# On Theoretical and Numerical Aspects of Bifurcations and Hysteresis Effects in Kinetic Energy Harvesters

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## Abstract

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## 1. Introduction

## 2. Mathematical Model

_{W}, f = Asin(ω

_{W}t). The differential equations of motion, taking into account the electromechanical coupling, have a general form of:

_{1}, b

_{B}, c

_{B}are effective (modal) beam parameters corresponding to the mass, damping and stiffness, respectively. c

_{1}and c

_{2}denote the coefficients of the magnetic force. C

_{P}and R

_{O}are parameters in the electrical circuit indicating capacity and resistance, respectively. Finally, k

_{p}is the coupling parameter and U

_{p}is the voltage output on the resistor [4,12,22].

_{0}represents the scaling parameter equal to the absolute value of the coordinate defining the position of the minimum potential barrier. On the other hand, the variable a occurring in the mathematical model reflects control quantity responsible for the shift of the operating point from one half of the potential barrier to the other. In general terms, the variable a takes the value of 1 or −1 depending on the hysteretic branch. Here, we limit ourselves to listing the numerical values of the mathematical model coefficients, based on which quantitative and qualitative computer simulations were carried out, i.e.,: δ = 0.05, ϑ = 0.5, θ = 0.05, σ = 0.05, α = 0.5.

## 3. The Results of Model Tests

^{−5}. To obtain a satisfactory resolution, the ranges of the control parameters ω and p were divided into 500 subintervals.

#### 3.1. Influence of the Parameter d on the Geometrical Structure of a Chaotic Attractor

_{C}> 1.5, harmonic components representing the frequency of the source of excitation dominate in the Fourier spectra. We deal with such geometrical structures of Poincaré cross-sections in the widest zone of chaotic solutions, located in the band ω ϵ [1.4, 1.9]. However, in the range of low ω values, the correlation dimension of the Poincaré cross-sections D

_{C}<1.5. At the same time, in the amplitude–frequency spectra, it is represented by the domination of harmonic components that are a multiple of the frequency of mechanical vibrations affecting the energy-harvesting system. We deal with such a sequence of dominant harmonic components of the Fourier spectrum when the correlation dimension of Poincaré cross-sections is in the D

_{C}range of [1.1, 1.5]. At this point, it is worth noting that such spectra occur both in the case of continuous and smooth (p = 1, d = 0, ω = 0.42) and discontinuous (p = 1, d = 0.3, ω = 0.26) characteristics representing the potential barrier. On the basis of the presented results of numerical experiments, it is also possible to state that for correlation dimensions D

_{C}< 1.1, in the amplitude–frequency spectra, the dominant harmonic components are the multiples of the combination of the two fundamental frequencies ω and ω1. As in the previous example, this is the case with both continuous and smooth (p = 1, d = 0, ω = 0.21) and discontinuous (p = 1, d = 0.3, ω = 0.48) characteristics representing the barrier potentials.

#### 3.2. Identification of Multiple Solutions

_{2}branch that appears in the diagrams identified for p = 2.

