Skip to main content

Advertisement

SpringerLink
Log in

Discovering governing equation from data for multi-stable energy harvester under white noise

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

It is sometimes difficult to model the stochastic differential equations for strongly nonlinear multi-stable vibration energy harvesters, especially for those under additive and multiplicative white noises, because of the existing challenges in quantifying noise intensities, nonlinear stiffness coefficients and damping coefficient. From the perspective of machine learning, a sparse identification method is devised to discover the general governing equation of energy harvester by using observed data on system state time series. With the observed data, the drift term and the diffusion term can be learned and then the stochastic differential equation can be identified. A penta-stable vibration energy harvester is taken as an example to verify the feasibility and effectiveness of the devised sparse identification method, which indicates that the method can be successfully applied to model the governing equation of a multi-stable vibration energy harvesting system under random excitation. Based on the learned data-driven stochastic differential equation for energy harvester, the stochastic dynamics can be further explored by appropriately adjusting the system parameters to improve energy harvesting performance and optimize the miniaturization design.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Panyam, M., Daqaq, M.F.: Characterizing the effective bandwidth of tri-stable energy harvesters. J. Sound Vib. 386, 336–358 (2017)

    Article  Google Scholar 

  2. Zhang, Y., Jin, Y., Xu, P., Xiao, S.: Stochastic bifurcations in a nonlinear tri-stable energy harvester under colored noise. Nonlinear Dyn. 99, 879–897 (2018)

    Article  Google Scholar 

  3. Huang, D., Zhou, S., Litak, G.: Analytical analysis of the vibrational tristable energy harvester with a RL resonant circuit. Nonlinear Dyn. 97, 663–677 (2019)

    Article  Google Scholar 

  4. Harne, R.L., Wang, K.W.: A review of the recent research on vibration energy harvesting via bistable systems. Smart Mater. Struct. 22, 023001 (2013)

    Article  Google Scholar 

  5. Zhang, Y., Jin, Y.: Stochastic dynamics of a piezoelectric energy harvester with correlated colored noises from rotational environment. Nonlinear Dyn. 98, 501–515 (2019)

    Article  Google Scholar 

  6. Zhang, Y., Jin, Y., Li, Y.: Enhanced energy harvesting using time-delayed feedback control from random rotational environment. Phys. D Nonlinear Phenomena. 422, 132908 (2021)

    Article  MathSciNet  Google Scholar 

  7. Zhou, Z., Qin, W., Zhu, P.: A broadband quad-stable energy harvester and its advantages over bi-stable harvester: simulation and experiment verification. Mech. Syst. Signal Process. 84, 158–168 (2017)

    Article  Google Scholar 

  8. Zhou, Z., Qin, W., Yang, Y., Zhu, P.: Improving efficiency of energy harvesting by a novel penta-stable configuration. Sens. Actuat. A Phys. 265, 297–305 (2017)

    Article  Google Scholar 

  9. Huang, D., Zhou, S., Litak, G.: Theoretical analysis of multi-stable energy harvesters with high-order stiffness terms. Commun. Nonlinear Sci. Numer. Simul. 69, 270–286 (2019)

    Article  MathSciNet  Google Scholar 

  10. Mei, X., Zhou, S., Yang, Z., Kaizuka, T., Nakano, K.: Enhancing energy harvesting in low-frequency rotational motion by a quad-stable energy harvester with time-varying potential wells. Mech. Syst. Signal Process. 148, 107167 (2021)

    Article  Google Scholar 

  11. Foupouapouognigni, O., Nono Dueyou Buckjohn, C., Siewe Siewe, M., Tchawoua, C.: Hybrid electromagnetic and piezoelectric vibration energy harvester with Gaussian white noise excitation. Phys. A Stat. Mech. Appl. 509, 346–360 (2018)

    Article  MathSciNet  Google Scholar 

  12. Ramakrishnan, S., Edlund, C.: Stochastic stability of a piezoelectric vibration energy harvester under a parametric excitation and noise-induced stabilization. Mech. Syst. Signal Process. 140, 106566 (2020)

