# Two-Degree-of-Freedom Piezoelectric Energy Harvesting from Vortex-Induced Vibration

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Wake Oscillator Model for VIVPEH

_{x}and F

_{y}are, respectively, the drag and lift force on the bluff body under the action of airflow.

#### 2.1. 1DOF VIVPEH

_{D}and C

_{L0}are the mean drag coefficient and the amplitude of the fluctuating lift force coefficient, respectively. ρ is the air density. D and L are the geometric dimensions of the cylindrical bluff body. $q\left(t\right)$ is the variable to describe the motion of the near wake generated behind the bluff body. It can be deemed as an imagined variable related to the averaged transverse component of the flow [29] or an assumed variable proportional to the transverse velocity of the near-wake fluid [30]. The governing equation of $q\left(t\right)$ follows a similar form as that of the van der Pol oscillator:

_{s}is the vortex shedding frequency:

_{L}, the circuit equation can then be written as:

_{p}is the clamped capacitance of the piezoelectric transducer.

#### 2.2. 2DOF VIVPEH

## 3. Numerical Simulations

#### 3.1. Numerical Method

_{a}

_{1}and C

_{a}

_{2}are expressed as:

_{a}

_{2}·q and the aerodynamics drag force as C

_{a}

_{1}·$\dot{y}$. In this way, the matrix differential equation can be analyzed and solved using the Runge–Kutta method. The MATLAB Ode45 function adopts the Runge–Kutta method with an adaptive step size control strategy. Based on the experimental data of the 1DOF VIVPEH model from the literature [15,28] and the parameters of an example MDOF GPEH model in the references [20,26], the masses, damping coefficients, stiffnesses, and piezoelectric properties for the 2DOF VIVPEH under investigation in this paper are listed in Table 1. The model proposed in this paper represents a conceptual design. To avoid making the conceptual design too arbitrary and unrealistic, the system parameters, including the equivalent masses, stiffnesses, damping ratios, and electromechanical coupling factor, are determined by referring to a 2DOF GPEH model [26]. Regarding the wake oscillator model, we assume the working condition is similar to that in [28] and adopt those coefficients in the equation from that article. According to the parameters listed in Table 1, the first- and second-order natural frequencies of the 2DOF VIVPEH under free undamped vibration were calculated to be 4.645 Hz and 10.292 Hz, respectively. In the simulation, the initial condition for Equation (9) was set at [0.0001 0 0 0 0 0 0]

^{T}, which meant the initial displacement of the cylindrical bluff body was 0.1 mm (only used to initiate the vibration).

#### 3.2. Preliminary Numerical Results

#### 3.3. Aerodynamic Characteristics

## 4. Parametric Study

#### 4.1. Effects of Circuit Parameters

^{4}Ω. Hence, it is known that 5 × 10

^{4}Ω is the optimal resistance of the 2DOF VIVPEH. As there is only a single peak in the power plot, it can be inferred that this 2DOF VIVPEH belongs to a weakly coupled piezoelectric system [33]. In other words, the change in the circuit parameters will not bring any significant influence on the dynamics of the mechanical structure. Figure 16 illustrates the effect of the load resistance on the bluff body displacement magnitude of the 2DOF VIVPEH. As predicted by the coupling strength theory, the variation of the bluff body displacement magnitude is minor and negligible in response to the change in the load resistance.

^{5}Ω. When U = 3.5 m/s, 4 m/s, and 4.5 m/s, the optimal load resistance becomes 5 × 10

^{4}Ω. This is because the excitation modes corresponding to region I and region II are different. According to the well-known formula to estimate the optimal resistance, i.e., R

_{opt}= 1/(ω

_{n}C

_{p}), the optimal resistances corresponding to the different modes are different. Moreover, the optimal resistance for the higher-order mode should be smaller. That explains why the optimal resistance in region II is smaller.

