# Analysis of a Cantilevered Piezoelectric Energy Harvester in Different Orientations for Rotational Motion

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{0}.

#### 2.1. Modeling of the Proposed PEH

_{d}is the rotational speed of the motor; L

_{1}is the length of the first segment; r

_{0}is the distance between the fixed end of the beam and the rotating axis; θ is the tilt angle of the beam; P is the axial load due to the centrifugal force caused by the tip mass and can be expressed as:

_{t}is the tip mass; and L is the total length of the beam.

_{ki}(x) is the mode shape function of the k-th segment and the i-th mode; η

_{i}(t) is the temporal function. In this study, only the first mode was considered. The mode shape function can be further represented as:

_{n}is the resonant frequency of the beam.

_{0}is the damping ratio of the system and f(t) is the normalized coefficient for excitation force caused by the gravitational force of the tip mass:

_{p}is the voltage across the resistive load R; C

_{p}is the capacitance of the piezoelectric layer. The coupling coefficient can be written as:

#### 2.2. Experimental Setup

## 3. Results and Discussion

_{0}. In the outward orientation, the centrifugal force resulted in tensile force on the beam so the resonant frequency increased as the driving frequency rose. The matching frequencies, which are located at intersections between the solid lines and the dashed line, indicate that the resonant frequency of the PEH matched the driving frequency. Note that the resonant frequency increased as r

_{0}was enlarged under the same driving frequency. That is because the centrifugal force caused by the tip mass was enhanced when r

_{0}increased. The slope of the curve became steeper when r

_{0}increased. Figure 6 depicts the simulation and experimental results of the voltage responses. The experimental results match the simulation well. The numerical peak voltage and resonant frequency were close to the experimental results. As predicted by the model, the experimental result showed that the matching frequency rose when r

_{0}was enlarged.

_{0}. The resonant frequency decreased as the driving frequency increased because the centrifugal force worked as compressive force on the beam to lower its stiffness when the beam was placed in the inward orientation. The slope of the curve became steeper when r

_{0}increased. The matching frequency decreased when r

_{0}was enlarged. Figure 8 depicts the frequency responses of the PEH with different r

_{0}. The resonant frequency was raised as r

_{0}decreased in both the simulation and experimental results.

_{0}was set to 30 mm. The PEH was tested with different tilt angles. It was expected that the tilt angle would change the magnitude of the component of centrifugal force in the longitudinal direction of the beam. The result showed that the resonant frequency decreased as the tilt angle increased, because a large tilt angle reduced the tensile axial load caused by the centrifugal force. The slope of the curve became less steep when the angle increased. A similar trend of the resonant frequency was seen in both the simulation and the experimental results shown in Figure 10. The resonant frequency reduced as the tilt angle increased in the simulation and experimental results. The peak voltage was enhanced when the resonant frequency was decreased. Note that there was a mismatch between the numerical and experimental resonant frequency when the tilt angle was 90°. The reason for this is that the component of the centrifugal force in the transverse direction of the beam made the beam deflect. The deflection would strengthen the axial load, as indicated in Figure 11, since the deflection resulted in a larger component of the centrifugal force in the axial direction. Therefore, the experimental stiffness would be higher than the stiffness estimated by the model and consequently the experimental resonant frequency would be higher than the numerical one.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The piezoelectric energy harvester (PEH) under rotational motion in different orientations: (

**a**) the outward configuration, (

**b**) the inward configuration, and (

**c**) the tilted configuration.

**Figure 6.**The simulation and experimental voltage responses of the PEH in the outward orientation with different r

_{0}.

**Figure 8.**The simulation and experimental voltage responses of the PEH in the inward orientation with different r

_{0}.

**Figure 10.**The simulation and experimental voltage responses of the PEH in the tilted orientation with different θ.

Symbol | Description | Value |
---|---|---|

L | Length (beam) | 75 mm |

b_{s} | Width (beam) | 12.7 mm |

h_{s} | Thickness (beam) | 0.1 mm |

ρ_{s} | Density (beam) | 7930 kg/m^{3} |

E_{s} | Young’s modulus (beam) | 193 GPa |

L_{1} | Length (MFC) | 28 mm |

b_{p} | Width (MFC) | 7 mm |

h_{p} | Thickness (MFC) | 0.3 mm |

ρ_{p} | Density (MFC) | 5440 kg/m^{3} |

E_{p} | Young’s modulus (MFC) | 30.336 GPa |

M_{t} | Tip mass | 3.04 g |

R | Load resistance | 1 MΩ |

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**MDPI and ACS Style**

Su, W.-J.; Lin, J.-H.; Li, W.-C. Analysis of a Cantilevered Piezoelectric Energy Harvester in Different Orientations for Rotational Motion. *Sensors* **2020**, *20*, 1206.
https://doi.org/10.3390/s20041206

**AMA Style**

Su W-J, Lin J-H, Li W-C. Analysis of a Cantilevered Piezoelectric Energy Harvester in Different Orientations for Rotational Motion. *Sensors*. 2020; 20(4):1206.
https://doi.org/10.3390/s20041206

**Chicago/Turabian Style**

Su, Wei-Jiun, Jia-Han Lin, and Wei-Chang Li. 2020. "Analysis of a Cantilevered Piezoelectric Energy Harvester in Different Orientations for Rotational Motion" *Sensors* 20, no. 4: 1206.
https://doi.org/10.3390/s20041206