three bluff bodies including two square cylinders and a circular cylinder. A tip magnet and two fixed magnets are added to introduce multi-stability. As the air flow passes through, the proposed harvester could begin executing snap-through at low wind speed and sustain this motion over a wide range of wind speed, thus generating a large output. Especially, the coherence resonance could be excited and sustained over a wide range of wind speed, by which the high output voltage can be generated and kept. For validation, the experiments were conducted. The results show that the proposed configuration could give a large output within the wind speed between 2.0 m/s to 7.0 m/s. This proposed structure may provide a new scenario to improve the performance of wind energy harvesting by combining the advantages of multi-stability and flow-induced instability. Keywords: energy harvesting; snap-through; air flow excitation; vortex-induced vibration; galloping 1. Introduction Harvesting ambient environmental energy has drawn increasing attention since it could realize the self-powered sensors and transducers with low consumption [1, 2]. In this field, many research works are devoted to generating inexhaustible electricity by utilizing the natural energy sources, such as solar energy [3], wind energy [4], noise energy [5], water flow energy [6], mechanical vibration [7], thermal reservoir [8] and ocean wave [9]. Of these various energy sources, wind energy is a practical and vital natural renewable energy source due to its eco-friendly characteristics and wide ∗
Corresponding author.
E-mail address: [email protected] (Z. Zhou)
1
distribution in environment [10, 11]. Wind energy harvesters are generally designed to convert the air flow energy into the electrical energy through flow-induced vibration by electromagnetic, electrostatic, or piezoelectric transduction mechanisms [12-14]. Aeroelastic effect has been widely exploited in flow energy harvesting through vortex-induced vibration (VIV) [15], galloping [16, 17], flutter [18] and buffeting [19]. Among these flow-induced vibrations, VIV and galloping are of great interest to the researchers due to their excellent performance in harvesting wind energy, which can be realized by adopting simple bluff bodies. Up till now, extensive studies have focused on converting flow energy to electrical power based on VIV and galloping [20]. VIV scenario generally uses the dynamic instability of a bluff body to generate Kármán vortices shedding from either side of the bluff body. Bernitsas et al [21] designed a VIV-based energy harvester to successfully convert ocean/river current hydro-kinetic energy to electric energy. Abdelkefi et al. [22] numerically investigated the energy harvesting from VIV oscillations of rigid circular cylinder and proposed a model with a modified van der Pol equation and the Gauss law. Akaydin et al. [23] proposed to utilize a flexible piezoelectric cantilever beams in the wake of a circular cylinder to harvest unsteady, turbulent fluid flow energy. Mehmood et al. [24] numerically investigated the energy harvesting from VIV of a circular cylinder for different Reynolds covering pre-synchronization, synchronization, and post-synchronization regimes. Dai et al. [25] experimentally compared the four distinct VIV-based piezoelectric energy harvesters in different orientations. Hu et al. [26] conducted a modeling and experimental study of piezoelectric energy to predict the optimum position for vortex shedding-induced vibration. The sensitivity of such energy harvester to the properties of cylinders such as cross sections [27], mass ratio [28], bending stiffness [29] and Reynolds number [30] has been analyzed to further improve the efficiency of energy extraction. However, typical VIV-based energy harvesters are designed to possess a linear restoring force. For such energy harvester, the efficiency of VIV-based generators will dramatically decrease when the vortex shedding frequency mismatches the natural frequency of system [31]. Therefore, the linear VIV-based scenarios have a very narrow frequency response bandwidth and are not desirable when operating in a wide flow speed range. On the other hand, some scholars have tried using galloping to scavenge the wind energy. Galloping is a kind of dynamic instability created by prismatic bluff bodies in flowing fluid, which might lead to a large-amplitude oscillation when the speed of incident flow exceeds the threshold one. Barrero-Gil et al. [32] exploited transverse galloping phenomena to extract flow energy and studied the effects of mass, mechanical properties, cross-section geometry and flow speed on the energy conversion efficiency. Abdelkefi et al. [33] investigated energy harvesting from transverse galloping of a bluff body with different cross-section geometries and explored the influence of parameters on the performance of output power. Sirohi and Mahadik [34, 35] compared the different galloping-based energy harvesters including D-shaped and triangular-cross-section geometries of bluff bodies. Zhao et al. [36] compared the modeling methods of galloping piezoelectric energy harvester including the lumped-parameter single-degree-of-freedom model and the Euler–Bernoulli 2
