Analysis and Modelling of Nonlinearties in Vibration Energy Harvesters

  • Peter Harte
  • Dimitri Galayko
  • Orla Feely
  • Elena BlokhinaEmail author


This chapter introduces the reader to a nonlinear mathematical model of an electrostatic vibration energy harvester and how it is developed. Semi-analytical techniques for analysing the dynamical behaviour and stability of the systems are introduced with particular emphasis on the harmonic balance method, the multiple scales method and the mechanical impedance method. These methods are compared and different ways to visualise and interpret the results of both numerical integration of the mathematical model, and the semi-analytical techniques are presented. Although the chapter is primarily focused on a gap closing eVEH operating in the constant charge mode, the material studied gives the reader the necessary tools to analyse and model any eVEH with nonlinearity present in its system.


Bifurcation Diagram Transducer Force Harmonic Balance Monodromy Matrix Mechanical Impedance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Meninger, S., Mur-Miranda, J., Amirtharajah, R., Chandrakasan, A., & Lang, J. (2001). Vibration-to-electric energy conversion. IEEE Transactions on Very Large Scale Integration (VLSI) Systems, 9(1), 64–76.CrossRefGoogle Scholar
  2. 2.
    Galayko, D., & Basset, P. (2011). A general analytical tool for the design of vibration energy harvesters (VEHs) based on the mechanical impedance concept. IEEE Transactions on Circuits and Systems I, 99, 299–311.MathSciNetCrossRefGoogle Scholar
  3. 3.
    Juillard, J. (2014). A comparative study of reduced-order modeling techniques for nonlinear mems beams. In DTIP 2014, pp. 261–265.Google Scholar
  4. 4.
    Nayfeh, A. H., & Balachandran, B. (2008). Applied nonlinear dynamics (Vol. 24). Wiley-VCH.Google Scholar
  5. 5.
    Nguyen, C. H., & Halvorsen, E. (2014). Harmonic-balance analysis of nonlinear energy harvester models In IEEE International Symposium on Circuits and Systems (ISCAS), pp. 2608–2611.Google Scholar
  6. 6.
    Nayfeh, A. (1993). Introduction to perturbation techniques. Wiley.Google Scholar
  7. 7.
    Blokhina, E., Galayko, D., Basset, P., & Feely, O. (2013). Steady-state oscillations in resonant electrostatic vibration energy harvesters. IEEE Transactions on Circuits and Systems I, 60, 875–884.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Blokhina, E., Fournier-Prunaret, D., Harte, P., Galayko, D., & Feely, O. (2013). Combined mechanical and circuit nonlinearities in electrostatic vibration energy harvesters. In Proceedings of the IEEE International Symposium on Circuits and Systems 2013, Beijing, China, 19–23 May 2013, 2013.Google Scholar
  9. 9.
    Blokhina, E., Galayko, D., Harte, P., Basset, P., & Feely, O. (2012). Limit on converted power in resonant electrostatic vibration energy harvesters. Applied Physics Letters, 101, 173904.CrossRefGoogle Scholar
  10. 10.
    Yen, B. C., & Lang, J. H. (2006). A variable-capacitance vibration-to-electric energy harvester. IEEE Transactions on Circuits and Systems I, 53, 288–295.CrossRefGoogle Scholar
  11. 11.
    Shu, Y., & Lien, I. (2006). Efficiency of energy conversion for a piezoelectric power harvesting system. Journal of Micromechanics and Microengineering, 16, 2429.CrossRefGoogle Scholar
  12. 12.
    Levitan, E. (1960). Forced oscillation of a spring-mass system having combined coulomb and viscous damping. The Journal of the Acoustical Society of America, 32, 1265.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Harte, P., Blokhina, E., Feely, O., Fournier-Prunaret, D., & Galayko, D. (2014). Electrostatic vibration energy harvesters with linear and nonlinear resonators. International Journal of Bifurcation and Chaos, 24(11), 1430030.CrossRefzbMATHGoogle Scholar
  14. 14.
    Hilborn, R. C. (2000). Chaos and nonlinear dynamics: An introduction for scientists and engineers. Oxford: Oxford University Press.
  15. 15.
    Blokhina, E., Galayko, D., Wade, R., Basset, P., & Feely, O. (2012). Bifurcations and chaos in electrostatic vibration energy harvesters. In Proceedings of the IEEE International Symposium on Circuits and Systems 2012, Seoul, Korea, 20–24 May 2012, 2012, pp. 397–400.Google Scholar
  16. 16.
    Galayko, D., Guillemet, R., Dudka, A., Basset, P. (2011). Comprehensive dynamic and stability analysis of electrostatic vibration energy harvester (E-VEH). In Proceedings of International Conference on Solid-State Sensors, Actuators and Microsystems (TRANSDUCERS), 2011, pp. 2382–2385.Google Scholar
  17. 17.
    Giaouris, D., Banerjee, S., Zahawi, B., & Pickert, V. (2008). Stability analysis of the continuous-conduction-mode buck converter via Filippov’s method. IEEE Transactions on Circuits and Systems I: Regular Papers, 55(4), 1084–1096.MathSciNetCrossRefGoogle Scholar
  18. 18.
    Filippov, A. (1998). Differential equations with discontinuous righthand sides: Control systems. Springer.Google Scholar
  19. 19.
    Utkin, V. (1992). Sliding modes in control optimization. Springer.Google Scholar
  20. 20.
    Leine, R., & Nijmeijer, H. (2006). Dynamics and bifurcations of non-smooth mechanical systems. Springer.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Peter Harte
    • 1
  • Dimitri Galayko
    • 1
  • Orla Feely
    • 1
  • Elena Blokhina
    • 1
    Email author
  1. 1.University College DublinDublinIreland

Personalised recommendations