# Piezoelectric Vibration-Based Energy Harvesting Enhancement Exploiting Nonsmoothness

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Vibration-Based Energy Harvesting System

## 3. Numerical Simulations

^{−4}s were assumed after a convergence analysis. Numerical simulations were carried out to provide a parametric analysis evaluating the energy harvesting capacity. Displacement, power, and efficiency were monitored as a function of forcing frequency and different values of support stiffness (${k}_{s}$) and gap ($g$). In this regard, it is interesting to define the nondimensional parameter $\beta $ that establishes a ratio between support and oscillator stiffness: $\beta ={k}_{s}/k.$

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Archetypal model of a vibration-based energy harvesting system with discontinuous support.

**Figure 2.**Maximum displacement versus forcing frequency: comparison between linear model without impacts ($g$ = 70 $\mathsf{\mu}$m) and a system incorporating impacts using $g$ = 50 $\mathsf{\mu}$m at different values of $\beta $.

**Figure 3.**Maximum displacement versus forcing frequency: comparison between linear model without impacts ($g$ = 70 $\mathsf{\mu}$m) and a system incorporating impacts with different values of $\beta $: (

**a**) $g$ = 30 $\mathsf{\mu}$m; (

**b**) $g=\text{}10\text{}\mathsf{\mu}\mathrm{m}.$

**Figure 4.**Maximum displacement versus forcing frequency $\beta =200$ with different gap values, $g$.

**Figure 5.**Phase spaces and Poincaré sections at resonance frequencies. A comparison between a linear model without impacts (WI, using $g$ = 70 $\mathsf{\mu}$m) and a system incorporating impacts with different values of $\beta $: (

**a**) $g$ = 10 $\mathsf{\mu}$m, (

**b**) $g$ = 30 $\mathsf{\mu}$m, and (

**c**) $g$ = 50 $\mathsf{\mu}$m.

**Figure 6.**Average power versus forcing frequency using $g$ = 70 $\mathsf{\mu}$m (without impact) and incorporating impacts with different values of $\beta $: (

**a**) $g$ = 50 $\mathsf{\mu}$m, (

**b**) $g$ = 30 $\mathsf{\mu}$m, and (

**c**) $g$ = 10 $\mathsf{\mu}$m.

**Figure 7.**Dynamical jumps of average power for frequency up-sweep and down-sweep assuming $\beta =100$: (

**a**) $g$ = 50 $\mathsf{\mu}$m, (

**b**) $g$ = 30 $\mathsf{\mu}$m, and (

**c**) $g$ = 10 $\mathsf{\mu}$m.

**Figure 8.**Efficiency versus forcing frequency: comparison between a linear model without impacts ($g$ = 70 $\mathsf{\mu}$m) and systems incorporating impacts with different values of support stiffness, ${k}_{s}$: (

**a**) $g$ = 50 $\mathsf{\mu}$m, (

**b**) $g$ = 30 $\mathsf{\mu}$m, and (

**c**) $g$ = 10 $\mathsf{\mu}$m.

**Figure 9.**Average harvested power output for different frequencies (interval of 500 to 1400. rad/s), for different values of gap, as a function of $\beta $.

**Figure 10.**Average efficiency for different frequencies (interval of 500 to 1400 rad/s), for different values of gap, as a function of $\beta $.

**Figure 11.**(

**a**) Bifurcation diagram for g = 50μm and β = 200. Phase space and Poincaré sections for (

**b**) ω = 708 rad/s and (

**c**) ω = 936 rad/s.

**Figure 12.**(

**a**) Bifurcation diagram for g = 10μm and β = 200. Phase spaces and Poincaré sections for (

**b**) ω = 536 rad/s, (

**c**) ω = 684 rad/s, and (

**d**) ω = 1228 rad/s.

**Table 1.**System parameters [23].

$\mathit{m}$ (kg) | $\mathit{k}(\mathbf{N}\xb7{\mathbf{m}}^{-1})$ | $\mathit{c}={\mathit{c}}_{\mathit{s}}(\mathbf{N}\xb7\mathbf{s}\xb7{\mathbf{m}}^{-1})$ | $\mathit{\delta}$$(\mathbf{m}/{\mathbf{s}}^{2})$ | ${\mathit{C}}_{\mathit{p}}\left(\mathbf{F}\right)$ | $\mathit{R}\left(\mathbf{\Omega}\right)$ | $\mathit{\Theta}(\mathbf{N}\xb7{\mathbf{V}}^{-1})$ |
---|---|---|---|---|---|---|

0.00878 | 4150 | $0.219$ | 2.5 | $4.194\times {10}^{-8}$ | $100\times {10}^{3}$ | −0.004688 |

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**MDPI and ACS Style**

Ai, R.; Monteiro, L.L.S.; Monteiro, P.C.C., Jr.; Pacheco, P.M.C.L.; Savi, M.A.
Piezoelectric Vibration-Based Energy Harvesting Enhancement Exploiting Nonsmoothness. *Actuators* **2019**, *8*, 25.
https://doi.org/10.3390/act8010025

**AMA Style**

Ai R, Monteiro LLS, Monteiro PCC Jr., Pacheco PMCL, Savi MA.
Piezoelectric Vibration-Based Energy Harvesting Enhancement Exploiting Nonsmoothness. *Actuators*. 2019; 8(1):25.
https://doi.org/10.3390/act8010025

**Chicago/Turabian Style**

Ai, Rodrigo, Luciana L. S. Monteiro, Paulo Cesar. C. Monteiro, Jr., Pedro M. C. L. Pacheco, and Marcelo A. Savi.
2019. "Piezoelectric Vibration-Based Energy Harvesting Enhancement Exploiting Nonsmoothness" *Actuators* 8, no. 1: 25.
https://doi.org/10.3390/act8010025