distributed parameter model. And the effects of wind exposure area and mass of the bluff body, load resistance, and length of piezoelectric sheets on output power performance have also been investigated. Bibo and Daqaq [37] established the universal design curves depending on the aerodynamic properties of the bluff bodies in order to optimize the performance of piezoelectric galloping-based energy harvesters. Abdelmoula and Abdelkefi [38] considered the potential of electrical impedance on the performance of galloping energy harvester with a square cross-section bluff body at its free end subjected to a uniform cross-flow. Hu et al. [39] investigated the effects of the inclining angles of the slender square-section cylinder on the galloping behavior. Based on Hu’s work, Javed and Abdelkefi [40] Further gave insights to the effects of damping-ratio, mass-ratio [41], the cross-sectional aspect ratio [42, 43], cross section [44] of a cylinder on the efficiency of galloping energy harvesters. Galloping featured sustained vibrations without self-limited amplitudes often occur at a speed higher than the flow speed of VIV at resonance. As a result, the piezoelectric galloping-based energy harvesters should work at a relatively high speed so as to get the high harvesting efficiency. Generally, many galloping configurations adopt the design of a bluff body connecting to a piezoelectric cantilever beam. Such similar designs have also appeared in many VIV designs [45]. However, for this kind of design, an unavoidable drawback exists that the energy efficiency could not sustain desirable for the flow with a variable speed, while in practice the fluctuation in wind speed is inevitable. Moreover, there have been many interests in exploiting nonlinear forces, e.g. magnetic forces or external axial loading forces, to excite the large-amplitude motion so as to obtain a high efficiency in energy harvesting. Nonlinear forces could produce mono-stable, bi-stable, tri-stable, even quad-stable motions, resulting in snap-through motion and thus large energy output. As a result some researchers have tried to employ such scenario in harvesting the fluid energy. Abdelkefi et al. [46] investigated the influences of nonlinear parameters of the piezoaeroelastic energy harvesters on the output powers. Bibo et al. [47] designed a galloping energy harvester with a quartic-potential-energy function to improve the output power and investigate the influence the nonlinearity on the energy harvesting performance. Huynh and Tjahjowidodo [48] simulated a nonlinear vortex induced vibration system and found the existence of chaotic vibrations in such structure. An experimental validation of bi-stable VIV system was carried out to investigate the effect of structural parameters including bi-stable gap and damping factor [49]. Zhang et al. [50] used nonlinear magnetic forces to enhance the output power form vortex-induced vibration. Naseer et al. [51] designed a mono-stable VIV-based piezoelectric energy harvester to widen the synchronization region. Alhadidi and Daqaq [52] proposed a wake-galloping flow energy harvester with bi-stable characteristics for broadband energy harvesting when flow speed varies. Usman et al. [53] designed an energy harvester employing the wake galloping phenomenon. The results of experimental validation indicates that there exists an optimal spacing distance between two cylinders. In the classical configurations, most flow-induced energy harvesters are designed by employing one of some types of instability phenomena, e,g., VIV, galloping or flutter [54]. 3
In this paper, the strategy for enhancing the performance of wind energy harvesting is proposed and realized by introducing nonlinear forces and using interactions among different instability phenomena. A novel bi-stable piezoelectric energy harvester (BPEH) is proposed to improve the efficiency of scavenging wind energy by introducing the dynamic multi-stability and incorporating the merits of VIV and galloping. For the purpose of working efficiently for variable speed winds, the configuration is designed to consist of a cruciform piezoelectric substrate with a circular, two square cylinders and three magnets. Different from the general scheme introduced in the literatures, this configuration owns two or three stable equilibria, between or among which the snap-through motion could be activated to occur. As a result, the proposed BPEH could generate a very large output voltage over a wide range of wind speed. 2. Proposed system
Fig. 1. The schematic diagram of the BPEH based on vortex-induced vibration and galloping.
