# An Analytical–Numerical Method for Simulating the Performance of Piezoelectric Harvesters Mounted on Wing Slats

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^{2}

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## Abstract

**:**

## 1. Introduction

^{®}, is fully analytical and is based on the modal expansion approach. It describes a whole slat excited by the broadband acceleration spectrum typical of wings, and it enables the calculation of the bending moment and shear force acting on the portion of the slat where the harvester is mounted. The contact between the slat and the wing edge is simulated by means of a distributed stiffness, and the variation in the value of the contact stiffness makes it possible to simulate both retracted and deployed slats.

^{®}, is numerical and is based on a multiphysics finite element (FE) method. It describes the sandwich structure of the harvester mounted on an equivalent portion of the slat, which is excited by the loads calculated by means of the large-scale model. This makes it possible to compute the voltage generated by the PE material considering the effects of the adhesive layer and of the other layers that compose the sandwich.

## 2. Integrated Analytical–Numerical Method

## 3. Input Data: Slat and PE Patch Properties

## 4. Large-Scale Analytical Model of the Slat

_{x}. Therefore, the interaction between the retracted slat and the wing determines an increase in the natural frequency of each mode. Figure 8 represents the natural frequencies of the first five modes of vibration of the retracted slat as a function of the distributed stiffness k

_{x}considering the mechanical properties of the composite slat shown in Table 1.

_{0}is

## 5. Small-Scale Model of a Portion of the Slat with the PE Patch

#### 5.1. Equivalent Model of a Portion of the Slat

_{R}and q by means of the equilibrium equations. Since the bending moment ${M}_{L}$ and shear force ${T}_{L}$ can also be exerted by the clamp of a cantilever beam, Figure 10b shows that the slat portion can be converted to an equivalent cantilever beam with the same length and forced by the same external loads acting on that portion of the slat. It is worth noticing that the slat portion and the cantilever beam have the same bending moment distribution. The same bending moment distribution causes the same curvature and the same strain and stress distribution, which guarantees the same performance of the PE patch.

#### 5.2. Finite Element Model

^{®}.

^{®}. In this analysis, the strain-charge form was adopted, since it leads to a drastic reduction in the number of material constants when some assumptions are made. The constitutive equations in the strain-charge form are as follows [12]:

- Electrodes of the PE material acting along the 3-axis of the local reference frame of the material, i.e., E
_{1}= E_{2}= 0, E_{3}≠ 0. - Thin beam, i.e., ${T}_{2}={T}_{3}={T}_{4}={T}_{5}={T}_{6}=0,\text{}{T}_{1}\ne 0$.
- Orthotropic material.

_{31}are provided in the datasheet of the considered PE patch [10]. The permittivity constant ${\epsilon}_{33}^{T}$ is not provided in the datasheet; however, it can be calculated as [4,6]

## 6. Numerical Results

#### 6.1. Validation of the Integrated Analytical–Numerical Method

#### 6.2. Effect of Contact Stiffness on Generated Voltage

_{x}on voltage output was only valuable above ${10}^{6}$ Nm

^{−2}. For large values of ${k}_{x}$ the generated voltage drastically decreased because the slat began to behave as a rigid body connected to the wing edge. These results agreed with the trend of the natural frequencies against contact stiffness, which is presented in Figure 8.

#### 6.3. Effect of Interposed Layers on Performance

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Vibrating slat. The inertia force causes an elastic deformation with respect to the rigid body motion.

**Figure 8.**Natural frequencies of the first five modes of vibration of the retracted slat as a function of the distributed stiffness ${k}_{x}$.

**Figure 10.**(

**a**) Equilibrium of a portion of the slat.; (

**b**) equilibrium of the equivalent cantilever beam.

**Figure 14.**Comparison between the magnitudes of the OCV FRFs obtained using the analytical and integrated models. (

**a**) ${k}_{x}=0{\mathrm{Nm}}^{-2}$; (

**b**) ${k}_{x}={10}^{6}{\mathrm{Nm}}^{-2}$.

**Figure 15.**${V}_{RMS}$ vs. ${k}_{x}$. The values of ${V}_{RMS}$ were derived from the numerical OCV FRFs obtained in COMSOL.

