I.А. Berezhnykh1

1 Voronezh State University (VSU) (Voronezh, Russia)
1 ignatosss@gmail.com

This article presents a analysis of the Bouc-Wen model, one of the most frequently used phenomenological hysteresis models, and its application in the context of energy harvesting systems. The Bouc-Wen model is based on a one-dimensional mechanical system where the restoring force applied to a material point is represented as the sum of two components: an elastic restoring force and a hysteresis term. The article also introduces a model of an energy harvester, focusing on the mechanical oscillatory system and the associated electromagnetic system that absorbs the energy of oscillations. The mechanical part of the energy converter is chosen as a pendulum, which is a common practical implementation in energy harvesting devices. The system is characterized by hysteretic dependencies in both the mechanical subsystem and the structural parts of the electromagnetic nature. The dynamic system is described by a set of ordinary differential equations. The method of small parameters is used to obtain an approximate analytical solution for the system. By substituting the series expansion in terms of the small parameter, the solution is obtained numerically. The results are presented in phase portraits and solutions of the Van der Pol equation, both with and without external excitation and hysteresis. Amlitude-frequency characteristic graph shows the dependence of the amplitude on the frequency of the external excitation on the Van der Pol oscillator. The article concludes with a discussion of the power of external excitation and the transmitted power in the case of a hysteretic element within the Bouc-Wen model. The results of the numerical modeling demonstrate the significance of nonlinear effects in energy harvesting systems. The inclusion of hysteresis nonlinearities significantly improves the adequacy of the models for the systems under consideration. The optimization of the system parameters, including hysteretic elements, identifies periodic regimes that correspond to optimal power transfer to the electrical subsystem. In summary, this article provides a detailed analysis of the Bouc-Wen model and its application in energy harvesting systems, highlighting the importance of considering nonlinear effects and hysteresis for improving the accuracy and efficiency of these systems.

Berezhnykh I.А. Dynamics of energy harvester with parametric excitation. Dynamics of complex systems. 2025. V. 19. № 4. P. 5−13. DOI: 10.18127/j19997493-202504-01 (in Russian).

