# Model Validation of a Porous Piezoelectric Energy Harvester Using Vibration Test Data

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Manufacture Of Porous Material

## 3. Theoretical Homogenisation

## 4. Numerical Model

^{®}was used to model the harvester. The homogenised material properties obtained from the Mori–Tanaka theory were modelled in the linear elastic range using 3D elements as Figure 1 shows. The element type used for the whole model was SOLID 227. The external resistors were modelled using the element CIRCU94, connecting the top surface of the piezoelectric disk and the bottom surface. The FE model was coupled with Matlab

^{®}which performed the homogenisation process and provided the material properties and geometrical parameters to ANSYS

^{®}. An in-house Matlab

^{®}code was developed to link ANSYS to MATLAB and can be used for optimisation/model updating purposes. The porous material was modelled using the homogenised parameters obtained in Section 3.

## 5. Experimental Validation

## 6. Experimental Results

## 7. Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Finite element model: (

**a**) general view of the cantilever beam energy harvester with the left side clamped; (

**b**) detailed view of the clamped side and the porous piezoelectric patch as well as the the resistor which connects the top surface with the bottom surface of the porous piezoelectric disk; and (

**c**) schematic view of the external circuit connected to the harvester.

**Figure 2.**(

**a**) Material coefficients ${d}_{31}$ (red) and ${\u03f5}_{33}$ (blue) predicted by homogenisation theory “Mori–Tanaka” (solid line) compared with the values measured experimentally (dots); (

**b**) view of the laboratory setup; and detailed view of a porous patch attached to the aluminium beam at the bottom right corner.

**Figure 3.**Dynamic test results performed on the non-porous patches. Solid lines correspond to simulation results and dashed lines correspond to experimental results.

**Figure 4.**Identification of possible sources of non linearities through sweep in the amplitude of the base excitation.

**Figure 6.**Dynamic test results (6–10) performed on the porous piezoelectric disks. FC, Free-casting; BS, BURPS. Solid lines correspond to simulation results and dashed line correspond to experimental measures.

**Figure 7.**Dynamic test (11–14) results performed on the porous piezoelectric disks. FC, Free-casting; BS, BURPS. Solid lines correspond to simulation results and dashed line correspond to experimental measures.

**Figure 8.**Comparison between the piezoelectric coefficients ${d}_{31}$, ${d}_{33}$ and ${k}_{33}$ for different inclusion shapes, sphere (solid line) and cylindrical (dashed line). Data were obtained using the Mori–Tanaka.

**Figure 9.**Uncertainty results for the voltage output of demonstrator Test 12. The simulation results for values between 40% and 120% of the nominal values of the piezoelectric coefficients are presented as a blue green area. Experimental results are given as blue line with blue circles and the results for the nominal values are given as dashed red line.

**Figure 10.**Power output for given porosity percentage on Barium Titanate test samples. The resistance is constant (see Figure 1c) and the frequency excitation is equal to the first natural frequency of each sample.

**Table 1.**Material properties of the non-porous piezoelectric patch supplied by Morgan Advance Ceramics PLC and the aluminium used for the beams.

PZT-5A | Aluminium | |||
---|---|---|---|---|

Density ($\mathrm{kg}/{\mathrm{m}}^{3}$) | 7750 | Density ($\mathrm{kg}/{\mathrm{m}}^{3}$) | 2700 | |

${S}_{33}^{E}$ (${10}^{-12}\phantom{\rule{4pt}{0ex}}{\mathrm{m}}^{2}/\mathrm{N}$) | 17.2 | Young’s Modulus (GPa) | 70 | |

${S}_{11}^{E}$ (${10}^{-12}\phantom{\rule{4pt}{0ex}}{\mathrm{m}}^{2}/\mathrm{N}$) | 16.7 | Poisson’s Ratio | 0.3 | |

${d}_{33}$ ($\mathrm{pC}/\mathrm{N}$) | 409 | |||

${d}_{31}$ ($\mathrm{pC}/\mathrm{N}$) | 176 | Beam Dimensions | ||

${\u03f5}_{33}^{T}/{\u03f5}_{0}$ | 1800 | Length × Width × Thickness (mm) | 375 × 16 × 1 |

Sample Number | Porosity (%) | Thickness (mm) | Diameter (mm) | Fabrication Method | Piezoelectric Coeff. ${\mathit{d}}_{31}$ ($\mathit{pC}/\mathit{N}$) | Relative Permittivity ${\mathit{\u03f5}}_{33}^{\mathit{T}}/{\mathit{\u03f5}}_{0}$ |
---|---|---|---|---|---|---|