## 4. Conclusions

- If the mean value of slope coefficients approximating the branches of the bifurcation diagram in the area of periodic solutions assumes positive values, then we deal with an increase in the effective value of the voltage induced on the piezoelectric electrodes. For its negative values, a decrease in the RMS voltage is observed at the piezoelectric electrodes.
- With increasing overlapping of the cut halves of the potential barrier, in a wide range of variability of the dimensionless excitation frequency, a reduction in the efficiency of energy harvesting is observed, which is confirmed by the diagrams of RMS voltage values presented in the graphs (Figure 4). The limitation of energy-harvesting efficiency, caused by the increase in the value of parameter d, is determined by the reduction in the width measured between the potential barriers.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Kaźmierski, T.J.; Beeby, S. Energy Harvesting Systems, Principles, Modelling and Applications; Springer: New York, NY, USA, 2011. [Google Scholar]
- Priya, S.; Inman, D.J. Energy Harvesting Technologies; Springer: Blacksburg, VA, USA, 2009. [Google Scholar]
- Erturk, A.; Inman, D.J. Piezoelectric Energy Harvesting; John Wiley and Sons: Chichester, UK, 2011. [Google Scholar] [CrossRef]
- Erturk, A.; Hoffmann, J.; Inman, D.J. A piezomagnetoelastic structure for broadband vibration energy harvesting. Appl. Phys. Lett.
**2009**, 94, 254102. [Google Scholar] [CrossRef] - Cottone, F.; Vocca, H.; Gammaitoni, L. Nonlinear energy harvesting. Phys. Rev. Lett.
**2009**, 102, 080601. [Google Scholar] [CrossRef] [Green Version] - Ferrari, M.; Ferrari, V.; Guizzetti, M.; Ando, 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] [Green Version] - Friswell, M.I.; Ali, S.F.; Bilgen, O.; Adhikari, S.; Lees, A.W.; Litak, G. Non-linear piezoelectric vibration energy harvesting from a vertical cantilever beam with tip mass. J. Intell. Mater. Syst. Struct.
**2012**, 23, 1505–1521. [Google Scholar] [CrossRef] - Andò, B.; Baglio, S.S.; Maiorca, F.; Trigona, C. Analysis of two dimensional, wide-band, bistable vibration energy harvester. Sens. Actuators A Phys.
**2013**, 202, 176–182. [Google Scholar] [CrossRef] - Harne, R.L.; Wang, K.W. A review of the recent research on vibration energy harvesting via bistable systems. Smart Mater. Struct.
**2013**, 22, 023001. [Google Scholar] [CrossRef] - Pellegrini, S.P.; Tolou, N.; Schenk, M.; Herder, J.L. Bistable vibration energy harvesters: A review. J. Intell. Mater. Syst. Struct.
**2013**, 24, 1303–1312. [Google Scholar] [CrossRef] - Huguet, T.; Lallart, M.; Badel, A. Orbit jump in bistable energy harvesters through buckling level modification. Mech. Syst. Signal Process.
**2019**, 128, 202–215. [Google Scholar] [CrossRef] - Litak, G.; Ambrożkiewicz, B.; Wolszczak, P. Dynamics of a nonlinear energy harvester with subharmonic responses. J. Phys. Conf. Ser.
**2021**, 1736, 012032. [Google Scholar] [CrossRef] - Daqaq, M.F.; Masana, R.; Erturk, A.; Quinn, D.D. On the role of nonlinearities in vibratory energy harvesting: A critical review and discussion. Appl. Mech. Rev.
**2014**, 66, 40801. [Google Scholar] [CrossRef] - Syta, A.; Litak, G.; Friswell, M.I.; Adhikari, S. Multiple solutions and corresponding power output of a nonlinear bistable piezoelectric energy harvester. Eur. Phys. J. B
**2016**, 89, 99. [Google Scholar] [CrossRef] [Green Version] - Huang, D.; Zhou, S.; Litak, G. Theoretical analysis of multi-stable energy harvesters with high order stiffness terms. Commun. Nonlinear Sci. Numer. Simul.
**2019**, 69, 270–286. [Google Scholar] [CrossRef] - Tan, T.; Wang, Z.; Zhang, L.; Liao, W.-L.; Yan, Z. Piezoelectric autoparametric vibration energy harvesting with chaos control feature. Mech. Syst. Signal Process.
**2021**, 161, 107989. [Google Scholar] [CrossRef] - Qian, F.; Hajj, M.R.; Zuo, L. Bio-inspired bi-stable piezoelectric harvester for broadband vibration energy harvesting. Energy Convers. Manag.
**2020**, 222, 113174. [Google Scholar] [CrossRef] - Montegiglio, P.; Maruccio, C.; Acciani, G.; Rizzello, G.; Seelecke, S. Nonlinear multi-scale dynamics modeling of piezoceramic energy harvesters with ferroelectric and ferroelastic, hysteresis. Nonlinear Dyn.
**2020**, 100, 1985–2003. [Google Scholar] [CrossRef] - Harris, P.; Bowen, C.R.; Kim, H.A.; Litak, G. Dynamics of a vibrational energy harvester with a bistable beam: Voltage response identification by multiscale entropy and “0–1” test. Eur. Phys. J. Plus
**2016**, 131, 109. [Google Scholar] [CrossRef] [Green Version] - Harris, P.; Arafa, M.; Litak, G.; Bowen, C.R.; Iwaniec, J. Output response identification in a multistable system for piezoelectric energy harvesting. Eur. Phys. J. B
**2017**, 90, 20. [Google Scholar] [CrossRef] [Green Version] - Strogatz, S.H. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering; Westview Press: Boulder, CO, USA, 2015. [Google Scholar]
- Margielewicz, J.; Gąska, D.; Litak, G.; Wolszczak, P.; Yurchenko, D. Nonlinear dynamics of a new energy harvesting system with quasi-zero stiffness. Appl. Energy
**2021**, 118159. [Google Scholar] [CrossRef] - Zhou, S.; Cao, J.; Inman, D.J.; Lin, J.; Liu, S.; Wang, Z. Broadband tristable energy harvester: Modeling and experiment verification. Appl. Energy
**2014**, 133, 33–39. [Google Scholar] [CrossRef] - Bernardini, D.; Litak, G. An overview of 0–1 test for chaos. J. Braz. Soc. Mech. Sci. Eng.
**2016**, 38, 1433–1450. [Google Scholar] [CrossRef] - Kantz, H. A robust method to estimate the maximal Lyapunov exponent of a time series. Phys. Lett. A
**1994**, 185, 77–87. [Google Scholar] [CrossRef] - Wolf, A.; Swift, J.B.; Swinney, H.L.; Vastano, J.A. Determining Lyapunov Exponens from a time series. Physica D
**1985**, 16, 285–317. [Google Scholar] [CrossRef] [Green Version] - Litak, G.; Margielewicz, J.; Gaska, D.; Wolszczak, P.; Zhou, S. Multiple solutions of the tristable energy harvester. Energies
**2021**, 14, 1284. [Google Scholar] [CrossRef] - Margielewicz, J.; Gaska, D.; Opasiak, T.; Litak, G. Multiple solutions and transient chaos in a nonlinear flexible coupling model. J. Braz. Soc. Mech. Sci. Eng.
**2021**, 43, 471. [Google Scholar] [CrossRef] - Syta, A.; Bowen, C.R.; Kim, H.A.; Rysak, A.; Litak, G. Experimental analysis of the dynamical response of energy harvesting devices based on bistable laminated plates. Meccanica
**2015**, 50, 1961–1970. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**System and its potential characteristics: (

