Nonlinear Dynamics

, Volume 80, Issue 1–2, pp 491–501 | Cite as

Nonlinear analysis of energy harvesting systems with fractional order physical properties

Original Paper


An electromechanical energy harvesting system with a fractional order current–voltage relationship for the electrical circuit and fractional power law in the restoring force of its mechanical part is considered to act as an energy harvester. Our results show that, under a single-well potential configuration, for a small amplitude of the perturbation, as the order of derivative increases, the resonant amplitude of mechanical vibration decreases while the bending degree (hardening case) remains fairly constant. For a large amplitude of the perturbation, the output power is increased due to the hardening effects. Under a double-well configuration, the fractional power stiffness \(\alpha \) strongly affects the crossing well dynamics (large amplitude motion) and consequently the output electrical power. The harvested electric power appears to be maximal for deterministic and random excitation for small \(\alpha \). High-level noise intensity is found to reduce the output power in the region of resonance and surprisingly increases the output in other regions of \(\alpha \). For sufficiently large amplitude of harmonic excitation, this effect is realized in a stochastic resonance.


Energy harvesting Bistability  Fractional order deflection Fractional derivative Stochastic resonance 



This work has been funded by the US Office of Naval research (CAKK and CN) under the Grant ONR N00014-08-1-0435 (Program manager: Mr. Anthony Seman III) and by the Polish National Science Center (GL) under Grant Agreement 2012/05/B/ST8/00080.


