Nonlinear Dynamics

, Volume 69, Issue 3, pp 1063–1079 | Cite as

On intentional introduction of stiffness nonlinearities for energy harvesting under white Gaussian excitations

  • Mohammed F. DaqaqEmail author
Original Paper


A significant body of the open literature on vibratory energy harvesting is currently focused on the concept of purposeful inclusion of stiffness nonlinearities for broadband transduction. When compared to their linear resonant counterparts, nonlinear energy harvesters have a wider steady-state frequency bandwidth, leading to the idea that they can be utilized to improve performance especially in random and non-stationary vibratory environments. To further investigate this common belief, this paper studies the response of vibratory energy harvesters to white Gaussian excitations. Both mono- and bi-stable piezoelectric Duffing-type harvesters are considered. The Fokker–Plank–Kolmogorov equation governing the evolution of the system’s transition probability density function is formulated and used to generate the moment differential equations governing the response statistics. The moment equations are then closed using a fourth-order cumulant-neglect closure scheme and the relevant steady-state response statistics are obtained. It is demonstrated that the energy harvester’s time constant ratio, i.e., the ratio between the nominal period of the mechanical subsystem and the time constant of the harvesting circuit, plays a critical role in characterizing the performance of nonlinear harvesters in a random environment. When the time constant ratio is large, stiffness-type nonlinearities have very little influence on the voltage response. In such a case, no matter how the potential function of the harvester is altered, it does not affect the average output power of the device. When the time constant ratio is small, the influence of the nonlinearity on the voltage output becomes more prevalent. In this case, a Duffing-type mono-stable harvester can never outperform its linear counterpart. A bi-stable harvester, on the other hand, can outperform a linear harvester only when it is designed with the proper potential energy function based on the known noise intensity of the excitation. Such conclusions hold for harvesters with nonlinearities appearing in the restoring force.


Energy harvesting Random Nonlinear White 



This material is based upon work supported by the National Science Foundation under CAREER Grant No. 1055419. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.


