# 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

- Hadas, Z.; Vetiska, V.; Huzlik, R.; Singule, V. Model-based design and test of vibration energy harvester for aircraft application. Microsyst. Technol.
**2014**, 20, 831–843. [Google Scholar] [CrossRef] - Silva, T.M.P.; De Marqui, C. Self-powered active control of elastic and aeroelastic oscillations using piezoelectric material. J. Intell. Mater. Syst. Struct.
**2017**, 28, 2023–2035. [Google Scholar] [CrossRef] - Arsalan, M.J.; Ahmad, T.; Saeed, S.A. Energy Harvesting for Downhole Applications in Open-hole Multilaterals. Soc. Pet. Eng.
**2018**. [Google Scholar] [CrossRef] - Tang, L.; Yang, Y.; Soh, C.K. Toward Broadband Vibration-based Energy Harvesting. J. Intell. Mater. Syst. Struct.
**2010**, 21, 1867–1897. [Google Scholar] [CrossRef][Green Version] - Daqaq, M.F.; Masana, R.; Erturk, A.; Dane Quinn, D.D. On the Role of Nonlinearities in Vibratory Energy Harvesting: A Critical Review and Discussion. ASME. Appl. Mech. Rev.
**2014**, 66, 040801–040824. [Google Scholar] [CrossRef] - Zhang, H.; Corr, L.R.; Ma, T. Issues in vibration energy Harvesting. J. Sound Vib.
**2018**, 421, 79–90. [Google Scholar] [CrossRef] - Mann, B.P.; Sims, N.D. Energy harvesting from the nonlinear oscillations of magnetic levitation. J. Sound Vib.
**2009**, 319, 515–530. [Google Scholar] [CrossRef][Green Version] - Sebald, G.; Kuwano, H.; Guyomar, D.; Ducharne, B. Experimental Duffing oscillator for broadband piezoelectric energy harvesting. Smart Mater. Struct.
**2011**, 20, 102001. [Google Scholar] [CrossRef][Green Version] - Erturk, A.; Inman, D.J. Broadband piezoelectric power generation on high-energy orbits of the bistable Duffing oscillator with electromechanical coupling. J. Sound Vib.
**2011**, 330, 2339–2353. [Google Scholar] [CrossRef] - Leadenham, S.; Erturk, A. Unified nonlinear electroelastic dynamics of a bimorph piezoelectric cantilever for energy harvesting, sensing, and actuation. Nonlinear Dyn.
**2015**, 79, 1727–1743. [Google Scholar] [CrossRef] - De Paula, A.S.; Inman, D.J.; Savi, M.A. Energy harvesting in a nonlinear piezomagnetoelastic beam subjected to random excitation. Mech. Syst. Signal Process.
**2015**, 54, 405–416. [Google Scholar] [CrossRef] - Challa, V.R.; Prasad, M.G.; Shi, Y.; Fisher, F.T. A Vibration Energy Harvesting Device with Bidirectional Resonance Frequency Tunability. Smart Mater. Struct.
**2008**, 17, 015035. [Google Scholar] [CrossRef] - Reissman, T.; Wolff, E.M.; Garcia, E. Piezoelectric Resonance Shifting Using Tunable Nonlinear Stiffness. In Proceedings of the SPIE Active and Passive Smart Structures and Integrated Systems, San Diego, CA, USA, 9–12 March 2009; Volume 7288, p. 72880G. [Google Scholar]
- Erturk, A.; Hoffmann, J.; Inman, D.J. A piezomagnetoelastic structure for broadband vibration energy harvesting. Appl. Phys. Lett.
**2009**, 94, 254102. [Google Scholar] [CrossRef][Green Version] - Erturk, A.; Inman, D.J. Piezoelectric Energy Harvesting; John Wiley & Sons Ltd.: Chichester, UK, 2011. [Google Scholar]
- Ferrari, M.; Ferrari, V.; Guizzetti, M.; Andò, B.; Baglio, S.; Trigona, C. Improved Energy Harvesting from Wideband Vibrations by Nonlinear Piezoelectric Converters. Sens. Actuators A Phys.
**2010**, 162, 425–431. [Google Scholar] [CrossRef] - Stanton, S.C.; Erturk, A.; Mann, B.P.; Inman, D.J. Nonlinear piezoelectricity in electroelastic energy harvesters: Modeling and experimental identification. J. Appl. Phys.