**a**) schematic diagram without indicating the origin of hysteresis, modelling of the hysteretic branches: (

**b**) potential without hysteresis loop, (

**c**) with hysteresis loop, d indicates the overlap shift, q is the displacement of the tip point of the beam. i is the current in the electrical circuit.

**Figure 2.**Influence of the parameter d and damping on the distribution of the largest Lyapunov exponent: (

**a**) d = 0, (

**b**) d = 0.3 with nodal initial conditions.

**Figure 3.**Influence of the parameter d on the geometric structure of the Poincaré cross-section: (

**a**) d = 0, (

**b**) d = 0.3. Starting from the top, the horizontal panels correspond to the Poincaré map and the corresponding time series for selected ω and amplitude–frequency spectrum. The bottom panels show the bifurcation diagrams based on the local minima (red) and the local maxima (blue) compared with RMS(u) for d = 0.3. To clarify the influence of the parameter d on bifurcations in the selected frequency interval, (

**c**,

**d**) magnify the difference between the cases d = 0 and d = 0.3, respectively. The simulation results were obtained assuming zero initial conditions.

**Figure 4.**Bifurcation diagram: effect of parameters d and p on energy recovery efficiency with various initial conditions considered simultaneously for the given system parameters. The parameters are indicated in the figures. For better clarity, the solutions are marked by nT

_{m}(n—the response period, m—the number of different solutions with the same response period). The bottom panels are the summery of the upper left and right parametric cases to show the tendencies of the system evolution with d parameter changes.

**Figure 5.**Influence of the parameters d and p on the efficiency of energy harvesting: 3D orbits with the vertical axis indicate the total mechanical energy. For clarity, the corresponding hysteretic potential is plotted with a division into two-color flaps with overlap. The system parameters are indicated in the figures.

**Figure 6.**Examples showing the influence of parameter d on coexisting solutions, plotted for: (

**a**) and (

**b**) coexisting solutions for p = 1, ω = 1.6, (

**c**) p = 1, ω = 2.0, (

**d**) p = 1, ω = 3.4.

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**MDPI and ACS Style**

Litak, G.; Margielewicz, J.; Gąska, D.; Rysak, A.; Trigona, C.
On Theoretical and Numerical Aspects of Bifurcations and Hysteresis Effects in Kinetic Energy Harvesters. *Sensors* **2022**, *22*, 381.
https://doi.org/10.3390/s22010381

**AMA Style**

Litak G, Margielewicz J, Gąska D, Rysak A, Trigona C.
On Theoretical and Numerical Aspects of Bifurcations and Hysteresis Effects in Kinetic Energy Harvesters. *Sensors*. 2022; 22(1):381.
https://doi.org/10.3390/s22010381

**Chicago/Turabian Style**

Litak, Grzegorz, Jerzy Margielewicz, Damian Gąska, Andrzej Rysak, and Carlo Trigona.
2022. "On Theoretical and Numerical Aspects of Bifurcations and Hysteresis Effects in Kinetic Energy Harvesters" *Sensors* 22, no. 1: 381.
https://doi.org/10.3390/s22010381