# System-Level Model and Simulation of a Frequency-Tunable Vibration Energy Harvester

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Dual Frequency Piezoelectric Energy Harvester

#### 2.1. Design Description

#### 2.2. Reduced Order Model of the Piezoelectric Energy Harvester

^{®}Mechanical (V2019, R1) was thoroughly described in [34]. Considering its high computational cost for a transient simulation, Krylov subspace-based model order reduction (MOR) methods, also known as rational interpolation [38,39,40] were implemented to generate a highly compact but accurate reduced order model (ROM). Furthermore, based on this ROM, a circuit-device co-simulation of the piezoelectric energy harvester became feasible in the system-level simulation.

#### Model Order Reduction

#### 2.3. System-Level Simulation

#### 2.3.1. Mechanical Resonator Reduced Order Model

#### 2.3.2. Piezoelectric Energy Harvester Reduced Order Model

#### 2.3.3. Electrical Simulation

#### Rectification and Filtering

#### Optimum Load

#### Voltage Regulation

#### 2.3.4. Energy Harvester Frequency Tuning

#### 2.3.5. Control Algorithm

## 3. Experimental Investigation

_{1 Exp}= 63.27 and f

_{2 Exp}= 76.35 Hz match the simulation results f

_{1 Sim}= 64.30 and f

_{2 Sim}= 77.50 Hz. However, a lower voltage output has been observed. We attribute this to the adhesive tape which attaches the patches to the steel. In our assembly this degrades the strain transfer between the steel resonator and the piezoelectric layers when compared to solid bonding, e.g., using glue. Our simulations considered the adhesive tape as a material of high compliance (E = 450 kPa). The FE model implements a constant damping ratio, which yields correct amplitudes at the first mode and does not describe the damping at the second mode. A mode-specific or even frequency dependent damping ratio shall be applied instead. Furthermore, the slight frequency shift (up to 1.63%) between the model and the experiment results is caused by the additional mass of the solder paste used to electrically connect the inner patch. The patch attachment procedure and the limited reproducibility of the magnets positioning contribute in turn to such a frequency shift.

## 4. Parametric Design Optimization

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Geometry description of the harvester design together with (

**b**) its simulated displacement, respectively power output. The transfer function illustrates the dual frequency operation of the structure under a base acceleration of 0.5 g. Simultaneously, it demonstrates comparable power output levels.

**Figure 2.**System-level model implemented in ANSYS twin builder, including a reduced order model of the harvester together with tuning actuation, the conditioning circuitry, and the tuning control algorithm.

**Figure 3.**(

**a**) The multiple-input multiple-output (MIMO) system of the mechanical resonator together with (

**b**) the resonator reduced model validation through a comparison with the full finite element (FE) model. Both models are subjected to an excitation amplitude of 10 μm.

**Figure 4.**(

**a**) Variation of the displacement amplitude of outer and (

**b**) inner beam during frequency tuning. An 18% of bidirectional frequency shift can be achieved. The data indicates that frequency tuning does not affect the other resonance frequency.

**Figure 5.**(

**a**) Experimental validation of the bidirectional frequency tuning simulation of first and (

**b**) second resonance frequency.

**Figure 6.**(

**a**) The MIMO system of the piezoelectric energy harvester and (

**b**) the harmonic response of the reduced order and full FE model subjected to a 10 μm excitation amplitude.

**Figure 7.**(

**a**) Simulation results of the AC voltage output before rectification and (

**b**) the filtered DC voltage output of the piezoelectric harvester subjected to 0.2 g base excitation.

**Figure 8.**(

**a**) Load matching to ensure maximum power output from the outer patches electrically connected in the series, (

**b**) respectively the inner patch.

**Figure 10.**(

**a**) Simulation results of the voltage regulation of the outer patches and (

**b**) the inner one under 0.2 g base excitation.

**Figure 11.**(

**a**) The voltage amplitude variation of the piezoelectric energy harvester under bidirectional magnetic frequency tuning of first and (

**b**) second resonance frequency at 0.2 g harmonic base excitation.

**Figure 13.**A control algorithm, which is based on maximum-voltage tracking, chooses the most effective tuning actuator.

**Figure 14.**Dual frequency piezoelectric energy harvester with macro fiber composite (MFC) patches on outer and inner beams. Permanent magnets are attached at the beam ends. The base excitation is applied to the clamped part on the left side.

