# A Dual-Beam Coupled System for Hybrid Galloping and Vortex-Induced Vibration Energy Harvesting

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## Abstract

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## 1. Introduction

## 2. System Overview and Governing Equations

_{A}and R

_{B}are the load resistances connected to the two piezoelectric patches. By adjusting the spring stiffness, the GPEH and the VIVPEH can interact with each other. M, K, and C are the effective mass, stiffness, and damping coefficient, respectively. The subscripts A and B denote the GPEH and the VIVPEH. K

_{C}means the stiffness of the string. X and Y denote the vertical displacements of the GPEH and the VIVPEH, respectively. Additionally, Z represents the base displacement.

_{A}= 150 mm, and the dimensions of the cross section are 40 mm × 40 mm. The cylinder bluff body has a length of L

_{B}= 120 mm, and the diameter D

_{B}is 35 mm.

#### 2.1. The GPEH Model

_{A}is the equivalent mass that includes the contributions of the beam and the bluff body; C

_{A}is the damping coefficient; K

_{A}is the equivalent stiffness of the beam; $x(t)$ is the displacement of the bluff body; and F

_{A}(t) is the aerodynamic force. F

_{A}(t) can be expressed as:

_{a}is the air density; U is the wind velocity; ${S}_{A}$ is the windward area of the bluff body and ${C}_{FZ}$ is the total aerodynamic force coefficient. ${C}_{FZ}$ is a function of the attack angle α, and can be determined through experiments or CFD (Computational Fluid Dynamics) simulations. For simplicity, ${C}_{FZ}$ can be empirically expressed as a polynomial expansion:

#### 2.2. The VIVPEH Model

#### 2.3. The HWPEH Model

_{C}. The coupled system is referred to as the hybrid wind piezoelectric energy harvester (HWPEH). Based on the governing equations of the GPEH and the VIVPEH and considering the existence of the coupling spring, the governing equations of the HWPEH can be expressed as:

## 3. Parametric Studies

#### 3.1. Effect of Mass

_{A}, which is the mass of the square-sectioned bluff body. With the increase in M

_{A}, the output of the GPEH first decreases and then increases. At low wind speeds, as shown in the blue box in the figure, the output voltage at M

_{A}= 10 g, M

_{A}= 20 g, and M

_{A}= 30 g is considerably large. When M

_{A}is 20 g, there is an obvious drop point at point A, after which the voltage output rises steadily again. Figure 2b shows the change in the voltage output response of the GPEH-Component by varying M

_{B}, which is the mass of the cylinder bluff body. With the increase in M

_{B}, the voltage output from the GPEH-Component decreases. Similar to the phenomenon in Figure 2a, when M

_{B}is 2 g, the output voltage from the GPEH-Component has an obvious drop at point B. Figure 2c demonstrates how the voltage output response of the VIVPEH-Component varies with the change in M

_{A}. In the case of M

_{A}= 10 g, the output from the VIVPEH-Component is significantly large. When M

_{A}exceeds 20 g, the voltage output becomes remarkably smaller. For the three four cases, i.e., M

_{A}= 30, 40, and 50 g, the output of the VIVPEH-Component is almost unchanged and tends to be stable. When M

_{A}is 60 g, the locking interval of VIV is slightly changed. At point C, it drops in advance and tends to be stable. Figure 2d reveals the variation of the voltage output from the VIVPEH-Component in response to the change in M

_{B}. When M

_{B}is small, the output voltage is small, but when M

_{B}is increased to 7, 8, and 9 g, the output voltage increases remarkably.

#### 3.2. Effect of Stiffness

_{A}. The whole wind speed range can be divided into three regions. In region

**I**, when K

_{A}= 80 N/m, the GPEH voltage is approximately 90V, while when K

_{A}= 10 N/m, there is almost no output voltage. With the increase in output voltage, the cut-in wind speed decreases significantly. In region

**II**, as the stiffness increases, the slope of the curve decreases. For example, the curve corresponding to K

_{A}= 10 N/m has a larger output slope, and the output voltage increases rapidly with the wind speed. Within region

**II**, the voltage output may not monotonically increase with the wind speed, such as in the case of K

_{A}= 70 N/m. Region

**III**corresponds to the high wind speed range. Over this wind speed range, the voltage output steadily increases with the increase in the wind speed. Figure 3b shows the voltage outputs from the GPEH-Component for different K

_{B}. It can be seen in Figure 3b that increasing K

_{B}will raise the cut-in wind speed and reduces the output voltage. Figure 3c depicts the influence of K

_{A}on the VIVPEH output. The output voltage increases drastically when K

_{A}is tuned to 80 N/m. In other cases, the changes are small. The curves for the cases of 70 and 80 N/m exhibit slight drop points at wind speeds between 3 and 4 m/s, respectively. In Figure 3d, with the increase in K

_{B}, the cut-in wind speed increases obviously, and its locking interval is shortened first and then widened.

