# Design Scalability Study of the Γ-Shaped Piezoelectric Harvester Based on Generalized Classical Ritz Method and Optimization

^{1}

^{2}

^{*}

## Abstract

**:**

^{3}kg·s·m

^{−3}which is the highest among the reviewed PE harvesters. We discuss how the design parameters need to be determined at different harvester scales.

## 1. Introduction

- Development of a GCRM-P model to predict linear electromechanical behaviors of PE harvesters having multiple structural members,
- Experimental validation for the GCRM-P model in terms of energy harvesting performance,
- Study of design scalability—design optimization of $\mathsf{\Gamma}$EH under different mass scales, and comparison of the power output performance with the other recent PE harvester studies.

## 2. Generalized CRM for Piezoelectric Harvester (GCRM-P)

#### 2.1. Description for a Unit Element

^{(k)}) with the origin point of ${O}^{\left(k\right)}$. A generic point ${P}_{0}^{\left(k\right)}$ between two nodal points ${N}^{\left(n\right)}$ and ${N}^{\left(n+1\right)}$ lies on the neutral axis of the beam, and is distanced from ${O}^{\left(k\right)}$ by ${x}^{\left(k\right)}$ along ${\widehat{\mathrm{e}}}_{1}^{\left(k\right)}$. The displacement vectors of ${P}_{0}^{\left(k\right)}$, ${N}^{\left(n\right)}$, and ${N}^{\left(n+1\right)}$ are given as ${u}_{0}^{\left(k\right)}\left({x}^{\left(k\right)},t\right){\widehat{\mathrm{e}}}_{1}^{\left(k\right)}+{w}_{0}^{\left(k\right)}\left({x}^{\left(k\right)},t\right){\widehat{\mathrm{e}}}_{2}^{\left(k\right)}$, ${X}^{\left(n\right)}\left(t\right){\widehat{\mathrm{g}}}_{1}+{Y}^{\left(n\right)}\left(t\right){\widehat{\mathrm{g}}}_{2}$, and ${X}^{\left(n+1\right)}\left(t\right){\widehat{\mathrm{g}}}_{1}+{Y}^{\left(n+1\right)}\left(t\right){\widehat{\mathrm{g}}}_{2}$. The initial angle between ${\widehat{\mathrm{g}}}_{1}$ and ${\widehat{\mathrm{e}}}_{1}^{\left(k\right)}$ is denoted as ${\varphi}^{\left(k\right)}$, and the rotations at ${P}_{0}^{\left(k\right)}$, ${N}^{\left(n\right)}$, and ${N}^{\left(n+1\right)}$ are given as ${\theta}^{\left(k\right)}\left({x}^{\left(x\right)},t\right)$, ${\mathsf{\Theta}}^{\left(n\right)}\left(t\right)$, and ${\mathsf{\Theta}}^{\left(n+1\right)}\left(t\right)$. The nodal masses at ${N}^{\left(n\right)}$ and ${N}^{\left(n+1\right)}$ are denoted as ${M}^{\left(n\right)}$, ${M}^{\left(n+1\right)}$ and their moments of inertia along ${\widehat{n}}_{3}$ (=${\widehat{\mathrm{n}}}_{1}\times {\widehat{\mathrm{n}}}_{2}$) is given as ${I}^{\left(n\right)}$, ${I}^{\left(n+1\right)}$, respectively. The load resistance ${R}^{\left(k\right)}$ in Figure 2b can be connected in either serial or parallel depending on the choice of the poling direction of the piezoelectric material and the wiring between the layers.

#### 2.2. Electromechanical Energy Formulations

#### 2.3. Constraint Equations

#### 2.4. Spatial Discretization and Electromechanical Equations of Motion

## 3. System Descriptions for the $\mathsf{\Gamma}$EH

#### 3.1. Uniform Strain Distribution in $\Gamma $-Shaped Structure

#### 3.2. Configuration of $\Gamma $-Shaped Harvester

^{2}), or

## 4. Shape Optimization

#### 4.1. Experimental Validation of the GCRM-P Model for $\Gamma $EH

#### 4.2. Design Formulation

#### 4.3. Shape Optimization Results and Discussions

^{3}) in all of the cases. This implies that using the maximum allowable piezoelectric material does not necessarily guarantee the higher power output, because of increased stiffness and decreased strain generation in the piezoelectric material. The harvester clearance from the ground (${d}_{ver}$, Figure 12e) is found positive from all of the cases by satisfying Equation (35).

