# On the Effects of Structural Coupling on Piezoelectric Energy Harvesting Systems Subject to Random Base Excitation

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Modelling of the System

#### 2.1. Eigen-Value Problem

#### 2.2. Dynamic Analysis

#### 2.3. Validation

## 3. Random Vibration Analysis

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^{3}). Moreover, in this figure, ${a}_{0}$ is assumed to be 36.966 (corresponding to nominal value of 50 Hz) which is slightly lower than the fourth natural frequency of the system, 53.678 Hz. Since the maximum excitation frequency is between the third and the fourth natural frequencies of the system, it would be reasonable to include four modes in dynamic analysis. Thus, hereafter in this paper, unless otherwise mentioned, we will consider four modes in the simulations.

## 4. Parametric Study

## 5. Design Optimization

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Determination of Exact Natural Frequencies and Mode Shapes

## Appendix B. Determination of the Location of the Neutral Surface

## Appendix C. Deriving Frequency Response Function

## References

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**Figure 5.**Frequency response curve of ${v}_{1}$, of the sample system, due to harmonic base acceleration.

**Figure 6.**Frequency response curves of the normalized tip displacement of the thin and thick beams due to harmonic base acceleration.

**Figure 10.**Mean value of the harvested powers versus the thickness ratio of the beams ${h}_{s}^{\left(2\right)}/{h}_{s}^{\left(1\right)}$.

**Figure 11.**Normalized natural frequencies of the system versus the thickness ratio ${h}_{s}^{\left(2\right)}/{h}_{s}^{\left(1\right)}$.

**Figure 13.**Mean value of the harvested power versus the thickness of the piezoelectric layer, when ${R}_{l}=0.2\mathrm{M}\Omega $.

**Figure 14.**The electromechanical coupling factor of the thin beam, ${\vartheta}_{2}$, versus thickness of the piezoelectric layer.

**Figure 15.**Value of the harvested power versus the thickness of the piezoelectric layer, when ${R}_{l}=0.453\mathrm{M}\Omega .$

Parameter | Symbol | Unit | Value |
---|---|---|---|

Length of the beam | ${L}_{s}$ | mm | 305 |

Thickness of the thick beam | ${h}_{s}^{\left(1\right)}$ | mm | 0.5 |

Thickness of the thin beam | ${h}_{s}^{\left(2\right)}$ | mm | 0.25 |

Width of the beams | ${b}_{s}$ | mm | 16 |

Young’s modulus | ${E}_{s}$ | GPa | 210 |

Mass density of beams | ${\rho}_{s}$ | kg/m^{3} | 7000 |

Mode Number | Analytical (rad/s) | FEA (rad/s) | Error (%) |
---|---|---|---|

1 | 25.6611 | 25.6610 | $0.0001$ |

2 | 71.7585 | 71.7585 | $0.0000$ |

3 | 167.1232 | 167.1232 | $0.0000$ |

Parameter | Symbol | Unit | Value |
---|---|---|---|

Width of the piezoelectric layers | ${b}_{p}$ | mm | 7 |

Young’s modulus of the piezoelectric layer | ${E}_{p}$ | GPa | 66 |

Permittivity | ${\epsilon}_{33}^{s}$ | nF/m | 15.93 |

Piezoelectric constant | ${d}_{31}$ | pm/v | −190 |

Mass density of the piezoelectric layer | ${\rho}_{p}$ | kg/m^{3} | 7800 |

Electrical resistance | ${R}_{l}$ | $\mathrm{M}\Omega $ | 0.2 |

Thickness of the piezoelectric layers | ${h}_{p}$ | mm | 0.25 |

Parameter | Symbol | Unit | Value |
---|---|---|---|

Length of the beam | ${L}_{s}$ | mm | 100 |

Thickness of the thick beam | ${h}_{s}^{\left(1\right)}$ | mm | 0.5 |

Thickness of the thin beam | ${h}_{s}^{\left(2\right)}$ | mm | 0. 5 |

Width of the beams | ${b}_{s}$ | mm | 20 |

Young’s modulus of the beams | ${E}_{s}$ | GPa | 100 |

Mass density of beams | ${\rho}_{s}$ | kg/m^{3} | 7165 |

Parameter | Symbol | Unit | Value |
---|---|---|---|

Width of the piezoelectric layers | ${b}_{p}$ | mm | 20 |

Young’s modulus of the piezoelectric layer | ${E}_{p}$ | GPa | 66 |

Permittivity | ${\epsilon}_{33}^{s}$ | nF/m | 15.93 |

Piezoelectric constant | ${d}_{31}$ | pm/v | −190 |

Mass density of the piezoelectric layer | ${\rho}_{p}$ | kg/m^{3} | 7800 |

Natural Frequency | Present Study (Hz) | Ref. [26] (Hz) | Error (%) |
---|---|---|---|

First Mode | 48.06 | 47.8 | $0.543$ |

Second Mode | 299.7 | 299.6 | $0.033$ |

Third mode | 838.9 | 838.2 | $0.083$ |

Natural Frequency | Present Study (Hz) | Ref. [26] (Hz) | Error (%) |
---|---|---|---|

First Mode | 48.83 | 48.8 | $0.0614$ |

Second Mode | 301.4 | 301.5 | $-0.031$ |

Third mode | 840.9 | 839.2 | $0.202$ |

Parameter | Optimum Value | Unit |
---|---|---|

${h}_{s}^{\left(2\right)}/{h}_{s}^{\left(1\right)}$ | 0.0835 | – |

$k$ | 665.9402 | N/m |

${h}_{p}$ | 2.8 | mm |

${b}_{p}$ | 4.3 | mm |

${b}_{s}$ | 57.8 | mm |

${R}_{l}$ | 1.635 × 10^{7} | Ω |

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**MDPI and ACS Style**

Masoumi, H.; Moeenfard, H.; Haddad Khodaparast, H.; Friswell, M.I.
On the Effects of Structural Coupling on Piezoelectric Energy Harvesting Systems Subject to Random Base Excitation. *Aerospace* **2020**, *7*, 93.
https://doi.org/10.3390/aerospace7070093

**AMA Style**

Masoumi H, Moeenfard H, Haddad Khodaparast H, Friswell MI.
On the Effects of Structural Coupling on Piezoelectric Energy Harvesting Systems Subject to Random Base Excitation. *Aerospace*. 2020; 7(7):93.
https://doi.org/10.3390/aerospace7070093

**Chicago/Turabian Style**

Masoumi, Hamidreza, Hamid Moeenfard, Hamed Haddad Khodaparast, and Michael I. Friswell.
2020. "On the Effects of Structural Coupling on Piezoelectric Energy Harvesting Systems Subject to Random Base Excitation" *Aerospace* 7, no. 7: 93.
https://doi.org/10.3390/aerospace7070093