The configuration of proposed BPEH is shown in Fig. 1. It consists of a cruciform cantilever substrate (including a horizontal part and a vertical part) covered with a piezoelectric patch. A circular cylinder, two square cylinders and a tip magnet are attached to the cruciform substrate, while two external magnets are fixed opposite to the tip magnet to introduce attractive forces. The square and circular cylinder bluff bodies subjected to the flow can induce two different types of vibrations: galloping and VIV, respectively. The cruciform cantilever substrate and the attached cylinders are designed such as to induce interactions between VIV and galloping. Owing to the existence of magnetic attractive force, when the wind speed is low, the harvester will stabilize on one stable position near a fixed magnet. Then with the wind speed increasing, the flow passing through the circular cylinder will create vortex and cause the cantilever beam to oscillate; furthermore, the two square cylinders will introduce the galloping phenomenon. These two kinds of vibration could both facilitate the beam to leave the original stable positions, then the beam will start to oscillate with the release of bending potential energy. This process could 4
repeatedly happen and perform an inter-well vibration that will induce high energy output. In practice, the onset speed of galloping is relatively high. Therefore, for the low speed wind, the snap-through will only be triggered by VIV; while the performance of VIV often degrades out of its best speed. To overcome this defect, the square cylinders were designed so that the increasing wind speed could exceed a critical value to enable the galloping to be excited. As a consequence the beam could maintain the snap-through motion, or inter-well vibration and produce a high energy output. Using the interaction between VIV and galloping and the resulted bi-stability, the proposed system can maintain a nearly periodic snap-through over a wide range of wind speed and thus generate high output voltages. It should be noted that in our configuration, the nonlinear forces could turn the system to be a bi-stable one, which plays an important role in improving the harvesting efficiency. Nomenclature Ls
Length of horizontal substrate
bs
Width of horizontal substrate
hs
Thickness of horizontal substrate
ρs
Density of horizontal substrate
Es
Young’s modulus of horizontal substrate
Lsv
Length of vertical substrate
bsv
Width of vertical substrate
hsv
Thickness of vertical substrate
ρsv
Density of vertical substrate
Esv
Young’s modulus of vertical substrate
Lp
Length of piezoelectric layer
bp
Width of piezoelectric layer
hp
Thickness of piezoelectric layer
Ep
Young’s modulus of piezoelectric layer
ρp
Density of piezoelectric layer
Ip
Piezoelectric moment of inertia
ezx
Young’s modulus of piezoelectric layer
V
Voltage of generated by piezoelectric layer
Ls
Length of square cylinder
Ws
Width of square cylinder
Hs
Height of square cylinder
ρs
Density of square cylinder
dc
Diameter of circular cylinder
Lc
Length of circular cylinder
ρc
Density of circular cylinder
Is
Substrate moment of inertia
a1
Effective magnetic moments of tip magnet
a2 (a3)
Effective magnetic moments of fixed magnet
µ0
Permeability constant
w(x,t)
The transverse displacement of cantilever beam at distance x and instant t 5
d
Separation distance
dg
Gap distance
σε
Variance of strain
Since the potential energy and its shape could reflect the multi-stable characteristic of system, we will calculate the potential energy and plot its shape first. The total potential energy could be subdivided into two parts: the magnetic potential energy and the elastic potential energy. The elastic potential energy can be formulated by the Euler-Bernoulli theory. Euler-Bernoulli theory states that the linear inextensible strain through the substrate is a result of bending motion only and proportional to the second spatial derivative of the bending coordinate. Therefore, the elastic potential energy can be given as [55]
= ′′, + ′′,
(1)
where the prime notation ′′ represents ⁄ . In this study, the permanent magnets are modeled as the dipoles in calculating magnetic potential energy, which can be expressed as [56, 57] =−
"# $%
%$&
-
'() , − * + + ,
. /
−
"# $% $.
&
'() , + * + + ,
-
The definition of parameters and their values are listed in Table 1. Table 1 Model parameters used for numerical and experimental studies. Symbol
Parameter
Value
Substrate properties Ls (Lsv)
Length
0.500 m
bs (bsv)
Width
0.01 m
hs (hsv)
Thickness
0.001 m
ρs (ρsv)
Density
7800 kg/m3
Es (Esv)
Young’s modulus
205 Gpa
Piezoelectric laminate properties Lp
Length
0.005 m
bp
Width
0.005 m
hp
Thickness
0.00015 m
Ep
Young’s modulus
2 Gpa
dc
Diameter
0.05 m
Lc
Length
0.25 m
ρc
Density
750 kg/m3
Circular cylinder
Square cylinder 6
. /
(2)
Ls
Length
0.25 m
Ws
Width
0.25 m
Hs
Height
0.05 m
ρs
Density
750 kg/m3
0 (0 , 01 )
Effective magnetic moments
0.213 Am2
µ0
Permeability constant
4π×10-7 NA-2
Potential energy (mJ)
Magnet properties
Fig. 2. Potential energy functions.