#### 4.2. Effect of Mechanical Parameters

_{1}, m

_{2}, k

_{1}, k

_{2}, c

_{1}, c

_{2}) on the “lock-in” regions. The two boundary wind speeds of lock-in region I and II in each state were selected as the ordinate. At the same time, the same lock-in region was filled with solid color, so that the lock-in region is shown as a colorful ribbon. The difference between the upper and lower bounds of each ribbon represents the width of the lock-in region under that condition. Therefore, these three figures show the changing trend of the boundary and width of the two lock-in regions under the change in mechanical parameters.

_{1}and m

_{2}on the formation of the two “lock-in” regions, respectively. It is obvious to see that the effects of m

_{1}and m

_{2}are different. With the increase in m

_{1}, the boundaries of region I and II both decrease, and the upper boundary of region II changes most dramatically. This results in a rapid reduction in the bandwidth of region II. While the bandwidth of region I decreases at a glacial pace, in contrast, the influence brought by the increase in m

_{2}has a more significant effect on region I rather than II. Therefore, we can conclude that regions I and II are, respectively, more sensitive to m

_{2}and m

_{1}.

_{1}and c

_{2}. The mechanical damping coefficient c

_{1}has a mild effect on the locking region: the bandwidth and position of both region I and II almost only slightly change in response to the change in c

_{1}. Unlike c

_{1}, increasing c

_{2}lifts up the two “lock-in” regions, leading to an increase in the cut-in wind speeds. This can be explained by the instability theory of self-excited oscillation. The VIVPEH starts to vibrate when the effective linear damping coefficient becomes negative. Increasing c

_{2}raises the threshold to turn the effective linear damping coefficient from positive to negative.

_{1}and k

_{2}on the two locking regions are demonstrated in Figure 21a,b. Compared with the results shown in Figure 19 and Figure 20, it can be seen that the influences of stiffnesses on the two lock-in region are much more profound. In particular, any increase in either k

_{1}or k

_{2}results in the upward movement and enlargement of region II (the blue and cyan-colored areas in Figure 21. However, regarding region I, k

_{1}and k

_{2}show completely different influences: increasing k

_{2}widens region I, while k

_{1}does the opposite.

_{1}and m

_{2}, the two convexes on the voltage response curve form due to the lock-in regions gradually moving to the left, which is consistent with the results in Figure 19. Meanwhile, the mass variation also causes a change in the output voltage. However, the change in m

_{1}and m

_{2}brings different consequences to the different lock-in regions. With the increase in m

_{1}, the convex height of region I slightly increases, while that of region II significantly decreases. In contrast, increasing m

_{2}leads to a significant decrease in the convex height of region I, with a slight increase in the convex height of region II. Therefore, from the voltage amplitude perspective, the previous conclusion that regions I and II are, respectively, more sensitive to m

_{2}and m

_{1}is still valid.

_{1}leads to a significant decrease in the RMS voltage in region I, but a slight increase in the RMS voltage in region II. In contrast, the increase in k

_{2}brings an increase in the RMS voltage in region I, while the RMS voltage in region II is almost unaffected. The reasons for these phenomena are as follows: on the one hand, the increase in mechanical stiffness makes it more difficult to excite the vibration, which is manifested as the increase in the cut-in wind speed. On the other hand, the first- and second-order modes of the 2DOF VIVPEH have different sensitivities to the mechanical stiffnesses k

_{1}and k

_{2}. That is why increasing k

_{1}and k

_{2}causes the system to take different dynamic behavior changes.