1. Daqaq M.F. On intentional introduction of stiffness nonlinearities for energy harvesting under white Gaussian excitations. Nonlinear Dyn. 2012, 69, 1063–1079. https://doi.org/10.1007/s11071-012-0327-0
2. 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, 040801. https://doi.org/10.1115/1.4026278
3. Ghouli Z., Belhaq M. Energy harvesting in a delay-induced parametric van der Pol–Duffing oscillator. Eur. Phys. J. Spec. Top. 2021, 230, 3591–3598. https://doi.org/10.1140/epjs/s11734-021-00243-5
4. Renno J.M., Daqaq M.F., Inman D.J. On the optimal energy harvesting from a vibration source. J. Sound Vib. 2009, 320, 386–405. https://doi.org/10.1016/j.jsv.2008.07.029
5. Belhaq M., Hamdi M. Energy harvesting from quasi-periodic vibrations. Nonlinear Dyn. 2016, 86, 2193–2205. https://doi.org/10.1007/ s11071-016-2668-6
6. Ghouli Z., Hamdi M., Belhaq M. Energy Harvesting in a Duffing Oscillator with Modulated Delay Amplitude, 2021. In IUTAM Symposium on Exploiting Nonlinear Dynamics for Engineering Systems; IUTAM Bookseries; Kovacic, I., Lenci, S. eds; Springer: Cham, Switzerland, 2018; Volume 37. https://doi.org/10.1007/978-3-030-23692-2_11
7. Cottone F., Vocca H., Gammaitoni L. Nonlinear Energy Harvesting. Phys. Rev. Lett. 2009, 102, 080601-1–080601-4. https://doi.org/10.1103/ PhysRevLett.102.080601
8. Challa V., Prasad M., Shi Y., Fisher F. A vibration energy harvesting device with bidirectional resonance frequency tunability. Smart Mater. Struct. 2008, 75, 1–10. https://doi.org/10.1088/0964-1726/17/01/015035
9. Duan G., Li Y., Tan C. A Bridge-Shaped Vibration Energy Harvester with Resonance Frequency Tunability under DC Bias Electric Field. Micromachines 2022, 13, 1227. https://doi.org/10.3390/mi13081227
10. Ganapathy S.R., Salleh H., Azhar M.K.A. Design and optimisation of magnetically-tunable hybrid piezoelectric-triboelectric energy harvester. Sci. Rep. 2021, 11, 4458. https://doi.org/10.1038/s41598-021-83776-y
11. Ibrahim P., Arafa M., Anis Y. An Electromagnetic Vibration Energy Harvester with a Tunable Mass Moment of Inertia. Sensors 2021, 21, 5611. https://doi.org/10.3390/s21165611
12. Le Scornec J., Guiffard B., Seveno R., Le Camb V. Frequency tunable, flexible and low cost piezoelectric micro-generator for energy harvesting. Sens. Actuators A Phys. 2020, 312, 112148. https://doi.org/10.1016/j.sna.2020.112148
13. Wang Y., Li L., Hofmann D., Andrade J.E., Daraio C. Structured fabrics with tunable mechanical properties. Nature 2021, 596, 238–243. https://doi.org/10.1038/s41586-021-03698-7
14. Xie L., Li J., Li Xi., Huang L., Cai S. Damping-tunable energy-harvesting vehicle damper with multiple controlled generators: Design, modeling and experiments. Mech. Syst. Signal Process. 2018, 99, 859–872. https://doi.org/10.1016/j.ymssp.2017.07.005
15. Wang Z., He L., Zhang Z., Zhou Z., Zhou J., Cheng G. Research on a Piezoelectric Energy Harvester with Rotating Magnetic Excitation. J. Electron. Mater. 2021, 50, 3228–3240. https://doi.org/10.1007/s11664-021-08910-y
16. Kovacova V., Glinsek S., Girod S., Defay E. High Electrocaloric Effect in Lead Scandium Tantalate Thin Films with Interdigitated Electrodes. Sensors 2022, 22, 4049. https://doi.org/10.3390/s22114049
17. Bouhedma S., Hu S., Schütz A., Lange F., Bechtold T., Ouali M., Hohlfeld D. Analysis and Characterization of Optimized Dual-Frequency Vibration Energy Harvesters for Low-Power Industrial Applications. Micromachines 2022, 13, 1078. https://doi.org/10.3390/mi13071078
18. Zhang Y., Jin Y. Stochastic dynamics of a piezoelectric energy harvester with correlated colored noises from rotational environment. Nonlinear Dyn. 2019, 98, 501–515. https://doi.org/10.1007/s11071-019-05208-x
19. Damjanovic D. Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin films and ceramics. Rep. Prog. Phys. 1998, 61, 1267–1324. https://doi.org/10.1088/0034-4885/61/9/002
20. Dawber M. Physics of thin-film ferroelectric oxides. Rev. Mod. Phys. 2005. 77, 1083. https://doi.org/10.1103/RevModPhys.77.1083
21. Ceponis A., Mažeika D., Jˇur¯enas V., Deltuvien˙e D., Bareikis R. Ring-shaped piezoelectric 5-DOF robot for angular-planar ˙ motion. Micromachines 2022, 13, 1763. https://doi.org/10.3390/mi13101763.
22. Semenov M., Solovyov A., Meleshenko P., Balthazar J. Nonlinear Damping: From Viscous to Hysteretic Dampers. Springer Proceedings in Physics. 2017, 199, 259-275. https://doi.org/10.1007/978-3-319-63937-6_15
23. Semenov M., Reshetova O., Solovyov A., Tolkachev A., Meleshenko P. Oscillations Under Hysteretic Conditions: From Simple Oscillator to Discrete Sine-Gordon Model. Springer Proceedings in Physics. 2019, 228, 229-253. https://doi.org/10.1007/978-981-13-9463-8_12
24. Semenov M.E., Shevlyakova D.V., Kanishheva O.I., Grachikov D.V. Stabilizaciya perevyornutogo mayatnika vertikal`ny`mi oscillyaciyami s pomoshh`yu gisterezisnogo upravleniya. Naukoemkie texnologii. 2012. 3. 27–34.
25. Semenov M., Reshetova O., Borzunov S., Meleshenko P. Self-oscillations in a system with hysteresis: the small parameter approach. The European Physical Journal. Special Topics. 2021, 230, 3565–3571. https://doi.org/10.1140/epjs/s11734-021-00237-3
26. Semenov M., Borzunov S., Meleshenko P. Stochastic Preisach operator: definition within the design approach. Nonlinear Dynamics. 2020, 101, 2599-2614. https://doi.org/10.1007/s11071-020-05907-w
27. Ismail M., Ikhouane F., Rodellar J. The hysteresis Bouc–Wen model, a survey. Arch. Comput. Methods Eng. 2009, 16, 161–188. https://doi.org/10.1007/s11831-009-9031-8