1 | 66 | 2.07 | 11.137 | BURPS | −10 | 290 |

2 | 55 | 2.00 | 11.240 | BURPS | −60 | 445 |

3 | 50 | 1.86 | 11.273 | BURPS | −120 | 526 |

4 | 32 | 1.50 | 11.247 | BURPS | −190 | 808 |

5 | 20 | 1.27 | 11.260 | BURPS | −250 | 1199 |

6 | 45 | 1.78 | 9.940 | Freeze cast | −340 | 563 |

7 | 35 | 2.06 | 10.230 | Freeze cast | −390 | 702 |

8 | 31 | 1.70 | 10.640 | Freeze cast | −300 | 788 |

**Table 3.**Dynamic tests performed for the different values of porosity, fabrication method and resistance configuration.

Test Number | Sample Number | Method Fabrication | Porosity | Main Resistor in Series | Resistor in Parallel |
---|---|---|---|---|---|

1 | #Ref | - | 0.0% | 10.042 M$\Omega $ | 0.9951 k$\Omega $ |

2 | #Ref | - | 0.0% | 14.677 M$\Omega $ | 0.9951 k$\Omega $ |

3 | #Ref | - | 0.0% | 14.677 M$\Omega $ | 9.987 k$\Omega $ |

4 | #Ref | - | 0.0% | 10.042 M$\Omega $ | 9.987 k$\Omega $ |

5 | #Ref | - | 0.0% | 10.042 M$\Omega $ | 9.987 k$\Omega $ |

6 | 7 | Free-casting | 35.0% | 10.042 M$\Omega $ | 9.987 k$\Omega $ |

7 | 8 | Free-casting | 31.0% | 10.042 M$\Omega $ | 9.987 k$\Omega $ |

8 | 3 | BURPS | 50.0% | 10.042 M$\Omega $ | 9.987 k$\Omega $ |

9 | 1 | BURPS | 66.0% | 10.042 M$\Omega $ | 9.987 k$\Omega $ |

10 | 2 | BURPS | 55.0% | 10.042 M$\Omega $ | 9.987 k$\Omega $ |

11 | 7 | Free-Casting | 35.0% | 10.042 M$\Omega $ | 9.987 k$\Omega $ |

12 | 2 | BURPS | 55.0% | 10.042 M$\Omega $ | 0.9951 k$\Omega $ |

13 | 6 | Free-Casting | 45.0% | 10.042 M$\Omega $ | 9.987 k$\Omega $ |

14 | 5 | BURPS | 20.0% | 10.042 M$\Omega $ | 9.987 k$\Omega $ |

**Table 4.**Statistical values for the porous piezoelectric materials manufactured using the BURPS method. From [23].

${\mathit{d}}_{33}$ (pC/m) | ${\mathit{d}}_{31}$ (pC/m) | |||
---|---|---|---|---|

Percentage | Mean | S.D. | Mean | S.D. |

30 | 50.74 | 5.67 | −0.69 | 0.45 |

40 | 76.54 | 9.83 | −4.86 | 1.20 |

50 | 90.48 | 5.65 | −11.37 | 1.81 |

60 | 97.08 | 7.23 | −17.65 | 1.52 |

70 | 99.01 | 4.85 | −19.40 | 1.39 |

80 | 91.90 | 5.36 | −22.56 | 0.48 |

90 | 100.35 | 9.09 | −30.30 | 11.12 |

Nominal | 149 | −59 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Martínez-Ayuso, G.; Haddad Khodaparast, H.; Zhang, Y.; Bowen, C.R.; Friswell, M.I.; Shaw, A.D.; Madinei, H.
Model Validation of a Porous Piezoelectric Energy Harvester Using Vibration Test Data. *Vibration* **2018**, *1*, 123-137.
https://doi.org/10.3390/vibration1010010

**AMA Style**

Martínez-Ayuso G, Haddad Khodaparast H, Zhang Y, Bowen CR, Friswell MI, Shaw AD, Madinei H.
Model Validation of a Porous Piezoelectric Energy Harvester Using Vibration Test Data. *Vibration*. 2018; 1(1):123-137.
https://doi.org/10.3390/vibration1010010

**Chicago/Turabian Style**

Martínez-Ayuso, Germán, Hamed Haddad Khodaparast, Yan Zhang, Christopher R. Bowen, Michael I. Friswell, Alexander D. Shaw, and Hadi Madinei.
2018. "Model Validation of a Porous Piezoelectric Energy Harvester Using Vibration Test Data" *Vibration* 1, no. 1: 123-137.
https://doi.org/10.3390/vibration1010010