    Article  Google Scholar 

  13. Daqaq, M.F.: Transduction of a bistable inductive generator driven by white and exponentially correlated Gaussian noise. J. Sound Vib. 330, 2554–2564 (2011)

    Article  Google Scholar 

  14. Jin, Y.F., Xiao, S.M., Zhang, Y.X.: Enhancement of tristable energy harvesting using stochastic resonance. J. Stat. Mech. Theory Exp. 2018, 123211 (2018)

    Article  MathSciNet  Google Scholar 

  15. Li, Y., Duan, J.: A data-driven approach for discovering stochastic dynamical systems with non-Gaussian Lévy noise. Phys. D Nonlinear Phenom. 417, 132830 (2021)

    Article  Google Scholar 

  16. Brunton, S.L., Proctor, J.L., Kutz, J.N.: Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl. Acad. Sci. USA 113, 3932–3937 (2016)

    Article  MathSciNet  Google Scholar 

  17. Champion, K., Lusch, B., Kutz, J.N., Brunton, S.L.: Data-driven discovery of coordinates and governing equations. Proc. Natl. Acad. Sci. USA 116, 22445–22451 (2019)

    Article  MathSciNet  Google Scholar 

  18. Boninsegna, L., Nuske, F., Clementi, C.: Sparse learning of stochastic dynamical equations. J. Chem. Phys. 148, 241723 (2018)

    Article  Google Scholar 

  19. Napoletani, D., Sauer, T.D.: Reconstructing the topology of sparsely connected dynamical networks. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 77, 026103 (2008)

  20. Wang, W., Lai, Y.C., Grebogi, C.: Data based identification and prediction of nonlinear and complex dynamical systems. Phys. Rep. 644, 1–76 (2016)

    Article  MathSciNet  Google Scholar 

  21. Yao, C., Bollt, E.M.: Modeling and nonlinear parameter estimation with Kronecker product representation for coupled oscillators and spatiotemporal systems. Phys. D Nonlinear Phenom. 227, 78–99 (2007)

    Article  MathSciNet  Google Scholar 

  22. Verdejo, H., Awerkin, A., Kliemann, W., Becker, C.: Modelling uncertainties in electrical power systems with stochastic differential equations. Int. J. Elect. Power Energy Syst. 113, 322–332 (2019)

    Article  Google Scholar 

  23. Klus, S., Nüske, F., Peitz, S., Niemann, J.H., Schütte, C.: Data-driven approximation of the Koopman generator: model reduction, system identification, and control. Phys. D Nonlinear Phenom. 406, 132416 (2020)

    Article  MathSciNet  Google Scholar 

  24. Harirchi, F., Kim, D., Khalil, O., Liu, S., Violi, A.: On sparse identification of complex dynamical systems: A study on discovering influential reactions in chemical reaction networks. Fuel 279, 118204 (2020)

    Article  Google Scholar 

  25. Sapena-Bano, A., Chinesta, F., Puche-Panadero, R., Martinez-Roman, J., Pineda-Sanchez, M.: Model reduction based on sparse identification techniques for induction machines: towards the real time and accuracy-guaranteed simulation of faulty induction machines. Int. J. Elect. Power Energy Syst. 125, 106417 (2021)

    Article  Google Scholar 

  26. Gagne, D.J., Christensen, H.M., Subramanian, A.C., Monahan, A.H.: Machine learning for stochastic parameterization: generative adversarial networks in the Lorenz '96 model. J. Adv. Model. Earth Syst. 12, e2019MS001896 (2020)

  27. Canhoto, A.I.: Leveraging machine learning in the global fight against money laundering and terrorism financing: An affordances perspective. J. Bus. Res. (2020)

  28. Christian, R., Schwantes, Vijay, S.: Pande: modeling molecular kinetics with tICA and the Kernel trick. J. Chem. Theory Comput. 11, 600–608 (2015)