**Figure 15.**(

**a**) Experimentally obtained voltage at the excitation levels of 0.5 and 1.0 g. Comparison of experimental data and (

**b**) simulation results for an excitation level of 0.5 g.

**Figure 16.**(

**a**) Power management boards 2151A from analog devices and (

**b**) bq2557OEVM-206 from Texas Instruments used as power management circuits.

**Figure 18.**Parameterization of the reference geometry (light grey corresponds to steel, black to NdFeB, and yellow to PIC255).

**Figure 19.**(

**a**) Reference geometry and (

**b**) the three optimized designs with thicknesses of $t=0.5$, (

**c**) $t=1,$ and (

**d**) $t=1.5$ mm.

**Figure 20.**Power density of the reference design and the optimized designs. The co-resonance results in an extended operative bandwidth at comparable power levels.

Material Properties | Value |
---|---|

Mass density (kg/m^{3)} | 5440 |

Tensile modulus, E_{1} (rod direction) (GPa) | 30.34 |

Tensile modulus, E_{1} (electrode direction) (GPa) | 15.86 |

Poisson’s ratio, v_{12} | 0.31 |

Poisson’s ratio, v_{21} | 0.16 |

Shear modulus, G_{12} (GPa) | 5.515 |

d_{33} (rod direction) (pC/N) | 400 |

d_{31} (electrode direction) (pC/N) | −170 |

**Table 2.**Harvester characterization at 0.5 g excitation level, using the 2151A power management board.

Patch | f (Hz) | V_{out} (V) | I_{out} (μA) | R (kΩ) | P_{out} (μW) |
---|---|---|---|---|---|

Outer (// connection) | 65.27 | 2.615 | 125.0 | 21.00 | 653.8 |

Inner | 1.0 | 0.256 | 21.00 | 0.512 | |

Outer (// connection) | 78.35 | 0.381 | 18.20 | 22.00 | 13.87 |

Inner | 2.305 | 105.1 | 22.00 | 484.5 |

**Table 3.**Efficiency comparison of the power management boards used as conditioning circuits for the designed harvester.

Board Type | V_{in} (V) | I_{in} (μA) | R (kΩ) | V_{out} (V) | I_{out} (μA) | Efficiency (%) |
---|---|---|---|---|---|---|

bq25570EVM-206 | 3.73 | 134.5 | 13.0 | 1.8 | 137.5 | 49.4 |

2151A | 4.42 | 106.0 | 14.5 | 1.8 | 124.5 | 47.8 |

Parameter | Reference Value (mm) | Lower Bound (mm) | Upper Bound (mm) |
---|---|---|---|

${L}_{o}$ | 80 | 40 | 120 |

${b}_{o}$ | 10 | 5.0 | 15.0 |

${L}_{c}$ | 10 | 5.0 | 15.0 |

${b}_{c}$ | 1.0 | 0.5 | 1.50 |

${L}_{i}$ | 60 | 23 | 113 |

${b}_{i}$ | 9.0 | 5.0 | 15.0 |

t ^{1} | 1.0 | 0.5 | 1.50 |

^{1}Discrete parameter since it is limited to commercial sheet metal; step size 0.5 mm.

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**MDPI and ACS Style**

Bouhedma, S.; Rao, Y.; Schütz, A.; Yuan, C.; Hu, S.; Lange, F.; Bechtold, T.; Hohlfeld, D.
System-Level Model and Simulation of a Frequency-Tunable Vibration Energy Harvester. *Micromachines* **2020**, *11*, 91.
https://doi.org/10.3390/mi11010091

**AMA Style**

Bouhedma S, Rao Y, Schütz A, Yuan C, Hu S, Lange F, Bechtold T, Hohlfeld D.
System-Level Model and Simulation of a Frequency-Tunable Vibration Energy Harvester. *Micromachines*. 2020; 11(1):91.
https://doi.org/10.3390/mi11010091

**Chicago/Turabian Style**

Bouhedma, Sofiane, Yongchen Rao, Arwed Schütz, Chengdong Yuan, Siyang Hu, Fred Lange, Tamara Bechtold, and Dennis Hohlfeld.
2020. "System-Level Model and Simulation of a Frequency-Tunable Vibration Energy Harvester" *Micromachines* 11, no. 1: 91.
https://doi.org/10.3390/mi11010091