# Design and Analysis of an Extended Simply Supported Beam Piezoelectric Energy Harvester

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Design and Modeling

_{1}), covered with a PVDF layer for energy generation, and an extended beam (L

_{2}) attached at the pin end. Torsional springs are attached to the roller and pin joints of the simply supported beam. A tip mass (M

_{t}) is also added to the free end of the extended beam for frequency adjustment.

_{n}is the bending stiffness; P is the axial compressive preload, which is below the critical load in this paper; w is the transverse displacement; m

_{n}is the mass density per unit length; the subscripts r and n indicate the mode and segment number, respectively. The transverse displacement can be expressed as:

_{rn}is the eigenvalue, which is a function of the undamped natural frequency ω

_{r}:

_{rn}, B

_{rn}, C

_{rn}, and D

_{rn}are the coefficients to be determined by the boundary and continuous conditions. The boundary and continuous conditions of the proposed PEH are presented in Equations (5)–(12).

_{t}and M

_{t}are the moment of inertia and the mass of the tip mass, respectively; k

_{1}and k

_{2}are the stiffnesses of the torsional springs at the roller and the revolute joints, respectively. For the cases with no spring attached, simply set the stiffnesses to zero. The boundary and continuous conditions can be rewritten as the matrix form:

_{r}is the damping ratio; Q

_{r}is the electro-mechanical coupling coefficient:

_{p}, b

_{p}, h

_{p}, and d

_{31}are the Young’s modulus, width, thickness, and piezoelectric constant of the piezoelectric layer, respectively; h

_{c}and h

_{d}are the distances from the neutral axis of the composite beam to the bottom and top of the piezoelectric layer, respectively. The dimensions of the composite beam can be found in the cross-section view shown in Figure 2 The normalized external force f can be written as:

_{b}is the displacement of the base excitation.

_{p}is the capacitance of the PVDF layer and κ

_{r}is the modal coupling term and can be written as:

## 3. Experiment

## 4. Results

_{31}is obtained by fitting the simulated and experimental voltages.

#### 4.1. ESSB PEH under Base Excitations

_{31}is acquired by fitting to the experimental voltage. It is worth noting that each configuration of the extended beam has its own damping ratio, but all the configurations share the same piezoelectric constant.

#### 4.2. ESSB PEH with Torsional Springs under Base Excitations

#### 4.3. ESSB PEH with Axial Preload under Base Excitations

#### 4.4. Comparison between the ESSB PEH and the Cantilevered PEH

#### 4.5. Influence of Tip Mass on Strain Distribution of the ESSB PEH

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 6.**Simulation and experimental results of the proposed PEH with different L

_{2}(

**a**) Displacement, L

_{2}= 20 mm, M

_{t}= 4.9 g (

**b**) Voltage, L

_{2}= 20 mm, M

_{t}= 4.9 g (

**c**) Displacement, L

_{2}= 30 mm, M

_{t}= 1.9 g (

**d**) Voltage, L

_{2}= 30 mm, M

_{t}= 1.9 g (

**e**) Displacement, L

_{2}= 40 mm, M

_{t}= 0.9 g (

**f**) Voltage, L

_{2}= 40 mm, M

_{t}= 0.9.