  1. 1.
    Roundy, S., Wright, P.K., Rabaey, J.: A study of low level vibrations as a power source for wireless sensor nodes. Comput. Commun. 26, 1131–1144 (2003) CrossRefGoogle Scholar
  2. 2.
    Roundy, S., Zhang, Y.: Toward self-tuning adaptive vibration-based micro-generators. In: Smart Materials, Nano- and Micro-Smart Systems, Sydney, Australia (2005) Google Scholar
  3. 3.
    Wu, W., Chen, Y., Lee, B., He, J., Peng, Y.: Tunable resonant frequency power harvesting devices. In: Proceedings of Smart Structures and Materials Conference, SPIE, San Diego, CA, p. 61690A (2006) Google Scholar
  4. 4.
    Challa, V., Prasad, M., Shi, Y., Fisher, F.: A vibration energy harvesting device with bidirectional resonance frequency tunability. Smart Mater. Struct. 75, 1–10 (2008) Google Scholar
  5. 5.
    Shahruz, S.M.: Design of mechanical band-pass filters for energy scavenging. J. Sound Vib. 292, 987–998 (2006) CrossRefGoogle Scholar
  6. 6.
    Shahruz, S.M.: Limits of performance of mechanical band-pass filters used in energy harvesting. J. Sound Vib. 294, 449–461 (2006) CrossRefGoogle Scholar
  7. 7.
    Baker, J., Roundy, S., Wright, P.: Alternative geometries for increasing power density in vibration energy scavenging for wireless sensors. In: Proceedings of the Third International Energy Conversion Conference, San Francisco, CA, pp. 959–970 (2005) Google Scholar
  8. 8.
    Rastegar, J., Pereira, C., Nguyen, H.L.: Piezoelectric-based power sources for harvesting energy from platforms with low frequency vibrations. In: Proceedings of Smart Structures and Materials Conference, SPIE, San Diego, CA, p. 617101 (2006) Google Scholar
  9. 9.
    McInnes, C.R., Gorman, D.G., Cartmell, M.P.: Enhanced vibrational energy harvesting using nonlinear stochastic resonance. J. Sound Vib. 318, 655–662 (2008) CrossRefGoogle Scholar
  10. 10.
    Barton, D., Burrow, S., Clare, L.: Energy harvesting from vibrations with a nonlinear oscillator. J. Vib. Acoust. 132, 0210091 (2010) CrossRefGoogle Scholar
  11. 11.
    Mann, B., Sims, N.: Energy harvesting from the nonlinear oscillations of magnetic levitation. J. Sound Vib. 319, 515–530 (2008) CrossRefGoogle Scholar
  12. 12.
    Masana, R., Daqaq, M.F.: Electromechanical modeling and nonlinear analysis of axially-loaded energy harvesters. J. Vib. Acoust. 133, 011007 (2011) CrossRefGoogle Scholar
  13. 13.
    Quinn, D., Triplett, L., Vakakis, D., Bergman, L.: Comparing linear and essentially nonlinear vibration-based energy harvesting. J. Vib. Acoust. 133, 011001 (2011) CrossRefGoogle Scholar
  14. 14.
    Erturk, A., Hoffman, J., Inman, D.J.: A piezo-magneto-elastic structure for broadband vibration energy harvesting. Appl. Phys. Lett. 94, 254102 (2009) CrossRefGoogle Scholar
  15. 15.
    Cottone, F., Vocca, H., Gammaitoni, L.: Nonlinear energy harvesting. Phys. Rev. Lett. 102, 080601 (2009) CrossRefGoogle Scholar
  16. 16.
    Daqaq, M.F., Stabler, C., Seuaciuc-Osorio, T., Qaroush, Y.: Investigation of power harvesting via parametric excitations. J. Intell. Mater. Syst. Struct. 20, 545–557 (2009) CrossRefGoogle Scholar
  17. 17.
    Stanton, S.C., McGehee, C.C., Mann, B.P.: Nonlinear dynamics for broadband energy harvesting: investigation of a bistable piezoelectric inertial generator. Physica D, Nonlinear Phenom. 239, 640–653 (2010) zbMATHCrossRefGoogle Scholar
  18. 18.
    Daqaq, M.F., Bode, D.: Exploring the parametric amplification phenomenon for energy harvesting. J. Syst. Control Eng. 225, 456–466 (2010) Google Scholar
  19. 19.
    Abdelkefi, A., Nayfeh, A.H., Hajj, M.: Global nonlinear distributed-parameter model of parametrically excited piezoelectric energy harvesters. Nonlinear Dyn. 67(2), 1147–1160 (2011) CrossRefGoogle Scholar
  20. 20.
    Abdelkefi, A., Nayfeh, A.H., Hajj, M.: Effects of nonlinear piezoelectric coupling on energy harvesters under direct excitation. Nonlinear Dyn. 67(2), 1221–1232 (2011) CrossRefGoogle Scholar
  21. 21.
    Mann, B.P., Owens, B.A.: Investigations of a nonlinear energy harvester with a bistable potential well. J. Sound Vib. 329, 1215–1226 (2010) CrossRefGoogle Scholar
  22. 22.
    Moon, F.C., Holmes, P.J.: A magnetoelastic strange attractor. J. Sound Vib. 65, 275 (2009) CrossRefGoogle Scholar
  23. 23.
    Masana, R., Daqaq, M.F.: Relative performance of a vibratory energy harvester in mono- and bi-stable potentials. J. Sound Vib. (2011). doi: 10.1016/j.jsv.2011.07.031 Google Scholar
  24. 24.
    Masana, R., Daqaq, M.F.: Exploiting super-harmonic resonances of a bi-stable axially-loaded beam for energy harvesting under low-frequency excitations. In: Proceedings of the ASME 2011 International Design Engineering Technical Conference and Computers and Information in Engineering Conference, Washington, DC (2011) Google Scholar
  25. 25.
    Adhikari, S., Friswell, M.I., Inman, D.J.: Piezoelectric energy harvesting from broadband random vibrations. Smart Mater. Struct. 18, 115005 (2009) CrossRefGoogle Scholar
  26. 26.
    Seuaciuc-Osorio, T., Daqaq, M.F.: Energy harvesting under excitations of time-varying frequency. J. Sound Vib. 329, 2497–2515 (2010) CrossRefGoogle Scholar
  27. 27.
    Barton, D., Burrow, S., Clare, L.: Energy harvesting from vibrations with a nonlinear oscillator. In: Proceedings of the ASME 2009 International Design Engineering Technical Conference and Computers and Information in Engineering Conference, San Diego, CA (2009) Google Scholar
  28. 28.
    Daqaq, M.F.: Response of uni-modal duffing type harvesters to random forced excitations. J. Sound Vib. 329, 3621–3631 (2010) CrossRefGoogle Scholar
  29. 29.
    Gammaitoni, L., Neri, I., Vocca, H.: Nonlinear oscillators for vibration energy harvesting. Appl. Phys. Lett. 94, 164102 (2009) CrossRefGoogle Scholar
  30. 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) CrossRefGoogle Scholar
  31. 31.
    Renno, J., Daqaq, M.F., Inman, D.J.: On the optimal energy harvesting from a vibration source. J. Sound Vib. 320, 386–405 (2009) CrossRefGoogle Scholar
  32. 32.
    Ito, K.: Stochastic integral. Proc. Imp. Acad. (Tokyo) 20, 519–524 (1944) MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Jazwinski, A.H.: Stochastic Processes and Filtering Theory. Academic Press, San Diego (1970) zbMATHGoogle Scholar
  34. 34.
    Ibrahim, R.A.: Parametric Random Vibrations. Research Studies Press, New York (1985) Google Scholar
  35. 35.
    Wojtkiewicz, S., Spencer, B., Bergman, L.A.: On the cumulant-neglect closure method in stochastic dynamics. Int. J. Non-Linear Mech. 95, 657–684 (1995) Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Nonlinear Vibrations and Energy Harvesting Lab. (NOVEHL), Department of Mechanical EngineeringClemson UniversityClemsonUSA

Personalised recommendations