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Stochastic bifurcations in a nonlinear tri-stable energy harvester under colored noise

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Abstract

In this paper, the stochastic bifurcations and the performance analysis of a strongly nonlinear tri-stable energy harvesting system with colored noise are investigated. Using the stochastic averaging method, the averaged Fokker–Plank–Kolmogorov equation and the stationary probability density (SPD) of the amplitude are obtained, respectively. Meanwhile, the Monte Carlo simulations are performed to verify the effectiveness of the theoretical results. D-bifurcation is studied through the largest Lyapunov exponent calculations, which implies the system undergoes D-bifurcation twice with increasing the nonlinear stiffness coefficients. The effects of the nonlinear stiffness coefficients, noise intensity and correlation time on P-bifurcation are discussed by the qualitative changes of the SPD. Moreover, the relationship between D- and P-bifurcation is explored. If the strength of stochastic jump has obvious gap with respect to the two statuses before and after the occurrence of P-bifurcation, the D-bifurcation will happen, too. Finally, the performance and the capability of harvesting energy from ambient random excitation are analyzed.

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References

  1. Mitcheson, P.D., Yeatman, E.M., Rao, G.K., Holmes, A.S., Green, T.C.: Energy harvesting from human and machine motion for wireless electronic devices. Proc IEEE 96, 1457–1486 (2008)

    Google Scholar 

  2. Roundy, S., Wright, P.K.: A piezoelectric vibration based generator for wireless electronics. Smart Mater. Struct. 13, 1131 (2004)

    Google Scholar 

  3. Mitcheson, P.D., Miao, P., Stark, B.H., Yeatman, E.M., Holmes, A.S., Green, T.C.: MEMS electrostatic micropower generator for low frequency operation. Sens. Actuators A 115, 523–529 (2004)

    Google Scholar 

  4. Beeby, S.P., Torah, R.N., Tudor, M.J., Glynne-Jones, P., O’Donnell, T., Saha, C.R., et al.: A micro electromagnetic generator for vibration energy harvesting. J. Micromech. Microeng. 17, 1257–1265 (2007)

    Google Scholar 

  5. Erturk, A., Inman, D.J.: Piezoelectric Energy Harvesting. Wiley, New York (2011)

    Google Scholar 

  6. Triplett, A., Quinn, D.D.: The effect of non-linear piezoelectric coupling on vibration based energy harvesting. J. Intell. Mater. Syst. Struct. 20, 1959–1967 (2009)

    Google Scholar 

  7. Mann, B.P., Owens, B.A.: Investigations of a nonlinear energy harvester with a bistable potential well. J. Sound Vib. 329, 1215–1226 (2010)

    Google Scholar 

  8. Masana, R., Daqaq, M.F.: Relative performance of a vibratory energy harvester in mono-and bi-stable potentials. J. Sound Vib. 330, 6036–6052 (2011)

    Google Scholar 

  9. Stanton, S.C., Mcgehee, C.C., Mann, B.P.: Nonlinear dynamics for broadband energy harvesting: investigation of a bistable piezoelectric inertial generator. Phys. D Nonlinear Phenom. 239, 640–653 (2010)

    MATH  Google Scholar 

  10. Masana, R., Daqaq, M.F.: Electromechanical modeling and nonlinear analysis of axially-loaded energy harvesters. ASME J. Vib. Acoust. 133, 011007 (2011)

    Google Scholar 

  11. Barton, D.A.W., Burrow, S.G., Clare, L.R.: Energy harvesting from vibrations with a nonlinear oscillator. ASME J. Vib. Acoust. 132, 427–436 (2009)

    Google Scholar 

  12. Guyomar, D., Lallart, M.: Nonlinear energy harvesting. In: Iop Conference Series Materials Science and Engineering, vol. 18, p. 092006 (2011)

    MATH  Google Scholar 

  13. Quinn, D., Triplett, A.L., Bergman, L.A., Vakakis, A.F.: Comparing linear and essentially nonlinear vibration-based energy harvesting. ACSM J. Vib. Acoust. 133, 377–378 (2010)

    Google Scholar 

  14. Sebald, G., Kuwano, H., Guyomar, D., Ducharne, B.: Simulation of a doffing oscillator for broadband piezoelectric energy harvesting. Smart Mater. Struct. 20, 075022 (2011)

    Google Scholar 

  15. Cammarano, A., Burrow, S.G., Barton, D.: Modelling and experimental characterization of an energy harvester with bi-stable compliance characteristics. Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng. 225, 475–484 (2011)

    Google Scholar 

  16. Erturk, A., Inman, D.J.: Broadband piezoelectric power generation on high-energy orbits of the bistable doffing oscillator with electromechanical coupling. J. Sound Vib. 330, 2339–2353 (2011)

    Google Scholar 

  17. Stanton, S.C., Mann, B.P.: Harvesting energy from the nonlinear oscillations of a bistable piezoelectric inertial energy generator. In: Proceedings of the ASME; 2009 International Design Engineering Technical Conference and Computers and Information in Engineering Conference, San Diego CA, pp. 447–456 (2009)

  18. He, Q.F., Daqaq, M.F.: Influence of potential function asymmetries on the performance of nonlinear energy harvesters under white noise. J. Sound Vib. 333, 3479–3489 (2014)

    Google Scholar 

  19. Panyam, M., Masana, R., Daqaq, M.F.: On approximating the effective bandwidth of bi-stable energy harvesters. Int. J. Nonlinear Mech. 67, 153–163 (2014)

    Google Scholar 

  20. Zhou, S.X., Cao, J.Y., Inman, D.J., Lin, J., Liu, S.S., Wang, Z.Z.: Broadband tristable energy harvester: modeling and experiment verification. Appl. Energy. 133, 33–39 (2014)

    Google Scholar 

  21. Zhou, S.X., Cao, J.Y., Lin, J., Wang, Z.Z.: Exploitation of a tristable nonlinear oscillator for improving broadband vibration energy harvesting. Eur. Phys. J. Appl. Phys. 67, 30902 (2014)

    Google Scholar 

  22. Cao, J.Y., Zhou, S.X., Wang, W., Lin, J.: Influence of potential well depth on nonlinear tristable energy harvesting. Appl. Phys. Lett. 106, 173903 (2015)

    Google Scholar 

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

    Google Scholar 

  24. 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. 66, 040801 (2014)

    Google Scholar 

  25. Xiao, S.M., Jin, Y.F.: Response analysis of the piezoelectric energy harvester under correlated white noise. Nonlinear Dyn. 90, 2069–2082 (2017)

    Google Scholar 

  26. Jiang, W.A., Chen, L.Q.: An equivalent linearization technique for nonlinear piezoelectric energy harvesters under Gaussian White noise. Commun. Nonlinear Sci. Numer. Simulat. 19, 2897–2904 (2014)

    Google Scholar 

  27. Jin, X.L., Wang, Y., Xu, M., Huang, Z.L.: Semi-analytical solution of random response for nonlinear vibration energy harvesters. J. Sound Vib. 340, 267–282 (2015)

    Google Scholar 

  28. Xu, M., Jin, X.L., Wang, Y., Huang, Z.L.: Stochastic averaging for nonlinear vibration energy harvesting system. Nonlinear Dyn. 78, 1451–1459 (2014)

    MathSciNet  Google Scholar 

  29. Cottone, F., Gammaitoni, L., Vocca, H., Ferrari, M., Ferrari, V.: Piezoelectric buckled beams for random vibration energy harvesting. Smart Mater. Struct. 21, 035021 (2012)

    Google Scholar 

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

    Google Scholar 

  31. Bobryk, R.V., Yurchenko, D.: On enhancement of vibration-based energy harvesting by a random parametric excitation. J. Sound Vib. 366, 407–417 (2016)

    Google Scholar 

  32. Liu, D., Xu, Y., Li, J.L.: Probabilistic response analysis of nonlinear vibration energy harvesting system driven by Gaussian colored noise. Chaos Solitons Fractals 104, 806–812 (2017)

    MATH  Google Scholar 

  33. Mokem, F.I.S., Nono, C., Siewe, M., Tchawoua, C.: Probabilistic distribution and stochastic P-bifurcation of a hybird energy harvester under colored noise. Commun. Nonlinear Sci. Numer. Simulat. 56, 177–197 (2018)

    Google Scholar 

  34. Schenk-Hoppe, K.R.: Bifurcation scenario of the noisy Duffing stochastic-Van der Pol oscillator. Nonlinear Dyn. 11, 255–274 (1996)

    MathSciNet  Google Scholar 

  35. Arnold, L.: Random Dynamical Systems. Springer, NewYork (1998)

    MATH  Google Scholar 

  36. Wei, J.G., Leng, G.: Lyapunov exponent and chaos of Duffing’s equation perturbed by white noise. Appl. Math. Comput. 88, 77–93 (1997)

    MathSciNet  MATH  Google Scholar 

  37. Cai, G.Q., Lin, Y.K.: On exact stationary solutions of equivalent non-linear stochastic systems. Int. J. Nonlinear Mech. 23, 315–325 (1988)

    MATH  Google Scholar 

  38. Sun, Z.K., Fu, J., Xiao, Y.Z., Xu, W.: Delay-induced stochastic bifurcations in a bistable system under white noise. Chaos 25, 083102 (2015)

    MathSciNet  MATH  Google Scholar 

  39. Zhu, W.Q., Cai, G.Q., Lin, Y.K.: On exact stationary solutions of stochastically perturbed Hamiltonian systems. Probab. Eng. Mech. 5, 84–87 (1990)

    Google Scholar 

  40. Xu, Y., Gu, R.C., Zhang, H.Q., Xu, W., Duan, J.Q.: Stochastic bifurcations in a bistable Duffing-Van der Pol oscillator with colored noise. Phys. Rev. E 83, 056215 (2011)

    Google Scholar 

  41. Baxendale, P.H.: Asymptotic Behaviour of Stochastic Flows of Diffeomorphisms, vol. 1203, pp. 1–19. Springer, Berlin (1986)

    MATH  Google Scholar 

  42. Crauel, H., Flandoli, F.: Additive noise destroys a pitchfork bifurcation. J. Dyn. Differ. Equ. 10, 259–274 (1998)

    MathSciNet  MATH  Google Scholar 

  43. Kumar, P., Narayanan, S., Gupta, S.: Investigations on the bifurcation of a noise Duffing-Van der Pol oscillator. Probab. Eng. Mech. 45, 70–86 (2016)

    Google Scholar 

  44. Roberts, J.B., Spanos, P.D.: Stochastic averaging: an approximate method of solving random vibration problems. Int. J. Nonlinear Mech. 21, 111–134 (1986)

    MathSciNet  MATH  Google Scholar 

  45. Zhu, W.Q., Lin, Y.K.: Stochastic averaging of energy envelope. ASCE J. Eng. Mech. 117, 1890–1905 (1991)

    Google Scholar 

  46. Zhu, W.Q., Huang, Z.L., Suzuki, Y.: Response and stability of strongly non-linear oscillators under wide-band random excitation. Int. J. Nonlinear Mech. 36, 1235–1250 (2001)

    MathSciNet  MATH  Google Scholar 

  47. Qiao, Y., Xu, W., Jia, W.T., Liu, W.Y.: Stochastic stability of variable-mass oscillator with mass disturbance modeled as Gaussian white noise. Nonlinear Dyn. 89, 607–616 (2017)

    MathSciNet  Google Scholar 

  48. Stephen, N.G.: On energy harvesting from ambient vibration. J. Sound Vib. 293, 409–425 (2006)

    Google Scholar 

  49. Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Phys. D 16, 285–317 (1985)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant No. 11772048.

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Authors and Affiliations

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

    Yanxia Zhang, Yanfei Jin, Pengfei Xu & Shaomin Xiao

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  1. Yanxia Zhang
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  2. Yanfei Jin
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  4. Shaomin Xiao
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Correspondence to Yanfei Jin.

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Zhang, Y., Jin, Y., Xu, P. et al. Stochastic bifurcations in a nonlinear tri-stable energy harvester under colored noise. Nonlinear Dyn 99, 879–897 (2020). https://doi.org/10.1007/s11071-018-4702-3

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