  29. Kutz, J.N., Brunton, S.L., Brunton, B.W., Proctor, J.L.: Dynamic mode decomposition: data-driven modeling of complex systems. Soc. Ind. Appl. Math. (2016)

  30. Jin, Y., Zhang, Y.: Dynamics of a delayed Duffing-type energy harvester under narrow-band random excitation. Acta Mech. 232, 1045–1060 (2021)

    Article  MathSciNet  Google Scholar 

  31. Zhang, Y., Jin, Y., Xu, P.: Dynamics of a coupled nonlinear energy harvester under colored noise and periodic excitations. Int. J. Mech. Sci. 172, 105418 (2020)

    Article  Google Scholar 

  32. Bonnin, M., Traversa, F.L., Bonani, F.: Analysis of influence of nonlinearities and noise correlation time in a single-DOF energy-harvesting system via power balance description. Nonlinear Dyn. 100, 119–133 (2020)

    Article  Google Scholar 

  33. Yu, H., Zhou, J., Yi, X., Wu, H., Wang, W.: A hybrid micro vibration energy harvester with power management circuit. Microelect. Eng. 131, 36–42 (2015)

    Article  Google Scholar 

  34. Risken, H.: The Fokker-Planck equation: methods of solution and applications. Springer, New York (1996)

    Book  Google Scholar 

  35. Lallart, M., Zhou, S., Yan, L., Yang, Z., Chen, Y.: Tailoring multistable vibrational energy harvesters for enhanced performance: theory and numerical investigation. Nonlinear Dyn. 96, 1283–1301 (2019)

    Article  Google Scholar 

  36. Wang, Z., Xu, Y., Yang, H.: Lévy noise induced stochastic resonance in an FHN model. Sci. China Technol. Sci. 59, 371–375 (2016)

    Google Scholar 

  37. Lu, F., Lin, K.K., Chorin, A.J.: Comparison of continuous and discrete-time data-based modeling for hypoelliptic systems. Commun. Appl. Math. Comput. Sci. 11, 187–216 (2016)

    Article  MathSciNet  Google Scholar 

  38. Pavliotis, G.A., Stuart, A.M.: Parameter estimation for multiscale diffusions. J. Stat. Phys. 127, 741–781 (2007)

    Article  MathSciNet  Google Scholar 

  39. Samson, A., Thieullen, M.: A contrast estimator for completely or partially observed hypoelliptic diffusion. Stoch. Processes Appl. 122, 2521–2552 (2012)

    Article  MathSciNet  Google Scholar 

  40. Zhang, Y., Jin, Y., Xu, P.: Stochastic resonance and bifurcations in a harmonically driven tri-stable potential with colored noise. Chaos 29, 023127sss (2019)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This study is supported by the National Natural Science Foundation of China (Nos. 12072025, 11772048). The first author warmly acknowledges the financial support of the China Scholarship Council (CSC No. 201906030059) and Excellent Doctoral Dissertation Seedling Fund of Beijing Institute of Technology (BIT) to enable this work.

Author information

Authors and Affiliations

  1. Department of Mechanics, Beijing Institute of Technology, Beijing, 100081, China

    Yanxia Zhang & Yanfei Jin

  2. Department of Applied Mathematics, College of Computing, Illinois Institute of Technology, Chicago, IL, 60616, USA

    Yanxia Zhang, Jinqiao Duan & Yang Li

  3. State Key Laboratory of Mechanics and Control of Mechanical Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Yang Li

Authors
  1. Yanxia Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jinqiao Duan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yanfei Jin
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yanfei Jin.

Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Availability of data and materials

The authors declare that datasets supporting the conclusions of this study are included within the paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, Y., Duan, J., Jin, Y. et al. Discovering governing equation from data for multi-stable energy harvester under white noise. Nonlinear Dyn 106, 2829–2840 (2021). https://doi.org/10.1007/s11071-021-06960-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-021-06960-9

Keywords

Search

Navigation