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Probabilistic solution of the vibratory energy harvester excited by Gaussian white noise

  • Wen-An Jiang
  • Peng Sun
  • Zhao-Wang Xia
Article

Abstract

This paper aims to investigate the statistical characteristics of strongly nonlinear vibratory energy harvesters under Gaussian white noise excitation. The high-dimensional Fokker–Planck–Kolmogorov (FPK) equation of the coupled electromechanical system is reduced to a low-dimensional equation by using the state-space-split method. The conditional moment given by the equivalent linearization method is employed to decouple the FPK equations of coupled system, and then obtained an equivalent nonlinear uncoupled subsystem. The exact stationary solution of the reduced FPK equation of the subsystem is established. The mean output power is derived by the second order conditional moment from the associated approximate probability density function of mechanical subsystem. The procedure is applied to mono- and bi-stable energy harvesters. Effectiveness of the probability density function of the proposed approach is examined via comparison with equivalent linearization method and Monte Carlo simulation. The effects of the system parameters on the mean-square displacement and the mean output power are discussed. The approximate analytical outcomes are qualitatively and quantitatively supported by the numerical simulations.

Keywords

Nonlinear State-space-split method Conditional moment Energy harvesting Fokker–Planck–Kolmogorov equation 

Notes

Acknowledgements

This work was supported by the National Natural Science of China (Nos. 11702119 and 11502071), the Natural Science Foundation of Jiangsu Province (No. BK20170565).

References

  1. 1.
    Elvin N, Erturk A (2013) Advances in energy harvesting methods. Springer, New YorkCrossRefGoogle Scholar
  2. 2.
    Erturk A, Inman DJ (2011) Piezoelectric energy harvesting. Wiley, New YorkCrossRefGoogle Scholar
  3. 3.
    Sodano HA, Park G, Inman DJ (2004) A review of power harvesting from vibration using piezoelectric materials. Shock Vib Digest 36:197–205CrossRefGoogle Scholar
  4. 4.
    Anton SR, Sodano HA (2007) A review of power harvesting using piezoelectric materials (2003–2006). Smart Mater Struct 16:R1–R21CrossRefGoogle Scholar
  5. 5.
    Tang LH, Yang YW, Soh CK (2010) Toward broadband vibration-based energy harvesting. J Intell Mater Syst Struct 21:1867–1897CrossRefGoogle Scholar
  6. 6.
    Pellegrini SP, Tolou N, Schenk M, Herder JL (2013) Bistable vibration energy harvesters: a review. J Intell Mater Syst Struct 24:1303–1312CrossRefGoogle Scholar
  7. 7.
    Harne RL, Wang KW (2013) A review of the recent research on vibration energy harvesting via bistable systems. Smart Mater Struct 22:023001CrossRefGoogle Scholar
  8. 8.
    Daqaq MF, Masana R, Erturk A, Quinn DD (2014) On the role of nonlinearities in vibratory energy harvesting: a critical review and discussion. Appl Mech Rev 66:040801CrossRefGoogle Scholar
  9. 9.
    Erturk A, Inman DJ (2010) Broadband piezoelectric power generation on high-energy orbits of the bistable Duffing oscillator with electromechanical coupling. J Sound Vib 330:2339–2353CrossRefGoogle Scholar
  10. 10.
    Zou HX, Zhang WM, Li WB, Wei KX, Gao QH, Peng ZK, Meng G (2016) A compressive-mode wideband vibration energy harvester using a combination of bistable and flextensional mechanisms. J Appl Mech 83:121005CrossRefGoogle Scholar
  11. 11.
    Harne RL, Wang KW (2016) Axial suspension compliance and compression for enhancing performance of a nonlinear vibration energy harvesting beam system. J Vib Acoust 138:011004CrossRefGoogle Scholar
  12. 12.
    Zou HX, Zhang WM, Li WB, Wei KX, Gao QH, Peng ZK, Meng G (2017) Design and experimental investigation of a magnetically coupled vibration energy harvester using two inverted piezoelectric cantilever beams for rotational motion. Energy Convers Manag 148:1391–1398CrossRefGoogle Scholar
  13. 13.
    Cottone F, Vocca H, Gammaitoni L (2009) Nonlinear energy harvesting. Phys Rev Lett 102:080601CrossRefGoogle Scholar
  14. 14.
    Litak G, Friswell MI, Adhikari S (2010) Magnetopiezoelastic energy harvesting driven by random excitations. Appl Phys Lett 96:214103CrossRefGoogle Scholar
  15. 15.
    Gammaitoni L, Neri I, Vocca H (2009) Nonlinear oscillators for vibration energy harvesting. Appl Phys Lett 94:164102CrossRefGoogle Scholar
  16. 16.
    Daqaq MF (2010) Response of uni-modal Duffing-type harvesters to random forced excitations. J Sound Vib 329:3621–3631CrossRefGoogle Scholar
  17. 17.
    Daqaq MF (2011) Transduction of a bistable inductive generator driven by white and exponentially correlated Gaussian noise. J Sound Vib 330:2554–2564CrossRefGoogle Scholar
  18. 18.
    Green PL, Worden K, Atallah K, Sims ND (2012) The benefits of Duffing-type nonlinearities and electrical optimisation of a mono-stable energy harvester under white Gaussian excitations. J Sound Vib 331:4504–4517CrossRefGoogle Scholar
  19. 19.
    Adhikari S, Friswell MI, Inman DJ (2009) Piezoelectric energy harvesting from broadband random vibrations. Smart Mater Struct 18:115005CrossRefGoogle Scholar
  20. 20.
    Ali S, Adhikari FS, Friswell MI, Narayanan S (2011) The analysis of piezomagnetoelastic energy harvesters under broadband random excitations. J Appl Phys 109:074904CrossRefGoogle Scholar
  21. 21.
    Masana R, Daqaq MF (2013) Response of duffing-type harvesters to band-limited noise. J Sound Vib 332:6755–6767CrossRefGoogle Scholar
  22. 22.
    Daqaq MF (2012) On intentional introduction of stiffness nonlinearities for energy harvesting under white Gaussian excitations. Nonlinear Dyn 69:1063–1079CrossRefGoogle Scholar
  23. 23.
    Martens W, Wagner UV, Litak G (2013) Stationary response of nonlinear magnetopiezo-electric energy harvester systems under stochastic excitation. Eur Phys J Special Topics 222:1665–1673CrossRefGoogle Scholar
  24. 24.
    He QF, Daqaq MF (2014) Influence of potential function asymmetries on the performance of nonlinear energy harvesters under white noise. J Sound Vib 333:3479–3489CrossRefGoogle Scholar
  25. 25.
    He QF, Daqaq MF (2015) New insights into utilizing bistability for energy harvesting under white noise. J Vib Acoust 137:021009CrossRefGoogle Scholar
  26. 26.
    Jiang WA, Chen LQ (2014) Snap-through piezoelectric energy harvesting. J Sound Vib 333:4314–4325CrossRefGoogle Scholar
  27. 27.
    Xu M, Jin XL, Wang Y, Huang ZL (2014) Stochastic averaging for nonlinear vibration energy harvesting system. Nonlinear Dyn 78:1451–1459MathSciNetCrossRefGoogle Scholar
  28. 28.
    Kumar P, Narayanan S, Adhikari S, Friswell MI (2014) Fokker–Planck equation analysis of randomly excited nonlinear energy harvester. J Sound Vib 333:2040–2053CrossRefGoogle Scholar
  29. 29.
    Jin XL, Wang Y, Xu M, Huang ZL (2015) Semi-analytical solution of random response for nonlinear vibration energy harvesters. J Sound Vib 340:267–282CrossRefGoogle Scholar
  30. 30.
    De Paula AS, Inman DJ, Savi MA (2015) Energy harvesting in a nonlinear piezomagnetoe-lastic beam subjected to random excitation. Mech Syst Signal Process 54–55:405–416CrossRefGoogle Scholar
  31. 31.
    Yue XL, Xu W, Zhang Y, Wang L (2015) Global analysis of response in the piezomagnetoe-lastic energy harvester system under harmonic and Poisson white noise excitation. Commun Theor Phys 64:420–424CrossRefGoogle Scholar
  32. 32.
    Jiang WA, Chen LQ (2016) Stochastic averaging of energy harvesting systems. Int J NonLinear Mech 85:174–187CrossRefGoogle Scholar
  33. 33.
    Jiang WA, Chen LQ (2016) Stochastic averaging based on generalized harmonic functions for energy harvesting systems. J Sound Vib 377:264–283CrossRefGoogle Scholar
  34. 34.
    Liu D, Xu Y, Li JL (2017) Randomly-disordered-periodic-induced chaos in a piezoelectric vibration energy harvester system with fractional-order physical properties. J Sound Vib 399:182–196CrossRefGoogle Scholar
  35. 35.
    Xiao SM, Jin YF (2017) Response analysis of the piezoelectric energy harvester under correlated white noise. Nonlinear Dyn 90:2069–2082CrossRefGoogle Scholar
  36. 36.
    Er GK (2011) Methodology for the solutions of some reduced Fokker–Planck equations in high dimensions. Ann. Phys. (Berlin) 523:247–258MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Er GK, Iu VP, Wang K, Guo SS (2016) Stationary probabilistic solutions of the cables with small sag and modeled as MDOF systems excited by Gaussian white noise. Nonlinear Dyn 85:1887–1899CrossRefGoogle Scholar
  38. 38.
    Er GK, Iu VP (2012) State-space-split method for some generalized Fokker–Planck–Kolmogorov equations in high dimensions. Phys Rev E 85:067701CrossRefGoogle Scholar
  39. 39.
    Zhu HT (2015) Probabilistic solution of a multi-degree-of-freedom Duffing system under nonzero mean Poisson impulses. Acta Mech 226:3133–3149MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Engineering MechanicsJiangsu University of Science and TechnologyZhenjiangChina
  2. 2.College of Energy and Power EngineeringJiangsu University of Science and TechnologyZhenjiangChina

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