The geometric, material and magnetic parameters used in simulations and experiments are listed in Table 1. Fig. 2 shows the results of potential energy versus the tip deflection for different separate distance d. At d=90 mm, i.e., the distance is quite large, the magnetic field is weak and has little influence on the potential energy, thus the system behaves like a linear one. Namely, they own a narrow working bandwidth near the natural frequency and thus can’t work efficiently for the variable wind speed. Then, the distance is reduced to d=26 mm, at which two symmetric potential energy wells begin to emerge, with a potential barrier between them. In this case, if the bluff bodies undergo VIV and galloping oscillations, the system response could take snap-through between the two potential wells, thereby generating high energy output. With a further decrease in d (d=22 mm), the distance between the two potential well’s bottoms will increase, along with a rise of the potential barrier. Thus, the distance d is a key parameter for the system and has a large influence on the potential well and the barrier. The large distance could produce a low barrier and a small separation distance. With this merit, the system could execute snap-through easily, but the resulting amplitude will be small, thus the output is small; whereas the small distance leads to a high barrier and large separation distance, implying that the system needs more excitation energy to execute snap-through. Thus the distance d should be optimized so as to obtain the best harvesting performance. 3. Experimental verification To prove the advantages of the proposed BPEH, corresponding validation experiments were performed for different air flow speeds. The prototype of the proposed BPEH was fabricated, and the experimental setup was built as shown in Fig. 7
3. The dynamic strain response and the dynamic voltage across the piezoelectric patch are recorded by a data acquisition device (DH5922D, DONG HUA). A digital anemometer (AS8336, XI MA) is used to measure the air flow speed. Fig. 4 illustrates the static stable positions of BPEH. If the two fixed magnets are removed, the system becomes a linear piezoelectric energy harvester (LPEH). For comparison with BPEH, the corresponding static stable position of LPEH are shown as well in Fig. 5.
Fig. 3. Experimental setup of the BPEH.
Fig. 4. Static stable positions of BPEH.
8
RMS Voltage [V]
Variance of strain
Fig. 5. Static stable position of LPEH
Fig. 6. (a) Variance of strain (23 ) and (b) RMS voltage (45 ) versus wind speed (LPEH and BPEH). The bar graphs are presented as mean ± standard deviation (n=5).
The variance of dynamic strain (26 ) and open-circuit RMS voltage (45 ) are chosen as the significant quantities to describe the dynamic behaviors of LPEH and BPEH. Fig. 6 shows the results of 23 and 45 versus wind speed v, which varies from 1.0 m/s to 7.0m/s. In order to show potential error and uncertainty of the experimental results, the standard deviation of measured data (5 sets of 200-second responses) has been added in the form of error bars. In Fig. 6(a), we can see that there appears a sharp increase in 26 at v=1.5 m/s, which corresponds to the occurrence of snap-through. The generated 45 is shown in Fig. 5(b), we can see that the 45 is directly proportional to 23 , and the BPEH outperforms the LPEH significantly when the speed exceeds the threshold of snap-through. To show the dynamics of small-amplitude and large-amplitude motions clearly, the dynamic strain and resulting voltage, in terms of time histories, are illustrated in the insets of Figs. 7-19, as the wind speed increased from 1.0 to 7.0 m/s. Moreover, the spectra of dynamic strain are illustrated to show the frequency components.
9
10
Strain amplitude
Voltage [V] Strain
Voltage [V]
Strain
Fig. 7. (a) Strain of LPEH, (b) voltage of LPEH, (c) strain of BPEH, (d) voltage of BPEH and (e) strain FFT (v=1.0 m/s).
Voltage [V]
Strain
Voltage [V]
Strain
First, at a relatively low wind speed (v=1.0 m/s), the dynamic responses of LPEH and BPEH are shown in Fig. 7. Both of them undertake small-amplitude motions confined in a single potential well. From the corresponding spectra (Fig. 7 (e)), it can be seen that the LPEH owns a dominant nearly periodic component near f=1.16Hz, while the BPEH has a dominant component between f=1.18 Hz and f=1.53 Hz due to involvement of nonlinear restoring forces.