**2010**, 108, 074903. [Google Scholar] [CrossRef][Green Version] - Cammarano, A.; Neild, S.A.; Burrow, S.G.; Inman, D.J. The bandwidth of optimized nonlinear vibration-based energy harvesters. Smart Mater. Struct.
**2014**, 23, 055019–055028. [Google Scholar] [CrossRef] - Roundy, S.; Zhang, Y. Toward self-tuning adaptive vibration based micro-generators. In Proceedings of the SPIE Smart Structures, Devices, and Systems II, Singapore, 24–26 October 2005; Volume 5649, pp. 373–384. [Google Scholar]
- Dutoit, N.E.; Wardle, B.L. Performance of microfabricated piezoelectric vibration energy harvesters. Integr. Ferroelectr.
**2006**, 83, 13–32. [Google Scholar] [CrossRef] - Anton, S.R.; Sodano, H.A. A review of power harvesting using piezoelectric materials (2003–2006). Smart Mater. Struct.
**2007**, 16, R1. [Google Scholar] [CrossRef] - Erturk, A.; Vieira, W.G.R.; De Marqui, C., Jr.; Inman, D.J. On the energy harvesting potential of piezoaeroelastic systems. Appl. Phys. Lett.
**2010**, 96, 184103. [Google Scholar] [CrossRef][Green Version] - Kim, M.; Hoegen, M.; Dugundji, J.; Wardle, B.L. Modeling and experimental verification of proof mass effects on vibration energy harvester performance. Smart Mater. Struct.
**2010**, 19, 045023. [Google Scholar] [CrossRef] - Crawley, E.F.; Anderson, E.H. Detailed models of piezoceramic actuation of beams. J. Intell. Mater. Syst. Struct.
**1990**, 1, 4–25. [Google Scholar] [CrossRef] - Triplett, A.; Quinn, D.D. The Effect of Non-linear Piezoelectric Coupling on Vibration-based Energy Harvesting. J. Intell. Mater. Syst. Struct.
**2009**, 20, 1959–1967. [Google Scholar] [CrossRef] - Silva, L.L.; Monteiro, P.C.; Savi, M.A.; Netto, T.A. Effect of the piezoelectric hysteretic behavior on the vibration-based energy harvesting. J. Intell. Mater. Syst. Struct.
**2013**, 24, 1285. [Google Scholar] [CrossRef] - Silva, L.L.; Monteiro, P.C.; Savi, M.A.; Netto, T.A. On the Nonlinear Behavior of the Piezoelectric Coupling on Vibration-Based Energy Harvesters. Shock Vib.
**2015**, 2015, 739381. [Google Scholar] [CrossRef] - Avirovik, D.; Kumar, A.; Bodnar, R.J.; Priya, S. Remote light energy harvesting and actuation using shape memory alloy-piezoelectric hybrid transducer. Smart Mater. Struct.
**2013**, 22, 052001–052007. [Google Scholar] [CrossRef] - Silva, L.L.; Oliveira, S.A.; Pacheco, P.M.C.L.; Savi, M.A. Synergistic Use of Smart Materials for Vibration-Based Energy Harvesting. Eur. Phys. J. Spec. Top.
**2015**, 224, 3005–3012. [Google Scholar] [CrossRef] - Le, C.P.; Halvorsen, E.; Sørasen, O.; Yeatman, E.M. Wideband excitation of an electrostatic vibration energy harvester with power-extracting end-stops. Smart Mater. Struct.
**2013**, 22, 075020–075029. [Google Scholar] [CrossRef] - Bai, Y.; Carl, M.; Button, T.W. Investigation of using free-standing thick-film piezoelectric energy harvesters to develop wideband devices. Int. J. Struct. Stab. Dyn.
**2014**, 14, 1440016. [Google Scholar] [CrossRef] - Hu, H.P.; Cui, Z.J.; Cao, J.G. Performance of a piezoelectric bimorph harvester with variable width. J. Mech.
**2007**, 23, 197–202. [Google Scholar] [CrossRef] - Friswell, M.I.; Ali, S.F.; Adhikari, S.; Lees, A.W.; Bilgen, O.; Litak, G. Nonlinear piezoelectric vibration energy harvesting from a vertical cantilever beam with tip mass. J. Intell. Mater. Syst. Struct.
