# Investigation of Nonlinear Piezoelectric Energy Harvester for Low-Frequency and Wideband Applications

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

^{2}. A similar FUC mechanism that uses snap-through buckling has also been reported [26,27].

## 2. Design and Modeling

## 3. Result and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Schematic diagram of the proposed nonlinear PEH with an exsect-tapered FR4 spring, and (

**b**) top view of the nonlinear PEH.

**Figure 2.**FEM analysis of the exsect spring structure design depicting the first four modes of the nonlinear PEH. The color legend shows the deformation of the structure in the z-axis.

**Figure 3.**The resonant frequency of the nonlinear PEHs: (

**a**) rectangular, (

**b**) tapered, and (

**c**) exsect-tapered.

**Figure 4.**The distribution of stress in PZT-5H piezo film of the nonlinear PEHs: (

**a**) rectangular, (

**b**) tapered, and (

**c**) exsect-tapered.

**Figure 6.**(

**a**) The restoring force of spring vs. deflection amplitude corresponding to different spring thicknesses in the exsect-tapered PEH and (

**b**) potential energy of the monostable PEH with a spring thickness ts = 0.5 mm.

**Figure 9.**Numerically simulated frequency response (linear) of the PEHs for an input excitation of 0.001 g.

**Figure 10.**Numerically simulated frequency response (nonlinear) of the PEHs for an input excitation of 0.9 g. Solid lines represent forward sweep, and dashed lines represent backward sweep.

**Figure 11.**Analytical simulated output power for a varying frequency with load resistance R = 0.27 MΩ and different input excitations (acc) for forward (solid line) and backward (dash line) frequency sweeps in an exsect-tapered nonlinear PEH.

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

$\mathrm{The}\mathrm{effective}\mathrm{mass}\mathrm{of}\mathrm{rectangular}\mathrm{tapered}\mathrm{and}\mathrm{exsect}-\mathrm{tapered}\mathrm{PEH},m$ | 4.1 g, 4.7 g and 3.781 g |

Spring width at the fixed end | 8.6 mm |

Spring width at the guided end | 4.3 mm |

PZT-5H size | 5.6 mm × 3 mm × 0.2 mm |

$\mathrm{The}\mathrm{thickness}\mathrm{of}\mathrm{the}\mathrm{FR}4\mathrm{spring},{t}_{s}$ | 0.5 mm |

$\mathrm{The}\mathrm{density}\mathrm{of}\mathrm{FR}4,{\rho}_{s}$ | $1900(\mathrm{kg}/{\mathrm{m}}^{3}$) |

$\mathrm{Young}\mathrm{Modulus}\mathrm{of}\mathrm{FR}4,{E}_{s}$ | 22 (GPa) |

$\mathrm{Young}\mathrm{Modulus}\mathrm{of}\mathrm{NdFeB},{E}_{n}$ | 160 (GPa) |

$\mathrm{The}\mathrm{density}\mathrm{of}\mathrm{NdFeB},{\rho}_{n}$ | $7800(\mathrm{kg}/{\mathrm{m}}^{3}$) |

$\mathrm{The}\mathrm{density}\mathrm{of}\mathrm{PZT}-5\mathrm{H},{\rho}_{p}$ | $7500(\mathrm{kg}/{\mathrm{m}}^{3}$) |

$\mathrm{Young}\mathrm{Modulus}\mathrm{of}\mathrm{PZT}-5\mathrm{H},{E}_{p}$ | 64 (GPa) |

$\mathrm{Piezoelectric}\mathrm{constant},{d}_{31}$ | 750 (pC/N) |

$\mathrm{PZT}\mathrm{relative}\mathrm{Permittivity}\mathrm{constant},{\epsilon}_{ss}$ | 39.84 (pF/m) |

Damping ratio, D | 0.003 |

Coupling coefficient | 0.04156 |

$\mathrm{Piezoelectric}\mathrm{Capacitance},{C}_{p}$ | 5.65 (nF) |

Load Resistance, R | 0.27 MΩ |

PEH | Stress (MPa) | k_{L} (N/m) | k_{NL} (N/m^{3}) |
---|---|---|---|

Rectangular | 2.5 | 6847 | 1.73 × 10^{10} |

Tapered | 3 | 4867 | 9.25 × 10^{9} |

Exsect-Tapered | 6 | 3227 | 8.92 × 10^{9} |

Nonlinear PEH | Resonant Frequency (Hz) | Optimal Load (Ω) | Bandwidth (Hz) | $\mathbf{Piezoelectric}\mathbf{Power}{\mathit{P}}_{\mathit{P}}\left(\mathbf{mW}\right),\mathbf{acc}=0.9\mathbf{g}$ |
---|---|---|---|---|

Rectangular | 196.6 | 1.0 × 10^{5} | 7.8 | 1.8 |

Tapered | 179.5 | 1.4 × 10^{5} | 8.1 | 2.05 |

Exsect-Tapered | 150.3 | 1.7 × 10^{5} | 9 | 2.6 |

S. No. | Wideband Harvester | Bandwidth (Hz) | Input Excitation (g) | Device Volume (cm ^{3}) | Generated Power Output (μW) | Normalized Power Density $(\mathbf{NPD}=\mathsf{\mu}\mathbf{W}/\mathbf{c}{\mathbf{m}}^{3}{\mathbf{g}}^{2})$ |
---|---|---|---|---|---|---|

1. | Multimode [43] | 59 | 0.5 | 0.0041 | 0.61 | 595.12 |

2. | FUC [25] | 22 | 0.8 | 0.0161 | 0.19 | 18.43 |

3. | Clamped-Clamped [34] | 9.64 | 0.1 | 1.22 | 125 | 10245 |

4. | Rectangular nonlinear (Fixed-Guided) | 7.8 | 0.9 | 0.824 | 1800 | 2696.87 |

Tapered nonlinear (Fixed-Guided) | 8.1 | 0.779 | 2050 | 3248.86 | ||

Exsect-Tapered (This Work) | 9 | 0.753 | 2600 | 4262.78 |

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**MDPI and ACS Style**

Pertin, O.; Guha, K.; Jakšić, O.; Jakšić, Z.; Iannacci, J.
Investigation of Nonlinear Piezoelectric Energy Harvester for Low-Frequency and Wideband Applications. *Micromachines* **2022**, *13*, 1399.
https://doi.org/10.3390/mi13091399

**AMA Style**

Pertin O, Guha K, Jakšić O, Jakšić Z, Iannacci J.
Investigation of Nonlinear Piezoelectric Energy Harvester for Low-Frequency and Wideband Applications. *Micromachines*. 2022; 13(9):1399.
https://doi.org/10.3390/mi13091399

**Chicago/Turabian Style**

Pertin, Osor, Koushik Guha, Olga Jakšić, Zoran Jakšić, and Jacopo Iannacci.
2022. "Investigation of Nonlinear Piezoelectric Energy Harvester for Low-Frequency and Wideband Applications" *Micromachines* 13, no. 9: 1399.
https://doi.org/10.3390/mi13091399