**Figure 7.**The experimental results of the proposed PEH with different L

_{2}(

**a**) Displacement (

**b**) Voltage.

**Figure 8.**The strain distribution of PVDF on the proposed PEH with different L

_{2}at its resonant frequency (

**a**) theoretical strain distribution (

**b**) theoretical normalized strain distribution.

**Figure 9.**Simulation results of the ESSB PEH with torsional spring at the roller joint (

**a**) Strain (

**c**) Voltage (

**e**) Displacement at the revolute joint (

**b**) Strain (

**d**) Voltage (

**f**) Displacement.

**Figure 10.**Simulation results of the ESSB PEH with axial preload stretching load (

**a**) Strain (

**c**) Voltage (

**e**) Displacement compressing load (

**b**) Strain (

**d**) Voltage (

**f**) Displacement.

**Figure 11.**Experimental results of the ESSB PEH and a cantilevered PEH (

**a**) Displacement (

**b**) Voltage.

**Figure 12.**The strain distribution of PVDF on the ESSB PEH and a cantilevered PEH at their resonant frequencies (

**a**) theoretical strain distribution (

**b**) theoretical normalized strain distribution.

**Figure 13.**Power output with different load resistance (

**a**) Cantilevered PEH—simulation (

**b**) Cantilevered PEH—experiment (

**c**) ESSB PEH—simulation (

**d**) ESSB PEH—experiment.

**Figure 14.**The theoretical normalized strain distribution of PVDF on the ESSB PEH with different tip masses.

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

Config. 1 | Config. 2 | Config. 3 | ||

L_{2} | Length of the extended beam | 20 mm | 30 mm | 40 mm |

M_{t} | Tip mass | 4.9 g | 1.9 g | 0.9 g |

ζ | damping ratio | 0.035 | 0.0265 | 0.0245 |

b_{x} | Width of the extended beam | 12.7 mm | ||

h_{x} | Thickness of the extended beam | 0.1 mm | ||

ρ_{x} | Density of the extended beam | 7930 kg/m^{3} | ||

Y_{x} | Young’s modulus of the extended beam | 193 GPa | ||

L_{1} | Length of the main beam | 100 mm | ||

b_{s} | Width of the substrate | 12.7 mm | ||

h_{s} | Thickness of the substrate | 0.1 mm | ||

ρ_{s} | Density of the substrate | 7930 kg/m^{3} | ||

Y_{s} | Young’s modulus of the substrate | 193 Gpa | ||

b_{e} | Width of the epoxy | 10 mm | ||

h_{e} | Thickness of the epoxy | 0.09 mm | ||

ρ_{e} | Density of the epoxy | 1200 kg/m^{3} | ||

Y_{e} | Young’s modulus of the epoxy | 27 Mpa | ||

b_{p} | Width of the PVDF | 10 mm | ||

h_{p} | Thickness of the PVDF | 0.2 mm | ||

ρ_{p} | Density of the PVDF | 1780 kg/m^{3} | ||

Y_{p} | Young’s modulus of the PVDF | 2.9 Gpa | ||

C_{p} | Capacitance of the PVDF | 1 nF | ||

d_{31} | Piezoelectric constant of the PVDF | 16 pm/V |

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

Config. A | Config. B | Config. C | Config. D | ||

k_{1} | Stiffness | 0 N/rad | 10^{−3} N/rad | 10^{−2} N/rad | 10^{−1} N/rad |

L_{e} | Length of the extended beam | 21.9 mm | 22.1 mm | 23.9 mm | 26.2 mm |

M_{t} | Tip mass | 4.4 g | 4.33 g | 3.83 g | 3.33 g |

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

Config. A | Config. B | Config. C | Config. D | ||

k_{1} | Stiffness | 0 N/rad | 10^{−3} N/rad | 10^{−2} N/rad | 10^{−1} N/rad |

L_{e} | Length of the extended beam | 21.9 mm | 21.8 mm | 20 mm | 12.2 mm |

M_{t} | Tip mass | 4.4 g | 4.64 g | 7.4 g | 50.6 g |

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

Config. A | Config. B | Config. C | Config. D | ||

P_{s} | Pre-stretching load | 0 N | 0.05 N | 0.1 N | 0.2 N |

L_{e} | Length of the extended beam | 21.9 mm | 22.9 mm | 23.8 mm | 25.1 mm |

M_{t} | Tip mass | 4.4 g | 4.12 g | 3.9 g | 3.61 g |

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

Config. A | Config. B | Config. C | Config. D | ||

k_{1} | Pre-compressing load | 0 N | 0.05 N | 0.1 N | 0.2 N |

L_{e} | Length of the extended beam | 21.9 mm | 20.8 mm | 19.5 mm | 18.5 mm |

M_{t} | Tip mass | 4.4 g | 4.74 g | 5.2 g | 4.9 g |

Tip Mass | Mean of Normalized Strain |
---|---|

0·Mt | 0.6465 |

0.1·Mt | 0.7146 |

0.2·Mt | 0.7379 |

0.5·Mt | 0.6405 |

Mt | 0.5656 |

2·Mt | 0.5316 |

5·Mt | 0.5124 |

100·Mt | 0.5006 |

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## Share and Cite

**MDPI and ACS Style**

Su, W.-J.; Tseng, C.-H.
Design and Analysis of an Extended Simply Supported Beam Piezoelectric Energy Harvester. *Sensors* **2023**, *23*, 5895.
https://doi.org/10.3390/s23135895

**AMA Style**

Su W-J, Tseng C-H.
Design and Analysis of an Extended Simply Supported Beam Piezoelectric Energy Harvester. *Sensors*. 2023; 23(13):5895.
https://doi.org/10.3390/s23135895

**Chicago/Turabian Style**

Su, Wei-Jiun, and Chu-Hsiang Tseng.
2023. "Design and Analysis of an Extended Simply Supported Beam Piezoelectric Energy Harvester" *Sensors* 23, no. 13: 5895.
https://doi.org/10.3390/s23135895