## 5. Conclusions

- (1)
- The 2DOF VIVPEH has two “lock-in” regions, which is beneficial for broadband wind energy harvesting. In general, the bandwidth of region I is smaller than that of region II. Region I and region II correspond to the first- and second-order resonances, respectively.
- (2)
- The optimal load resistance corresponding to the maximum output power is almost the same within the same lock-in region. However, the optimal load resistance corresponding to region II is smaller than that corresponding to region I. This is because different vibration modes are activated in the different lock-in regions.
- (3)
- To reduce the cut-in wind speed, one can increase m
_{1}and m_{2}. However, considering the voltage output within the lock-in regions, m_{1}and m_{2}cannot be arbitrarily tuned. Regarding c_{1}and c_{2}, they should be reduced as much as possible to decrease the cut-in wind speed and enhance the voltage output. Certain environmental factors should be taken into account in selecting k_{1}and k_{2}. For example, when the ambient wind speed is low (2 m/s), one can select a relatively large k_{2}. When the ambient wind speed is high (5 m/s), it is suggested to increase k_{1}properly.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Physical prototype of a conventional 1DOF VIVPEH [15].

**Figure 6.**Fast Fourier transform (FFT) of the steady-state part of the voltage–time response. (U = 4.0 m/s).

**Figure 12.**Fast Fourier transform (FFT) of the steady-state part of the voltage–time response. (U = 1.8 m/s).

**Figure 13.**Fast Fourier transform (FFT) of the steady-state part of the voltage–time response under different wind speeds: (

**a**) 1.6 m/s, (

**b**) 1.8 m/s, (

**c**) 2.0 m/s, (

**d**) 3.4 m/s, (

**e**) 4.2 m/s, and (

**f**) 5.0 m/s.

**Figure 20.**Effects of damping coefficients on the “lock-in” regions of the 2DOF VIVPEH: (

**a**) c

_{1}and (

**b**) c

_{2}.

**Figure 21.**Effects of stiffnesses on the “lock-in” regions of the 2DOF VIVPEH: (

**a**) k

_{1}and (

**b**) k

_{2}.

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

Effective mass m_{1} (g) | 10 | Air density, ρ (kg × m^{−3}) | 1.204 |

Effective mass m_{2} (g) | 7.5 | Bluff body height, L (m) | 0.203 |

Effective stiffness k_{1} (N × m^{−1}) | 21.985 | Cross flow dimension, D (m) | 0.0396 |

Effective stiffness k_{2} (N × m^{−1}) | 12.15 | Amplitude of the fluctuating lift force coefficient, C_{L0} | 0.3 |

Effective damping c_{1} (N × s/m) | 0.0041 | ||

Effective damping c_{2} (N × s/m) | 0.0026 | Mean drag coefficient, C_{D} | 2 |

Electromechanical coupling θ (μN × V^{−1}) | 39.657 | Modal constants, λ | 0.3 |

Capacitance, C_{p} (nF) | 30.78 | Modal constants, A | 12 |

Resistive load, R_{L} (MΩ) | 1 | — | — |

Wind Speed | RMS Voltage | Wind Speed | RMS Voltage |
---|---|---|---|

(m/s) | (V) | (m/s) | (V) |

1.565 | 0.663 | 1.566 | 8.511 |

2.015 | 14.038 | 2.016 | 0.563 |

3.108 | 1.069 | 3.109 | 7.422 |

5.000 | 16.023 | 5.001 | 0.6627 |

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**MDPI and ACS Style**

Lu, D.; Li, Z.; Hu, G.; Zhou, B.; Yang, Y.; Zhang, G.
Two-Degree-of-Freedom Piezoelectric Energy Harvesting from Vortex-Induced Vibration. *Micromachines* **2022**, *13*, 1936.
https://doi.org/10.3390/mi13111936

**AMA Style**

Lu D, Li Z, Hu G, Zhou B, Yang Y, Zhang G.
Two-Degree-of-Freedom Piezoelectric Energy Harvesting from Vortex-Induced Vibration. *Micromachines*. 2022; 13(11):1936.
https://doi.org/10.3390/mi13111936

**Chicago/Turabian Style**

Lu, De, Zhiqing Li, Guobiao Hu, Bo Zhou, Yaowen Yang, and Guiyong Zhang.
2022. "Two-Degree-of-Freedom Piezoelectric Energy Harvesting from Vortex-Induced Vibration" *Micromachines* 13, no. 11: 1936.
https://doi.org/10.3390/mi13111936