#### 3.3. Effect of Damping Coefficient

_{A}on the GPEH- and VIVPEH-Components. As C

_{A}increases, the voltage outputs of both components decrease. When the damping coefficient is larger, for example, C

_{A}= 0.02 N/(m/s), the cut-in wind speed will increase to a certain level. As shown in Figure 4b, the influence of C

_{B}on the GPEH output voltage is mainly reflected in the cut-in wind speed. With the increase in C

_{B}, the cut-in wind speed of the GPEH-Component decreases obviously, but the output voltage becomes higher at low wind speeds. In Figure 4d, when C

_{B}increases, the output voltage from the VIVPEH decreases, the cut-in wind speed increases, and the locking interval shrinks.

#### 3.4. Discussion on the Parametric Studies

_{C}) on the HWPEH is shown in Figure 9. With the increase in K

_{C}, the cut-in wind speed of the GPEH-Component increases and tends to be stable. At high wind speeds, the vibration displacement of the GPEH-Component first rises and then stabilizes at approximately 50 mm. For the VIVPEH-Component, the displacement increases with the increase in K

_{C}, and the peak moves to the high wind speed direction. At the same time, the locking interval is significantly widened. As shown by the arrow in the figure, when the spring stiffness is 27 N/m, the displacements of both components suddenly drop.

## 4. Impedance Matching Analysis

_{A}keeps increasing. Figure 10b illustrates that with the increase in R

_{A}, the power of the GPEH-Component first increases and then decreases, and the peak is attained at approximately R

_{A}= 0.2 MΩ, which is the optimal resistance. From Figure 10c, it is noted that the output voltage of the VIVPEH-Component increases monotonically with the increase in its shunt resistance R

_{B}, which is similar to the phenomenon in Figure 10a. However, Figure 10d shows that under the same wind speed, the power output produced by the VIVPEH-Component first increases and then decreases. It can be identified that the optimal resistive load of the VIVPEH-Component is approximately 1.95 MΩ.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Effects of masses on the voltage of the HWPEH (K

_{A}= 52.01 N/m; K

_{B}= 9.894 N/m; C

_{A}= 0.0121 N/(m/s); C

_{A}= 0.0121 N/(m/s); K

_{C}= 5 N/m): (

**a**) M

_{A}on GPEH; (

**b**) M

_{B}on GPEH; (

**c**) M

_{A}on VIVPEH; (

**d**) M

_{B}on VIVPEH.

**Figure 3.**Effects of stiffness on the voltage of the HWPEH (M

_{A}= 28.16 g; M

_{B}= 4.316 g; C

_{A}= 0.0121 N/(m/s); C

_{A}= 0.0121 N/(m/s); K

_{C}= 5 N/m): (

**a**) K

_{A}on GPEH; (

**b**) K

_{B}on GPEH; (

**c**) K

_{A}on VIVPEH; (

**d**) K

_{B}on VIVPEH.

**Figure 4.**Effects of damping coefficient on the voltage of the HWPEH (M

_{A}= 28.16 g; M

_{B}= 4.316 g; K

_{A}= 52.01 N/m; K

_{B}= 9.894 N/m; K

_{C}= 5 N/m): (

**a**) C

_{A}on GPEH; (

**b**) C

_{B}on GPEH; (

**c**) C

_{A}on VIVPEH; (

**d**) C

_{B}on VIVPEH.

**Figure 5.**Phase portrait (U = 3 m/s): (

**a**) traditional GPEH; (

**b**) GPEH-Component of the HWPEH; (

**c**) traditional VIVPEH; (

**d**) VIVPEH-Component of the HWPEH.

**Figure 6.**The time-domain waveforms of the harvester (U = 3 m/s): (

**a**) traditional GPEH; (

**b**) GPEH-Component of the HWPEH; (

**c**) traditional VIVPEH; (

**d**) VIVPEH-Component of the HWPEH.

**Figure 7.**Fast Fourier transform (FFT) of the steady-state response: (

**a**) traditional GPEH versus GPEH-Component of the HWPEH at U = 3 m/s; (

**b**) traditional GPEH versus GPEH-Component of the HWPEH at U = 6 m/s; (

**c**) traditional VIVPEH versus VIVPEH-Component of the HWPEH at U = 3 m/s; (

**d**) traditional VIVPEH versus VIVPEH-Component of the HWPEH at U = 6 m/s.

**Figure 8.**The displacement of three HWPEH conditions: (

**a**) traditional GPEH versus GPEH-Component of the HWPEH when the cut-in wind speed of traditional VPEH is lower than that of the traditional GPEH. (