**2012**, 23, 1505–1521. [Google Scholar] [CrossRef] - Lesieutre, G.A.; Davis, C.L. Can a coupling coefficient of a piezoelectric actuator be higher than those of its active material? J. Intell. Mater. Syst. Struct.
**1997**, 8, 859–867. [Google Scholar] [CrossRef] - Leland, E.S.; Wright, P.K. Resonance Tuning of Piezoelectric Vibration Energy Scavenging Generators Using Compressive Axial Preload. Smart Mater. Struct.
**2006**, 15, 14131420. [Google Scholar] [CrossRef] - Betts, D.N.; Kim, H.A.; Bowen, C.R.; Inman, D.J. Optimal configurations of bistable piezo-composites for energy harvesting. Appl. Phys. Lett.
**2012**, 100, 114104. [Google Scholar] [CrossRef][Green Version] - Soliman, M.S.M.; Abdel-Rahman, E.M.; El-Saadany, E.F.; Mansour, R.R. A wideband vibration-based energy harvester. J. Micromech. Microeng.
**2008**, 18, 115021. [Google Scholar] [CrossRef] - Kaur, S.; Halvorsen, E. Parameter sensitivity of an in-plane gap closing electrostatic energy harvester with end-stop impacts. J. Intell. Mater. Syst. Struct.
**2016**, 1, 11. [Google Scholar] [CrossRef] - Blystad, L.C.J.; Halvorsen, E. A piezoelectric energy harvester with a mechanical end stop on one side. Microsyst. Technol.
**2011**, 17, 505–551. [Google Scholar] [CrossRef] - Vijayan, K.; Friswell, M.I.; Khodaparast, H.H.; Adhikari, S. Energy harvesting in a coupled system using nonlinear impact. Struct. Health Monit.
**2014**, 5, 255–261. [Google Scholar] - Basset, P.; Galayko, D.; Cottone, F.; Guillemet, R.; Blokhina, E.; Marty, F.; Bourouina, T. Electrostatic vibration energy harvester with combined effect of electrical nonlinearities and Mechanical impact. J. Micromech. Microeng.
**2014**, 24, 035001. [Google Scholar] [CrossRef] - Rysak, A.; Müller, M.; Borowiec, M.; Zubrzycki, J.; Litak, G.; Godlewska-Lach, A.; Wittstock, V. Broadband Concept of Energy Harvesting in Beam Vibrating Systems for Powering Sensors. Adv. Sci. Technol. Res. J.
**2014**, 8, 62–67. [Google Scholar] - Savi, M.A.; Divenyi, S.; Franca, L.F.P.; Weber, H.I. Numerical and experimental investigations of the on linear dynamics and chaos in non-smooth systems. J. Sound Vib.
**2007**, 30, 59–73. [Google Scholar] [CrossRef] - Divenyi, S.; Savi, M.A.; Franca, L.F.P.; Weber, H.I. Nonlinear dynamics and chaos in systems with discontinuous support. Shock Vib.
**2006**, 13, 315–326. [Google Scholar] [CrossRef] - Divenyi, S.; Savi, M.A.; Weber, H.I.; Franca, L.F.P. Experimental investigation of an oscillator with discontinuous support considering different system aspects. Chaos Solitons Fractals
**2008**, 38, 685–695. [Google Scholar] [CrossRef] - Jacquelin, E.; Adhikari, S.; Friswell, M.I. A piezoelectric device for impact energy harvesting. Smart Mater. Struct.
**2011**, 20, 105008–105020. [Google Scholar] [CrossRef] - Vijayan, K.; Friswell, M.I.; Khodaparast, H.H.; Adhikari, S. Non-linear energy harvesting from coupled impacting beams. Int. J. Mech. Sci.
**2015**, 96, 101–109. [Google Scholar] [CrossRef][Green Version] - Kaur, S.; Halvorsen, E.; Søråsen, O.; Yeatman, E.M. Numerical Analysis of Nonlinearities due to Rigid End-Stops in Energy Harvesters. In Proceedings of the Conference: Power MEMS Technical Digest Poster Sessions, Leuven, Belgium, 1–3 December 2010. [Google Scholar]

**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