  1. 1.
    Erturk, A., Inman, D.J.: Piezoelectric Energy Harvesting. Wiley, Chichester (2011)CrossRefGoogle Scholar
  2. 2.
    Shahruz, S.M.: Design of mechanical band-pass filters for energy scavenging. J. Sound Vib. 292(3–5), 987–998 (2006)CrossRefGoogle Scholar
  3. 3.
    Shahruz, S.M.: Increasing the efficiency of energy scavengers by magnets. J. Comput. Nonlinear Dyn. 3(4), 041,001 (2008)CrossRefGoogle Scholar
  4. 4.
    Ramlan, R., Brennan, M.J., Mace, B.R., Kovacic, I.: Potential benefits of a non-linear stiffness in an energy harvesting device. Nonlinear Dyn. 59, 545–558 (2010)CrossRefMATHGoogle Scholar
  5. 5.
    Stanton, S.C., Mann, B.P., Owens, B.A.: Melnikov theoretic methods for characterizing the dynamics of the bistable piezoelectric inertial generator in complex spectral environments. Phys. D Nonlinear Phenom. 241(6), 711–720 (2012)CrossRefGoogle Scholar
  6. 6.
    Gammaitoni, L., Hanggi, P., Jung, P., Marchesoni, F.: Stochastic resonance. Rev. Mod. Phys. 70, 223–288 (1998)CrossRefGoogle Scholar
  7. 7.
    Litak, G., Borowiec, M., Syta, A.: Vibration of generalized double well oscillators. ZAMM 87, 590–602 (2007)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Litak, G., Friswell, M.I., Adhikari, S.: Magnetopiezoelastic energy harvesting driven by random excitations. Appl. Phys. Lett. 96(21), 214103 (2010)CrossRefGoogle Scholar
  9. 9.
    Borowiec, M., Rysak, A., Betts, D.H., Bowen, C.R., Kim, H.A., Litak G.: Complex response of the bistable laminated plate: multiscale entropy analysis. Eur. Phys. J. Plus. 129, 211 (2014)Google Scholar
  10. 10.
    Kwuimy, C.A.K., Litak, G., Borowiec, M., Nataraj, C.: Performance of a piezoelectric energy harvester driven by air flow. Appl. Phys. Lett. 100(2), 024,103–3 (2012)Google Scholar
  11. 11.
    Tekam, G.O., Tchuisseu, E.T., Kwuimy, C., Woafo, P.: Analysis of an electromechanical energy harvester system with geometric and ferroresonant nonlinearities. Nonlinear Dyn. 76(2), 1561–1568 (2014)CrossRefGoogle Scholar
  12. 12.
    Owens, B.A., Mann, B.P.: Linear and nonlinear electromagnetic coupling models in vibration-based energy harvesting. J. Sound Vib. 331(4), 922–937 (2012)CrossRefGoogle Scholar
  13. 13.
    Li, C.: Keynote lecture: “Fractional dynamics: an overview and some challenges”. In: ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference ((IDETC)/CIE) (2013)Google Scholar
  14. 14.
    Debnath, L.: Recent applications of fractional calculus to science and engineering. Int. J. Math. Math. Sci. 54, 3413–3442 (2003)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Duarte, F., Machado, J.A.T.: Chaotic phenomena and fractional-order dynamics in the trajectory control of redundant manipulators. Nonlinear Dyn. 29(1–4), 315–342 (2002)CrossRefMATHGoogle Scholar
  16. 16.
    Agrawal, O.: Application of Fractional Derivatives in Thermal Analysis of Disk Brake. Nonlinear Dyn. 38, 191–206 (2004)CrossRefMATHGoogle Scholar
  17. 17.
    Ngueuteu, G.M., Woafo, P.: Dynamics and synchronization analysis of coupled fractional-order nonlinear electromechanical systems. Mech. Res. Commun. 46, 20–25 (2012)CrossRefGoogle Scholar
  18. 18.
    Cao, J., Zhou, S., Inman, D.J., Chen, Y.: Chaos in the fractionally damped broadband piezoelectric energy generator. Nonlinear Dyn. (in press) (2014)Google Scholar
  19. 19.
    Machado, J.A.T., Silva, M.F., Barbosa, R.S., Jesus, I.S., Reis, C.M., Marcos, M.G., Galhano, A.F.: Some applications of fractional calculus in engineering. Math. Probl. Eng. ID 639801, 1–34 (2010)Google Scholar
  20. 20.
    Silva, M.F., Machado, J.A.T.: Fractional order \(pd^{\mu }\) joint control of legged robots. J. Vib. Control 12(12), 1483–1501 (2006)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Syta, A., Litak, G., Lenci, S., Scheffler, M.: Chaotic vibrations of the duffing system with fractional damping. Chaos 24, 013,107 (2014)Google Scholar
  22. 22.
    Cveticanin, L., Zukovic, M.: Melnikov’s criteria and chaos in systems with fractional order deflection. J. Sound Vib. 326(3–5), 768–779 (2009)CrossRefGoogle Scholar
  23. 23.
    Cveticanin, L.: Oscillator with fraction order restoring force. J. Sound Vib. 320(4–5), 1064–1077 (2009)CrossRefGoogle Scholar
  24. 24.
