The Delayed van der Pol Oscillator and Energy Harvesting

  • Zakaria Ghouli
  • Mustapha Hamdi
  • Mohamed BelhaqEmail author
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 228)


In the first part of the chapter, we present some results on quasi-periodic (QP) vibration-based energy harvesting (EH) in a delayed van der Pol oscillator with modulated delay amplitude. Two examples are considered which include a delayed van der Pol harvester coupled either to a delayed or undelayed electromagnetic sub- system. The influence of delay parameters on the performance of the harvester has been examined. It is shown that a maximum amplitude of the response does not induce necessarily a maximum output power. In the second part, we investigate QP vibration-based EH in the case where the van der Pol oscillator is subjected to external harmonic excitation and coupled to a delayed piezoelectric component. Perturbation method is applied near a resonance to obtain approximation of the periodic and QP responses as well as the amplitude of the harvested powers. To guarantee the robustness of the QP vibration during energy extraction operation, a stability anal- ysis is performed and the QP stability chart is determined. Results show that in the presence of time delay in the electrical circuit of the excited van der Pol oscillator, it is possible to harvest energy from QP vibrations with a good performance over a broadband of system parameters away from the resonance. Numerical simulations are conducted to support the analytical predictions.


Energy harvesting Van der Pol oscillator Quasi-periodic vibrations Delayed piezoelectric coupling 


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Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Zakaria Ghouli
    • 1
  • Mustapha Hamdi
    • 2
  • Mohamed Belhaq
    • 1
    Email author
  1. 1.Faculty of Sciences Ain ChockUniversity Hassan II CasablancaCasablancaMorocco
  2. 2.University Mohammed I Oujda, FST-Al HoceimaAl-HoceimaMorocco

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