^{2}(coefficient of determination) of 0.999. Based on this formula, energy harvester designers can approximately estimate a total mass that will be needed to meet the power requirement. An additional optimization study at ${M}_{allowed}$ = 37.5 g found the power output 1.857 mW (the optimized design variables can be found in Table 4) and confirms this linear relation—only 0.6% different from a predicted value (1.845 mW).

^{3}~23.10 × 10

^{3}kg·s·m

^{−3}) than those of any other PE harvesters reviewed in this table (0.016 × 10

^{3}~11.55 × 10

^{3}kg·s·m

^{−3}). It is noted that the highest NPD among the referenced works, i.e., 11.55 × 10

^{3}kg·s·m

^{−3}by Yang et al. [63], was enabled by its large mass (100 g) and the high piezoelectric coupling constant of 275 pC/N. However, more than doubled NPD can be obtained from our design using a smaller mass (${M}_{allowed}$ = 60 g) and a smaller piezoelectric coupling constant (210 pC/N). In terms of the modified NPD, the proposed designs with ${M}_{allowed}$ = 45 g and 60 g have higher values than the design in [63]. This shows the effectiveness of the optimized $\mathsf{\Gamma}$EH. By the way, the NPD increases as the mass becomes larger, but shows a saturated trend (around 24 × 10

^{3}kg·s·m

^{−3}), because the constraints of vertical dimension and stress (Equations (34) and (36)) become active.

## 5. Conclusions

- (1)
- The accuracy of the proposed GCRM-P model used for the frequency response analysis of the $\mathsf{\Gamma}$EH was experimentally validated with the error 5.5% for the peak power frequency.
- (2)
- The proposed DE-based approach successfully provided the optimized solutions with the high NPDs, while satisfying the six design constraints. Specifically, we could obtain higher harvester NPDs than the multiple mesoscale PE harvesters from recent studies.
- (3)
- The linear relation between the harvester mass (M
_{allowed}) and power performance does not necessarily mean that all the design variables are linearly scaled—they need to be carefully chosen to maximize the power output while satisfying all of the constraints, especially the stress and the natural frequency measures.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Configurations for (

**a**) proposed $\mathsf{\Gamma}$-shaped piezoelectric energy harvester ($\mathsf{\Gamma}$EH) and (

**b**) cantilever-shaped piezoelectric energy harvester (CEH). Both systems consist of piezoelectric layers, substrate layer, and tip mass. The CEH consists of a single beam whereas the $\mathsf{\Gamma}$EH consists of two beams which are perpendicular to each other.

**Figure 2.**(

**a**) Configuration of the unit piezoelectric beam element before and after deformation. (

**b**) Two possible choices of electrical connection: series and parallel connections of the piezoelectric layers. The arrows show the poling directions of the piezoelectric layers.

**Figure 3.**Graphical representation for the first four Legendre polynomials used to approximate the deformation variables of the beam element.

**Figure 4.**Simplified structures for (

**a**) CEH and (

**b**) $\mathsf{\Gamma}$EH. L, E, I, F, $\delta $, and O are the length, Young’s modulus, second moment of the cross-section area, applied load, tip displacement, and origin point. The subscripts C, $\mathsf{\Gamma}$, $\mathsf{\Gamma},\mathrm{V}$, and $\mathsf{\Gamma},\mathrm{H}$ denote cantilever, $\mathsf{\Gamma}$-shaped structure, vertical beam of the $\mathsf{\Gamma}$-shaped structure, and horizontal beam of the $\mathsf{\Gamma}$-shaped structure.

**Figure 5.**Finite element analysis results for the simplified (

**a**) CEH and (

**b**) ΓEH. All structures are under the same transverse load $F$ acting downward and each root of the structure is assumed to be rigidly fixed to the ground. The color shows the von Mises strain level (high and low strain levels are colored by red and blue, respectively).

**Figure 6.**Side view of the $\mathsf{\Gamma}$EH. The nodes, elements, and design variables are given in the figure. The width of the beam (b), the width of the tip mass (b

_{T}), and the load resistance (R) connected to the energy harvester, which are not shown in the figure, are remaining design variables to be determined in the design problem.

**Figure 7.**(

**a**) Demonstration of the typical self-powered wireless sensor node and the outdoor condensing unit from which the electricity is generated by the energy harvester attached to the vibrating surface. (

**b**) FFT results for the acceleration data in three axes. The acceleration in the y-axis is the most dominant and the main frequency component has 40 Hz and 0.16 g.

**Figure 8.**Experimental setup used for measuring the electromechanical frequency response functions (FRFs).

**Figure 9.**Measured (symbol) and predicted (line) voltage and power output FRFs of the $\mathsf{\Gamma}$EH. Both quantities are normalized by gravitational acceleration (g). (

**a**) voltage output; (

**b**) power output.

**Figure 11.**Graphical demonstrations of the four optimized $\mathsf{\Gamma}$EHs having different allowed total mass: (

**a**) 15 g, (

**b**) 30 g, (

**c**) 45 g, and (

**d**) 60 g.

**Figure 12.**Results for the four optimized $\mathsf{\Gamma}$EHs: (

**a**) maximum stress induced by static, dynamic, and combined loadings, (

**b**) piezoelectric material volume, (

**c**) horizontal length, (

**d**) vertical length, (

**e**) closest distance between ground and tip mass, (

**f**) power output and NPD.

PIC151 | PLA | Lead | |
---|---|---|---|

Young’s modulus (GPa) | 66.67 | 2.5 | Not used |

Density (${\mathrm{kg}/\mathrm{m}}^{3}$) | 7800 | 1250 | 11,340 |

Transverse strain constant, d_{31} (10^{−12} C/N) | $-$210 | Not used | |

Relative permittivity at constant stress, ${\epsilon}_{33}^{T}$/${\epsilon}_{0}$ | 2400 |