11
Strain amplitude
Fig. 8. (a) Strain of LPEH, (b) voltage of LPEH, (c) strain of BPEH, (d) voltage of BPEH and (e) strain FFT (v=1.5 m/s).
Strain
Voltage [V]
Strain
Then adding a small increment to the wind speed, v=1.5 m/s, the BPEH could be triggered to undertake snap-through between the stable equilibrium positions and generate a quiet large output voltage. It can be found that there exists a large amplitude in the low frequency area other than at the first resonance frequency, implying that the snap-through could transfer the excitation energy to the low frequency component.
12
Voltage [V] Strain amplitude Voltage [V]
Strain
Fig. 9. (a) Strain of LPEH, (b) voltage of LPEH, (c) strain of BPEH, (d) voltage of BPEH and (e) strain FFT (v=2.0 m/s).
13
Strain Voltage [V] Strain amplitude Strain
Fig. 10. (a) Strain of LPEH, (b) voltage of LPEH, (c) strain of BPEH, (d) voltage of BPEH and (e) strain FFT (v=2.5 m/s).
14
Voltage [V] Strain Voltage [V] Strain amplitude
Fig. 11. (a) Strain of LPEH, (b) voltage of LPEH, (c) strain of BPEH, (d) voltage of BPEH and (e) strain FFT (v=3.0 m/s).
15
16
Strain amplitude
Voltage [V]
Strain
Voltage [V]
Strain
Voltage [V]
Strain
Voltage [V]
Strain
Fig. 12. (a) Strain of LPEH, (b) voltage of LPEH, (c) strain of BPEH, (d) voltage of BPEH and (e) strain FFT (v=3.5 m/s).
17
Strain amplitude
Fig. 13. (a) Strain of LPEH, (b) voltage of LPEH, (c) strain of BPEH, (d) voltage of BPEH and (e) strain FFT (v=4.0 m/s).
Strain
Voltage [V]
Strain
As the wind speed increases further and attains v=2.0-4.0 m/s, there appears a frequent snap-through motion, resulting in a frequent peak in the output voltage. As for the spectrum, as shown in Figs.9-13, the extremely low frequency components (0-1.3Hz) are dominant.
18
Voltage [V] Strain amplitude Voltage [V]
Strain
Fig. 14. (a) Strain of LPEH, (b) voltage of LPEH, (c) strain of BPEH, (d) voltage of BPEH and (e) strain FFT (v=4.5 m/s).
19
Strain Voltage [V] Strain amplitude Strain
Fig. 15. (a) Strain of LPEH, (b) voltage of LPEH, (c) strain of BPEH, (d) voltage of BPEH and (e) strain FFT (v=5.0 m/s).
20
Voltage [V] Strain Voltage [V] Strain amplitude
Fig. 16. (a) Strain of LPEH, (b) voltage of LPEH, (c) strain of BPEH, (d) voltage of BPEH and (e) strain FFT (v=5.5 m/s).
21
22
Voltage [V]
Strain
Voltage [V]
Strain
Strain amplitude
Fig. 17. (a) Strain of LPEH, (b) voltage of LPEH, (c) strain of BPEH, (d) voltage of BPEH and (e) strain FFT (v=6.0 m/s).
Voltage [V]
Strain
Further increasing the wind speed to v=4.5-6.0 m/s, the times of snap-through continue to increase. There occurs frequently jumping in the response, as shown in Figs. 14–17, which results in a large output, e.g. at v=6.0 m/s. The frequently jumping is desired in energy harvesting, because the voltage across the piezoelectric patch is proportional to the amplitude of vibration. From the spectrum, it can be found that the energy exhibits a likely uniform distribution over a region of low frequency (0-1.1Hz). This desired vibration is beneficial for energy harvesting.
23
Strain Voltage [V] Strain amplitude Strain
Fig. 18. (a) Strain of LPEH, (b) voltage of LPEH, (c) strain of BPEH, (d) voltage of BPEH and (e) strain FFT (v=6.5 m/s).
24
Voltage [V] Strain Voltage [V] Strain amplitude
Fig. 19. (a) Strain of LPEH, (b) voltage of LPEH, (c) strain of BPEH, (d) voltage of BPEH and (e) strain FFT (v=7.0 m/s).