    Lewis, G., Monasa, F.: Large deflections of cantilever beams of non-linear materials. Comput. Struct. 14, 357–360 (1981)CrossRefGoogle Scholar
  25. 25.
    Lee, K.: Large deflections of cantilever beams of non-linear elastic material under a combined loading. Int. J. Non-Linear Mech. 37, 439–443 (2002)CrossRefMATHGoogle Scholar
  26. 26.
    Bank, B., Zambon, S., Fontana, F.: A modal-based real-time piano synthesizer. IEEE Trans. Audio Speech Lang. Process. 18, 809–821 (2010)Google Scholar
  27. 27.
    Shatarat, N., Al-Sadder, S., Katkhuda, H., Qablan, H., Shatnawi, A.: Behavior of a rhombus frame of nonlinear elastic material under large deflection. Int. J. Mech. Sci. 51, 166–177 (2009)CrossRefMATHGoogle Scholar
  28. 28.
    Patten, W.N., Sha, S., Mo, C.: A vibration model of open celled polyurethane foam automotive seat cushions. J. Sound Vib. 217(1), 145–161 (1998)Google Scholar
  29. 29.
    Tripathy, M.C., Mondal, D., Biswas, K., Sen, S.: Experimental studies on realization of fractional inductors and fractional-order bandpass filters. Int. J. Circuit Theory Appl. (2014). doi: 10.1002/cta.2004
  30. 30.
    Kwuimy, C.A.K., Nbendjo, B.R.N.: Active control of horseshoes chaos in a driven Rayleigh oscillator with fractional order deflection. Phys. Lett. A 375(39), 3442–3449 (2011)CrossRefMATHGoogle Scholar
  31. 31.
    Kwuimy, C.A.K., Nbendjo, B.N., Woafo, P.: Optimization of electromechanical control of beam dynamics: analytical method and finite differences simulation. J. Sound Vib. 298(1–2), 180–193 (2006)CrossRefGoogle Scholar
  32. 32.
    Kwuimy, C.A.K., Woafo, P.: Dynamics, chaos and synchronization of self-sustained electromechanical systems with clamped-free flexible arm. Nonlinear Dyn. 53(3), 201–213 (2008)CrossRefMATHGoogle Scholar
  33. 33.
    Stanton, S.C., Owens, B.A., Mann, B.P.: Harmonic balance analysis of the bistable piezoelectric inertial generator. J. Sound Vib. 331(15), 3617–3627 (2012)CrossRefGoogle Scholar
  34. 34.
    Ducharne, B., Zhang, B., Guyomar, D., Sebald, G.: Fractional derivative operators for modeling piezoelectric polarization behaviours under dynamic mechanical stress excitation. Sensors Actuators A 189, 74–79 (2013)CrossRefGoogle Scholar
  35. 35.
    Nayfeh, A.H., Mook, D.: Nonlinear Oscillations. Wiley-Interscience, New York (1979)MATHGoogle Scholar
  36. 36.
    Chen, L., Zhu, W.: Stochastic jump and bifurcation of duffing oscillator with fractional derivative damping under combined harmonic and white noise excitations. Int. J. Non-Linear Mech. 46(10), 1324–1329 (2011)CrossRefGoogle Scholar
  37. 37.
    Leung, A., Guo, Z.: Forward residue harmonic balance for autonomous and non-autonomous systems with fractional derivative damping. Commun. Nonlinear Sci. Numer. Simul. 16(4), 2169–2183 (2011)CrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    Leung, A., Yang, H., Guo, Z.: The residue harmonic balance for fractional order van der Pol like oscillators. J. Sound Vib. 331(5), 1115–1126 (2012)CrossRefGoogle Scholar
  39. 39.
    Xiao, M., Zheng, W.X., Cao, J.: Approximate expressions of a fractional order van der pol oscillator by the residue harmonic balance method. Math. Comput. Simul. 89, 1–12 (2013)CrossRefMathSciNetGoogle Scholar
  40. 40.
    Litak, G., Borowiec, M.: On simulation of a bistable system with fractional damping in the presence of stochastic coherence resonance. Nonlinear Dyn. 77(3), 681–686 (2014)CrossRefMathSciNetGoogle Scholar
  41. 41.
    Shen, Y., Yang, S., Xing, H., Ma, H.: Primary resonance of Duffing oscillator with two kinds of fractional-order derivatives. Int. J. Non-Linear Mech. 47, 975–983 (2012)CrossRefGoogle Scholar
  42. 42.
    Ruzziconi, L., Litak, G., Lenci, S.: Nonlinear oscillations, transition to chaos and escape in the Duffing system with non-classical damping. J. Vib. Eng. 13, 22–38 (2011)Google Scholar
  43. 43.
    Litak, G., Borowiec, M., Friswell, M.I., Adhikari, S.: Energy harvesting in a magnetopiezoelastic system driven by random excitations with uniform and Gaussian distributions. J. Theor. Appl. Mech. 49, 757 (2011)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Center for Nonlinear Dynamics and Control, Department of Mechanical EngineeringVillanova UniversityVillanovaUSA
  2. 2.Faculty of Mechanical EngineeringTechnical University of LublinLublinPoland

Personalised recommendations