Nonlinear analysis and analog simulation of a piezoelectric buckled beam with fractional derivative

  • I. S. Mokem Fokou
  • C. Nono Dueyou Buckjohn
  • M. Siewe Siewe
  • C. Tchawoua
Regular Article

Abstract.

In this article, an analog circuit for implementing fractional-order derivative and a harmonic balance method for a vibration energy harvesting system under pure sinusoidal vibration source is proposed in order to predict the system response. The objective of this paper is to discuss the performance of the system with fractional derivative and nonlinear damping (\(\mu_{b}\)). Bifurcation diagram, phase portrait and power spectral density (PSD) are provided to deeply characterize the dynamics of the system. These results are corroborated by the 0-1 test. The appearance of the chaotic vibrations reduces the instantaneous voltage. The pre-experimental investigation is carried out through appropriate software electronic circuit (Multisim). The corresponding electronic circuit is designed, exhibiting periodic and chaotic behavior, in accord with numerical simulations. The impact of fractional derivative and nonlinear damping is presented with detail on the output voltage and power of the system. The agreement between numerical and analytical results justifies the efficiency of the analytical technique used. In addition, by combining the harmonic excitation with the random force, the stochastic resonance phenomenon occurs and improves the harvested energy. It emerges from these results that the order of fractional derivative μ and nonlinear damping \(\mu_{b}\) play an important role in the response of the system.

References

  1. 1.
    Richard Herrmann, Fractional Calculus: An Introduction for Physicists, 2nd edition (GigaHedron, Germany, 2013)Google Scholar
  2. 2.
    J. Liouville, J. Ecole Polytech. 13, 1 (1832)Google Scholar
  3. 3.
    N. Sugimoto, J. Fluid Mech. 225, 631 (1991)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    N. Engheta, IEEE Antennas Propag. Mag. 77, 35 (1997)ADSCrossRefGoogle Scholar
  5. 5.
    G. Joulin, Combust. Sci. Technol. 43, 99 (1985)CrossRefGoogle Scholar
  6. 6.
    A. Dzielinski, G. Sarwas, D. Sierociuk, Adv. Differ. Equ. 2011, 11 (2011)CrossRefGoogle Scholar
  7. 7.
    A. Dzielinski, G. Sarwas, D. Sierociuk, Time domain validation of ultracapacitor fractional order model, in 49th IEEE Conference on Decision and Control, December 15-17, Hilton Atlanta Hotel, Atlanta, GA, USA 2010 (IEEE, 2010) pp. 3730--3735Google Scholar
  8. 8.
    R. Martin, J. Jose Quintana, A. Ramos, I. de la Nuez, Electrotechnical Conference, MELECON 2008, in The 14th IEEE Mediterranean (2008) pp. 61--66Google Scholar
  9. 9.
    A. Burke, J. Power Sources 91, 37 (2000)ADSCrossRefGoogle Scholar
  10. 10.
    A. Jakubowska, J. Walczak, Analysis of the transient state in a circuit with supercapacitor, in Poznan University of Technology Academic Journals, no. 81, (2015) pp. 71--77Google Scholar
  11. 11.
    Y. Wang, T. Tom Hartley, F. Carl Lorenzo, L. Jay Adams, Modeling ultracapacitors as fractional-order systems, in New Trends in Nanotechnology and Fractional Calculus Applications, edited by Joan E. Carletta, Robert J. Veillette, D. Baleanu, Z.B. Güvenç, J.A. Tenreiro Machado (Springer Science, New York, 2010)Google Scholar
  12. 12.
    T.J. Freeborn, A.S. Elwakil, IEEE J. Emerg. S. T. CAS 3, 367 (2013)Google Scholar
  13. 13.
    A. Dzielinski, D. Sierociuk, Acta Montanistica Slov. Rocník 13, 136 (2008)Google Scholar
  14. 14.
    L. Debnath, Int. J. Math. Math. Sci. 2003, 3413 (2003)CrossRefGoogle Scholar
  15. 15.
    A.M. Tusset, J.M. Balthazar, D.G. Bassinello, B.R. Pontes Jr., Jorge Luis Palacios Felix, Nonlinear Dyn. 69, 1837 (2012)CrossRefGoogle Scholar
  16. 16.
    J. Cao, S. Zhou, D.J. Inman, Y. Chen, Nonlinear Dyn. 80, 1705 (2015)CrossRefGoogle Scholar
  17. 17.
    D. Li, J. Cao, S. Zhou, Y. Chen, Fractional order model of broadband piezoelectric energy harvesters, in Proceedings of the ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference IDETC/CIE 2015, August 2-5, Boston, Massachusetts, USA 2015Google Scholar
  18. 18.
    J.A.T. Machado, M.F. Silva, R.S. Barbosa, I.S. Jesus, C.M. Reis, M.G. Marcos, A.F. Galhano, Math. Probl. Eng. 2010, 639801 (2010)Google Scholar
  19. 19.
    F. Cottone, L. Gammaitoni, H. Vocca, M. Ferrari, V. Ferrari, Smart Mater. Struct. 21, 035021 (2012)ADSCrossRefGoogle Scholar
  20. 20.
    G. Litak, Nonlinear analysis of energy harvesting system with fractional order physical properties, in Proceeding of first international symposium on energy challenges and mechanics, 08-14 july 2014, Aberdeen, Scotland, UK 2014Google Scholar
  21. 21.
    A.E. Mohamed Herzallah, D. Baleanu, Nonlinear Dyn. 58, 385 (2009)CrossRefGoogle Scholar
  22. 22.
    A.E. Mohamed Herzallah, D. Baleanu, Nonlinear Dyn. 69, 977 (2012)CrossRefGoogle Scholar
  23. 23.
    F. Riewe, Phys. Rev. E 55, 3581 (1997)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    E. Popescu, Rom. Astron. J. 23, 8597 (2013)Google Scholar
  25. 25.
    L. Yabin, A.H. Sodano, J. Intell. Mater. Syst. Struct. 21, 149 (2010)CrossRefGoogle Scholar
  26. 26.
    S. Westerlund, Phys. Scr. 43, 174 (1991)ADSCrossRefGoogle Scholar
  27. 27.
    S. Westerlund, IEEE Trans. Dielectr. Electr. Insul. 1, 826 (1994)CrossRefGoogle Scholar
  28. 28.
    M.J. Curie, Ann. Chim. Phys. 6, 203 (1889)Google Scholar
  29. 29.
    E.R. Von Schweidler, Ann. Phys. 24, 711 (1907)CrossRefGoogle Scholar
  30. 30.
    G. Ala, M. Di Paola, E. Francomano, Li Yan, F.P. Pinnola, Viscoelasticity: An electrical point of view, in Proceeding of IEEE 2014 International Conference on Fractional Differentiation and Its Applications (ICFDA), 23-25 June 2014, Catania, Italy 2014 (IEEE, 2014) po. 1--6, DOI:10.1109/ICFDA.2014.6967407
  31. 31.
    E. von Schweidler, Die Anomalien der dielectrischen Erscheinugen, in Handbuch der Elektrizität und des Magnetismus, Vol. 1, Elektrizitätserregung und Elektrostatik, edited by Leo Graetz (Teubner, Leipzing Verlag, 1918) pp. 232--253Google Scholar
  32. 32.
    S. Priya, D. Viehland, A.V. Carazo, J. Ryu, K. Uchino, J. Appl. Phys. 90, 1469 (2001)ADSCrossRefGoogle Scholar
  33. 33.
    M.K. Samal, P. Seshu, S.K. Parashar, U. von Wagner, P. Hagedorn, B.K. Dutta, H.S. Kushwaha, Int. J. Solids Struct. 43, 1437 (2006)CrossRefGoogle Scholar
  34. 34.
    U. von Wagner, Int. J. Nonlinear Mech. 38, 565 (2003)CrossRefGoogle Scholar
  35. 35.
    O.P. Agrawal, J. Math. Anal. Appl. 272, 368 (2002)MathSciNetCrossRefGoogle Scholar
  36. 36.
    R. Caponetto, G. Dongola, L. Fortuna, I. Petras, Fractional order systems: modeling and control applications (World Scientific, New Jersey, 2010)Google Scholar
  37. 37.
    A. Leung, H. Yang, Z. Guo, J. Sound Vib. 331, 1115 (2012)ADSCrossRefGoogle Scholar
  38. 38.
    M. Xiao, W.X. Zheng, J. Cao, Math. Comput. Simul. 89, 1 (2013)CrossRefGoogle Scholar
  39. 39.
    I. Petras, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation (Springer, New York, 2010)Google Scholar
  40. 40.
    Mamat Mustafa, W.S. Mada Sanjaya, Maulana Dian Syah, Appl. Math. Sci. 7, 1 (2013)MathSciNetGoogle Scholar
  41. 41.
    Gutenbert Kenfack, Alain Tiedeu, J. Signal Inf. Process. 4, 158 (2013)Google Scholar
  42. 42.
    Tae H. Lee, Ju H. Park, Int. J. Phys. Sci. 5, 1183 (2010)Google Scholar
  43. 43.
    J.O. Maaita, I.M. Kyprianidis, Ch.K. Volos, E. Meletlidou, J. Eng. Sci. Technol. Rev. 6, 74 (2013)Google Scholar
  44. 44.
    Mada Sanjaya, Halimatussadiyah, Dian Syah Maulana, Int. Schol. Sci. Res. Innov. 5, 1064 (2011)Google Scholar
  45. 45.
    W.M. Ahmad, A.M. Harb, Chaos, Solitons Fractals 18, 693 (2003)ADSCrossRefGoogle Scholar
  46. 46.
    W.M. Ahmad, J.C. Sprott, Chaos, Solitons Fractals 16, 339 (2003)ADSCrossRefGoogle Scholar
  47. 47.
    G.A. Gottwald, I. Melbourne, SIAM J. Appl. Dyn. Syst. 8, 129 (2009)ADSMathSciNetCrossRefGoogle Scholar
  48. 48.
    G.A. Gottwald, I. Melbourne, Nonlinearity 22, 1367 (2009)ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    A. Syta, G. Litak, S. Lenci, M. Scheffler, Chaos 24, 013107 (2014)ADSMathSciNetCrossRefGoogle Scholar
  50. 50.
    A. Syta, C.R. Bowen, H.A. Kim, A. Rysak, G. Litak, Meccanica 50, 1961 (2015)CrossRefGoogle Scholar
  51. 51.
    Grzegorz Litak, Michael I. Friswell, Sondipon Adhikari, Eur. Phys. J. Plus 130, 103 (2015)CrossRefGoogle Scholar
  52. 52.
    S. Jeyakumari, V. Chinnathambi, S. Rajasekar, M.A.F. Sanjuan, Chaos 19, 043128 (2009)ADSCrossRefGoogle Scholar
  53. 53.
    T.L.M. Djomo Mbong, M. Siewe Siewe, C. Tchawoua, Commun. Nonlinear Sci. Numer. Simul. 22, 228 (2015)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • I. S. Mokem Fokou
    • 1
    • 2
  • C. Nono Dueyou Buckjohn
    • 1
  • M. Siewe Siewe
    • 1
  • C. Tchawoua
    • 1
  1. 1.University of Yaounde IFaculty of science, Department of Physics, Laboratory of Mechanics, Materials and StructuresYaoundeCameroon
  2. 2.The African Center of Excellence in Information and Communication Technologies (CETIC)YaoundeCameroon

Personalised recommendations