When the wind speed reaches a high value, e.g., v=6.5-7.0 m/s, as plotted in Figs.18 and 19, the main energy gradually concentrates at the first natural frequency, showing that the magnetic force is small compared to the aerodynamic force now and can’t trap the system. Nevertheless, the existence of two fixed magnets extend the amplitude of vibration effectively, as Figs. 19(a) and 19(c) illustrated. Thus the BPEH still could keep a high output. The maximum voltage could reach nearly 4.2 V now. 25
Strain Voltage [V]
Fig. 20. (a) Strain of LPEH and BPEH, (b) voltage of LPEH and BPEH (v=2.0 m/s).
Strain
In this system, owing to involvement of nonlinear forces and the combination of two kinds of flow-induced vibration: VIV and galloping, there appear some complicated nonlinear phenomena. In order to show the snap-through and its influence on the output voltage clearly, the time histories of response at v=2.0, 3.5, 5.5 and 7.0 m/s are magnified and shown in Figs. 20-23. Fig. 20 shows a short time period of strain and the corresponding output voltage at v=2.0 m/s. At the relatively low speed, the BPEH vibrate around one equilibrium point for a while and then jump to another, undergoing a snap-through between the two equilibrium positions.
26
Voltage [V] Voltage [V]
Strain
Fig. 21. (a) Strain of LPEH and BPEH, (b) voltage of LPEH and BPEH (v=3.5 m/s).
Fig. 22. (a) Strain of LPEH and BPEH, (b) voltage of LPEH and BPEH (v=5.5 m/s).
Strain
Figs. 21 and 22 shows the portions of strain and voltage for two large wind speeds (v=3.5 m/s and v=5.5 m/s). It can be seen that the oscillation of BPEH includes a large amplitude of low frequency vibration and some small high harmonic components. The large vibration corresponds to the bi-stable snap-through, i.e., the inter-well vibration. In this case, both the BPEH’s amplitude and output voltage outperform those of the LPEH.
27
Voltage [V]
Fig. 23. (a) Strain of LPEH and BPEH, (b) voltage of LPEH and BPEH (v=7.0 m/s).
At a quite high wind speed (e.g. v=7.0 m/s), the BPEH undertake nearly the similar oscillation as the LPEH; in contrast, compared to the LPEH, the BPEH’s amplitude is considerably large, thus it gives a quite high output, as shown in Fig. 23. As a summary, the proposed BPEH incorporates the VIV and galloping effects to make the system lose stability easily for the variable speed wind. To increase the amplitude and corresponding output, a multi-stable design is proposed and realized by magnetic attractive forces, by which the instability resulted from VIV or galloping could activate a snap-through between the equilibrium positions and generate a high output. Furthermore, this design could produce a frequent snap-through motions over a wide range of wind speed, which is desired in harvesting wind energy and could generate a large output. The experimental results proved this superior performance clearly. The proposed harvester may be more suitable for application in practical environment, owing to its ability to execute snap-through easily and keep this motion for the wind speed varying in a wide range. 4. Conclusions We proposed a novel harvester (BPEH) to scavenge wind energy, which could increase the harvesting efficiency by incorporating the merits of multi-stability and flow-induced instability, i.e., the vortex-induced vibration and galloping. By introducing the multi-stability and combining the advantages of VIV and galloping, the system could maintain a large amplitude vibration over a wide range of wind speed and generate a high electric output. For low wind speed, VIV takes effect and makes the system undergo snap-through; for high wind speed, galloping happens and drive the system to execute snap-through. Thus the BPEH could generate a high output for variable speed wind. Furthermore, it is proved that over a relatively wide range of wind speed, the BPEH could reach coherence resonance and thus generate a considerably large electric output. Compared to the classical wind energy harvester, our design could keep a stable output as the wind speed fluctuates in a wide range. Thus the proposed harvester is preferred for wind energy harvesting in practical environment. Acknowledgments The authors gratefully acknowledge the support by the National Science Foundation of China (Grant No. 11672237). 28
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Highlights
A design including bluff bodies and magnets is proposed to harvest wind energy. Flow-induced instability and bi-stability are incorporated in the design. Snap-through could be triggered by vortex-induced vibration and galloping. Coherence resonance could be excited to happen over a range of wind speed. The harvester can keep a large output in the environment with variable wind speed.
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: