# Experimentally Verified Analytical Models of Piezoelectric Cantilevers in Different Design Configurations

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model of Piezoelectric Vibration Energy Harvester

#### 2.1. Bimorph Cantilever Beam with Piezoelectric Layers in Series

_{E}; the remaining portion of piezoelectric layers is not polarized (mentioned as section V

_{R}), and thus is not affected by the piezoelectric effect. A tip mass ${M}_{\mathrm{t}}$ of negligible rotary inertia is attached to the free end of the beam spanning over the length ${L}_{\mathrm{M}\mathrm{t}}$—this section is denoted as V

_{R, Mt}. The bimorph model is reduced to a single DOF model which describes the movement of the bimorph’s free end q relative to the moving clamped end.

_{E}which is affected by the piezoelectric effect, section V

_{R}which is not affected by the piezoelectric effect and section V

_{R, Mt}with distributed tip mass attached, the displacement needs to be a piecewise function defined as

^{*}is the bimorph’s mass per unit of its length defined as

^{*}is bending stiffness of the non-polarized section of the beam (the rest of the beam outside the length ${L}_{\mathrm{E}}$) defined as

_{1,r}. This fact means that beam vibrations are composed mostly of the first vibrational mode and, as a consequence, the beam’s displacement relative to the base movement in all sections (V

_{E}, V

_{R}and V

_{R, Mt}) can be written as

#### 2.1.1. Effect of Chosen Mode Shape Function on Model Output

#### 2.1.2. Single DOF Model of Bimorph Configuration

#### 2.2. Modification of Single DOF Model for Unimorph Configuration

^{*}changes to

^{*}is defined as

## 3. Verification of Analytical Model Based on Experimental Results

#### 3.1. PZT-5A Bimorph with a Full Electrode Length and a Linear Response

_{SC}and the open-circuit frequency f

_{OC}of this coupled electromechanical system. The short-circuit and open-circuit frequency are the first resonant frequencies in case of ${R}_{\mathrm{l}}$ = 0 and ${R}_{\mathrm{l}}$ → ∞, respectively. The match of simulation results with the measured ones for various values of resistive load ${R}_{\mathrm{l}}$ and kinematic excitation at both the short-circuit frequency f

_{SC}and the open-circuit frequency f

_{OC}is shown in Figure 4. Both states correspond with operations slightly below and above the resonance excitation for various values of resistive load, which determine the value of actual resonance frequency.

#### 3.2. PZNN-PLZT Bimorph with Partial Electrode Length and Weak Non-Linear Response

#### 3.3. PVDF Unimorph with a Partial Electrode Length and a Linear Response

_{1,r}= 18.7 Hz) with a constant acceleration amplitude ${a}_{0}$ = 0.035 g. The unimorph had an experimentally determined damping ratio ${b}_{\mathrm{r}}$ = 0.0065 via an analysis of impulse response in the short-circuit state.

#### 3.4. Single DOF Model Parameters of Considered Harvesters

## 4. Comparison of Piezoelectric Materials for Kinetic Energy Harvesting Purposes

#### 4.1. Harmonic Vibrations Case

#### 4.2. Random Vibrations Case

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 2.**(

**a**) Comparison between approximation and true mode shape; (

**b**) comparison of slopes between approximation and true mode shape.

**Figure 3.**Comparison of electrical power and velocity of tip mass for both experimental results [26] and analytical model.

**Figure 4.**Peak power values as a function of resistive load upon excitation at short-circuit resonance frequency and the open-circuit resonance frequency.

**Figure 6.**Comparison of generated voltage and velocity of harvester’s tip mass obtained from measurement and developed analytical model for R

_{l}are 1 MΩ and 10 MΩ.

**Figure 8.**PVDF unimorph used in the experiment and a strip of PVDF foil used as the piezoelectric layer.

**Figure 9.**Comparison of output voltage and velocity of tip mass obtained from the measurement and by using the developed analytical model for R

_{l}= 10 MΩ.

**Figure 10.**Comparison of harvested electrical power among the considered materials for various resistive loads upon simple harmonic forcing at the first resonant frequency.

**Figure 11.**(

**a**) Measured acceleration of a random movement of a human wearable and (

**b**) its spectrogram; a comparison of harvested electrical energy from the human forearm movement among (

**c**) the piezoelectric harvesters used in the experiments and (

**d**) the tuned piezoelectric harvesters from Section 4.1 for different values of resistive load.

Harvester Type (Configuration) | L [mm] | ${\mathit{L}}_{\mathbf{E}}$ $\left[\mathbf{mm}\right]$ | ${\mathit{L}}_{\mathbf{Mt}}\phantom{\rule{0ex}{0ex}}\left[\mathbf{mm}\right]$ | B [mm] | ${\mathit{h}}_{\mathbf{s}}\phantom{\rule{0ex}{0ex}}\left[\mathbf{mm}\right]$ | ${\mathit{h}}_{\mathbf{p}}$ $\left[\mathbf{mm}\right]$ | ${\mathit{M}}_{\mathbf{t}}\phantom{\rule{0ex}{0ex}}\left[\mathbf{g}\right]$ |
---|---|---|---|---|---|---|---|

PZT-5A (bimorph) | 50.8 | 50.8 | – | 31.8 | 0.14 | 0.26 | 12 |

PZZN-PLZT (bimorph) | 40 | 25 | 15 | 10 | 0.1 | 0.2 | 10 |

PVDF (unimorph) | 71.9 | 49.2 | 4 | 10 | 0.3 | 0.13 | 2.6 |

**Table 2.**Material properties of piezoelectric layers and substrates for each harvester used in experiments.

Harvester Type (Configuration) | Material | Ρ [kg/m ^{3}] | Y [GPa] | ${\mathit{d}}_{31}$ $[\mathbf{C}/\mathbf{N}]$ | ${\mathit{\u03f5}}_{33}^{\mathit{S}}$$/{\mathit{\u03f5}}_{0}\phantom{\rule{0ex}{0ex}}[-]$ |
---|---|---|---|---|---|

PZT-5A (bimorph) | PZT-5A | 7800 | 66 | –190 × 10^{–12} | 1500 |

Brass shim | 9000 | 105 | – | – | |

PZZN-PLZT (bimorph) | PZNN-PLZT | 7800 | 62.5 | –195 × 10^{–12} | 1850 |

Steel shim | 7850 | 210 | – | – | |

PVDF (unimorph) | PVDF | 1760 | 2 | –19 × 10^{–12} | 12 |

Steel shim | 7850 | 210 | – | – |

Harvester Type | M_{eff}[g] | B_{eff}[Ns/m] | K_{eff}[N/m] | F_{eff}[N/g] | θ_{eff}[N/V] | C_{eq}[F] |
---|---|---|---|---|---|---|

PZT-5A | 14.1 | 2.24 × 10^{–1} | 1218.10 | 1.51 × 10^{–1} | 2.20 × 10^{–3} | 4.12 × 10^{–8} |

PZZN-PLZT | 6.10 | 5.13 × 10^{–2} | 164.56 | 7.90 × 10^{–2} | 6.03 × 10^{–5} | 3.65 × 10^{–9} |

PVDF | 2.90 | 4.40 × 10^{–3} | 40.38 | 3.27 × 10^{–2} | 1.21 × 10^{–6} | 3.08 × 10^{–10} |

**Table 4.**Tuned dimensions of harvesters and their equivalent single DOF model parameters used in the comparison.

Harvester Type(Configuration) | L[mm] | ${\mathit{L}}_{\mathbf{E}}$[mm] | ${\mathit{L}}_{\mathbf{Mt}}$[mm] | B[mm] | ${\mathit{h}}_{\mathbf{s}}$[mm] | ${\mathit{h}}_{\mathbf{p}}$[mm] | ${\mathit{M}}_{\mathbf{t}}$[g] |

PZT-5A (bimorph) | 68.8 | 40 | 5 | 10 | 0.15 | 0.26 | 3.67 |

PZZN-PLZT (bimorph) | 54.3 | 40 | 5 | 10 | 0.15 | 0.26 | 3.99 |

PVDF (unimorph) | 71.9 | 40 | 5 | 40 | 0.3 | 0.13 | 2.60 |

Harvester Type(Configuration) | ${\mathit{M}}_{\mathbf{eff}}$[g] | ${\mathit{B}}_{\mathbf{eff}}$[Ns/m] | ${\mathit{K}}_{\mathbf{eff}}$[N/m] | ${\mathit{f}}_{\mathbf{1}}$[Hz] | ${\mathit{F}}_{\mathbf{eff}}\mathbf{/}\mathbf{g}$[N/1g] | ${\mathit{\theta}}_{\mathbf{eff}}$[N/V] | ${\mathit{C}}_{\mathbf{eq}}$[F] |

PZT-5A (bimorph) | 4.21 | 4.45 × 10^{–2} | 161.25 | 31.13 | 5.02 × 10^{–2} | 5.14 × 10^{–4} | 1.02 × 10^{–8} |

PZZN-PLZT (bimorph) | 4.21 | 4.20 × 10^{–2} | 161.05 | 31.13 | 4.75 × 10^{–2} | 6.36 × 10^{–5} | 4.50 × 10^{–9} |

PVDF (unimorph) | 4.21 | 1.07 × 10^{–2} | 161.25 | 31.13 | 5.40 × 10^{–2} | 5.54 × 10^{–6} | 1.30 × 10^{–9} |

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**MDPI and ACS Style**

Machu, Z.; Rubes, O.; Sevecek, O.; Hadas, Z. Experimentally Verified Analytical Models of Piezoelectric Cantilevers in Different Design Configurations. *Sensors* **2021**, *21*, 6759.
https://doi.org/10.3390/s21206759

**AMA Style**

Machu Z, Rubes O, Sevecek O, Hadas Z. Experimentally Verified Analytical Models of Piezoelectric Cantilevers in Different Design Configurations. *Sensors*. 2021; 21(20):6759.
https://doi.org/10.3390/s21206759

**Chicago/Turabian Style**

Machu, Zdenek, Ondrej Rubes, Oldrich Sevecek, and Zdenek Hadas. 2021. "Experimentally Verified Analytical Models of Piezoelectric Cantilevers in Different Design Configurations" *Sensors* 21, no. 20: 6759.
https://doi.org/10.3390/s21206759