**Table 2.**Geometrical specifications of the $\mathsf{\Gamma}$EH used for the experimental validation.

Variable | Note | Unit | Value |
---|---|---|---|

${L}_{P}$ | Length of the piezoelectric layer | mm | 22.8 |

${L}_{S}$ | Length of the horizontal substrate layer | 38 | |

${h}_{{S}_{1}}$ | Thickness of the vertical substrate layer | 0.25 | |

${h}_{{S}_{2}}$ | Thickness of the horizontal substrate layer | 2.2 | |

b | Width of the beam | 10 | |

${L}_{T}$ | Length of the tip mass | 12 | |

${H}_{T}$ | Height of the tip mass | 9.5 | |

${b}_{T}$ | Width of the tip mass | 10 | |

${M}_{Total}$ | Total mass | g | 15 |

R | Load resistance | k$\mathsf{\Omega}$ | 930 |

**Table 3.**Optimized design variables depending on the allowed total mass (15 g, 30 g, 45 g and 60 g).

Variable | Unit | Allowed Total Mass | |||
---|---|---|---|---|---|

15 g | 30 g | 45 g | 60 g | ||

${L}_{P}$ | mm | 13.81 | 13.95 | 15.93 | 17.50 |

${L}_{S}$ | 26.86 | 29.64 | 32.64 | 32.72 | |

${h}_{{S}_{1}}$ | 0.52 | 0.72 | 1.06 | 1.25 | |

${h}_{{S}_{2}}$ | 4.62 | 5.65 | 7.10 | 7.50 | |

$b$ | 3.35 | 5.40 | 6.05 | 7.05 | |

${L}_{T}$ | 8.32 | 11.21 | 13.16 | 14.87 | |

${H}_{T}$ | 7.82 | 10.05 | 11.88 | 13.47 | |

${b}_{T}$ | 19.31 | 22.19 | 23.97 | 25.00 | |

$R$ | k$\mathsf{\Omega}$ | 977 | 667 | 511 | 405 |

Variable | Unit | Value |
---|---|---|

${L}_{P}$ | mm | 16.41 |

${L}_{S}$ | 34.36 | |

${h}_{{S}_{1}}$ | 1.09 | |

${h}_{{S}_{2}}$ | 7.49 | |

$b$ | 4.99 | |

${L}_{T}$ | 10.96 | |

${H}_{T}$ | 11.44 | |

${b}_{T}$ | 24.82 | |

$R$ | k$\mathsf{\Omega}$ | 623 |

References | Material (Piezoelectric Charge Constant (10 ^{−12} C/N)) | Excitation Amplitude (m/s ^{2}) | Piezoelectric Material Volume (mm ^{3}) | System Volume (mm ^{3}) | Power Output (mW) | Total Mass (g) | NPD (10 ^{3} kg·s·m^{−3}) | Modified NPD (10 ^{6} kg·s·m^{−3}) |
---|---|---|---|---|---|---|---|---|

Tang and Yang [61] | MFC (d _{31}: $-$170) | 2.83 | 58.8 | 29,167 | 1.43 | 10.42 | 3.04 | 6.12 |

Yang and Zu [62] | PZN-PT (d _{31}: $-$1346) | 2.94 | 28.8 | 4277 | 0.86 | 2.61 | 3.45 | 23.26 |

Yang et al. [63] | PZT-5H (d _{31}: $-$275) | 2.94 | 300 | 101,250 | 30 | 100 | 11.55 | 34.28 |

Pan and Dai [64] | PZT-5H (d _{31}: $-$275) | 29.43 | 20 | 10,400 | 31.1 | 7.6 | 1.80 | 3.45 |

Li et al. [65] | MFC (d _{31}: $-$174) | 0.98 | 300 | 25,232 | 0.427 | 11.57 | 1.48 | 17.62 |

Gao et al. [66] | PIN-PMN-PT (d _{15}: 3480) | 29.43 | 200 | 7500 | 2.756 | 8.5 | 0.016 | 0.42 |

Lee et al. [67] | PZT (N/A) | 0.98 | 17.78 | 6135 | 0.012 | N/A | 0.703 | 2.04 |

This study (${M}_{allowed}$: 15 g) | PIC151 (d _{31}: $-$210) | 1.57 | 19.43 | 12,684 | 0.74 | 15 | 15.49 | 23.67 |

This study (${M}_{allowed}$: 30 g) | 31.62 | 18,010 | 1.48 | 30 | 19.03 | 33.34 | ||

This study (${M}_{allowed}$: 45 g) | 40.55 | 25,685 | 2.22 | 45 | 22.22 | 35.06 | ||

This study (${M}_{allowed}$: 60 g) | 51.81 | 30,259 | 2.95 | 60 | 23.10 | 39.55 |

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**MDPI and ACS Style**

Jeong, S.; Lee, S.; Yoo, H. Design Scalability Study of the Γ-Shaped Piezoelectric Harvester Based on Generalized Classical Ritz Method and Optimization. *Electronics* **2021**, *10*, 1887.
https://doi.org/10.3390/electronics10161887

**AMA Style**

Jeong S, Lee S, Yoo H. Design Scalability Study of the Γ-Shaped Piezoelectric Harvester Based on Generalized Classical Ritz Method and Optimization. *Electronics*. 2021; 10(16):1887.
https://doi.org/10.3390/electronics10161887

**Chicago/Turabian Style**

Jeong, Sinwoo, Soobum Lee, and Honghee Yoo. 2021. "Design Scalability Study of the Γ-Shaped Piezoelectric Harvester Based on Generalized Classical Ritz Method and Optimization" *Electronics* 10, no. 16: 1887.
https://doi.org/10.3390/electronics10161887