**b**) Traditional VPEH versus VIVPEH-Component of the HWPEH when the cut-in wind speed of traditional VPEH is lower than that of the traditional GPEH. (

**c**) Traditional GPEH versus GPEH-Component of the HWPEH when the cut-in wind speed of traditional VPEH and traditional GPEH are approximately the same. (

**d**) Traditional VPEH versus VPEH-Component of the HWPEH when the cut-in wind speed of traditional VPEH and traditional GPEH are approximately the same. (

**e**) Traditional GPEH versus GPEH-Component of the HWPEH when the cut-in wind speed of traditional VPEH is higher than that of the traditional GPEH. (

**f**) Traditional VPEH versus VPEH-Component of the HWPEH when the cut-in wind speed of traditional VPEH is higher than that of the traditional GPEH.

**Figure 9.**Effects of K

_{C}on HWPEH: (

**a**) GPEH-Component displacement; (

**b**) VIVPEH-Component displacement.

**Figure 10.**Effects of the resistive load on the output of the HWPEH under different wind speeds: (

**a**) R

_{A}on GPEH voltage; (

**b**) R

_{A}on GPEH average power; (

**c**) R

_{B}on VPEH voltage; (

**d**) R

_{B}on VPEH average power.

Mechanical Parameters | Aerodynamic Parameters | ||
---|---|---|---|

GPEH effective mass ${M}_{A}$(g) | 28.16 | Air density, ρ (kg·m^{−3}) | 1.204 |

VIVPEH effective mass ${M}_{B}$(g) | 4.316 | Empirical aerodynamic coefficients of polynomial expansion of ${C}_{{}_{FZ}}$ | ${A}_{1}=2.3$ ${A}_{2}=0$ ${A}_{1}=-18$ |

GPEH effective stiffness ${K}_{A}$(N·m^{−1}) | 52.01 | ||

VIVPEH effective stiffness ${K}_{B}$ (N·m^{−1}) | 9.894 | ||

GPEH effective damping ${C}_{A}$ | 0.019 | Amplitude of the fluctuating lift force coefficient ${C}_{L0}$ | 0.490 |

VIVPEH effective damping ${C}_{B}$ | 0.01 | ||

GPEH electromechanical coupling ${\Theta}_{A}$(μN·V^{−1}) | 370.6 | Mean drag coefficient ${C}_{D}$ | 1.331 |

VIVPEH electromechanical coupling ${\Theta}_{B}$(μN·V^{−1}) | 98.94 | Strouhal number ${S}_{\mathrm{t}}$ | 0.128 |

GPEH capacitance, ${C}_{A}^{P}$(nF) | 180 | ||

VIVPEH capacitance, ${C}_{B}^{P}$(nF) | 12.2 | ||

GPEH resistive load, ${R}_{A}$ (MΩ) | 3.5 | ||

VIVPEH resistive load, ${R}_{B}$ (MΩ) | 3.5 |

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**MDPI and ACS Style**

Li, Z.; Liu, K.; Zhao, C.; Zhou, B.; Yang, Y.; Zhang, G.
A Dual-Beam Coupled System for Hybrid Galloping and Vortex-Induced Vibration Energy Harvesting. *Symmetry* **2022**, *14*, 2601.
https://doi.org/10.3390/sym14122601

**AMA Style**

Li Z, Liu K, Zhao C, Zhou B, Yang Y, Zhang G.
A Dual-Beam Coupled System for Hybrid Galloping and Vortex-Induced Vibration Energy Harvesting. *Symmetry*. 2022; 14(12):2601.
https://doi.org/10.3390/sym14122601

**Chicago/Turabian Style**

Li, Zhiqing, Kaihua Liu, Chaoyang Zhao, Bo Zhou, Yaowen Yang, and Guiyong Zhang.
2022. "A Dual-Beam Coupled System for Hybrid Galloping and Vortex-Induced Vibration Energy Harvesting" *Symmetry* 14, no. 12: 2601.
https://doi.org/10.3390/sym14122601