**Figure 17.**(

**a**) RMS value of OCV vs. thickness ${H}_{a}$ of the interposed layer and the Young’s modulus ${E}_{a}$ of the interposed layer; (

**b**) distance ${z}_{n,eq}$ of the center of the PE patch from the neutral axis.

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

Young’s modulus ($E$) | $\mathrm{GPa}$ | $45$ |

Density ($\rho $) | ${\mathrm{kg}\text{}\mathrm{m}}^{-3}$ | $1800$ |

Cross-section area ($A$) | ${\mathrm{m}}^{2}$ | $2347.6\times {10}^{-6}$ |

Cross-section moment of inertia ($I$) | ${\mathrm{m}}^{4}$ | $\mathrm{2,615,710.7}\times {10}^{-12}$ |

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

Patch overall length (${L}_{p,o}$) | $\mathrm{m}$ | $0.100$ |

Patch active length (L_{p,a}) | $\mathrm{m}$ | $0.085$ |

Patch overall width (${w}_{p,o}$) | $\mathrm{m}$ | $0.018$ |

Patch active width (${w}_{p,a}$) | $\mathrm{m}$ | $0.014$ |

Patch thickness (H_{p}) | $\mathrm{m}$ | $0.0003$ |

Coordinate ${x}_{p1}$ | $\mathrm{m}$ | $1.4075$ |

Coordinate ${x}_{pc}$ | $\mathrm{m}$ | $1.450$ |

Coordinate ${x}_{p2}$ | $\mathrm{m}$ | $1.4925$ |

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

Compliance constant along 1-axis at constant electric field (${s}_{11}^{E}$) | ${\mathrm{GPa}}^{-1}$ | $3.296\times {10}^{-11}$ |

Piezoelectric strain constant (${d}_{31}$) | C N^{−1} | $-210\times {10}^{-12}$ |

Capacitance (${C}_{p}$) | F | 84.04 × 10 ^{−9} |

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

Patch length (${L}_{p}$) | $\mathrm{m}$ | 0.085 |

Patch width (${w}_{p}$) | $\mathrm{m}$ | 0.014 |

Patch thickness (${H}_{p}$) | $\mathrm{m}$ | 0.0003 |

Beam length (L_{eq}) | $\mathrm{m}$ | $0.090$ |

Beam width (${w}_{eq}$) | $\mathrm{m}$ | $0.014$ |

Beam thickness (${H}_{eq}$) | $\mathrm{m}$ | $0.002$ |

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

Young’s modulus (${E}_{eq}$) | $\mathrm{GPa}$ | $5.00\times {10}^{14}$ |

Density (${\rho}_{eq}$) | ${\mathrm{kg}\text{}\mathrm{m}}^{-3}$ | $1.51\times {10}^{5}$ |

Cross-section area (${A}_{eq}$) | ${\mathrm{m}}^{2}$ | $2.80\times {10}^{-5}$ |

Cross-section moment of inertia (${I}_{eq}$) | m^{4} | $9.33\times {10}^{-12}$ |

Natural frequency (${f}_{n,eq}$) | $\mathrm{Hz}$ | $2297$ |

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

${\epsilon}_{33}^{T}$ | ${\mathrm{F}\text{}\mathrm{m}}^{-1}$ | $2544{\epsilon}_{0}$ |

$\mathrm{Density}\text{}({\rho}_{p}$) | ${\mathrm{kg}\text{}\mathrm{m}}^{-3}$ | $5440$ |

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

Material 1 | Material 2 | ||

$\mathrm{Length}\text{}\left({L}_{a}\right)$ | $\mathrm{m}$ | $0.085$ | $0.085$ |

$\mathrm{Width}\text{}({w}_{a}$) | $\mathrm{m}$ | $0.014$ | $0.014$ |

$\mathrm{Young}\u2019\mathrm{s}\text{}\mathrm{modulus}\text{}({E}_{a}$) | $\mathrm{GPa}$ | $2$ | $0.5$ |

$\mathrm{Poisson}\u2019\mathrm{s}\text{}\mathrm{ratio}\text{}({\nu}_{a}$) | $0.3$ | $0.3$ | |

$\mathrm{Density}\text{}({\rho}_{a}$) | ${\mathrm{kg}\text{}\mathrm{m}}^{-3}$ | $1100$ | $1100$ |

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**MDPI and ACS Style**

Tommasino, D.; Moro, F.; Zumalde, E.; Kunzmann, J.; Doria, A.
An Analytical–Numerical Method for Simulating the Performance of Piezoelectric Harvesters Mounted on Wing Slats. *Actuators* **2023**, *12*, 29.
https://doi.org/10.3390/act12010029

**AMA Style**

Tommasino D, Moro F, Zumalde E, Kunzmann J, Doria A.
An Analytical–Numerical Method for Simulating the Performance of Piezoelectric Harvesters Mounted on Wing Slats. *Actuators*. 2023; 12(1):29.
https://doi.org/10.3390/act12010029

**Chicago/Turabian Style**

Tommasino, Domenico, Federico Moro, Eneko Zumalde, Jan Kunzmann, and Alberto Doria.
2023. "An Analytical–Numerical Method for Simulating the Performance of Piezoelectric Harvesters Mounted on Wing Slats" *Actuators* 12, no. 1: 29.
https://doi.org/10.3390/act12010029