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Energy harvesting in a Mathieu–van der Pol–Duffing MEMS device using time delay

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Abstract

This paper investigates quasi-periodic vibration-based energy harvesting in a delayed nonlinear MEMS device consisting of a delayed Mathieu–van der Pol–Duffing type oscillator coupled to a delayed piezoelectric coupling mechanism. We use the multiple scales method to approximate the quasi-periodic response and the related power output near the principal parametric resonance. The effect of time delay on the energy harvesting performance is studied. It is shown that for appropriate combination of time delay parameters, there exists an optimum range of excitation frequency beyond the resonance where quasi-periodic vibration-based energy harvesting is maximum. Numerical simulations are performed to confirm the analytical predictions.

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Authors and Affiliations

  1. Faculty of Sciences Aïn Chock, University Hassan II of Casablanca, Casablanca, Morocco

    Mohamed Belhaq & Zakaria Ghouli

  2. Faculty of Science and Technology-Al Hoceima, University Mohammed I, Oujda, Morocco

    Mustapha Hamdi

Authors
  1. Mohamed Belhaq
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  2. Zakaria Ghouli
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  3. Mustapha Hamdi
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Corresponding author

Correspondence to Mohamed Belhaq.

Appendix

Appendix

$$\begin{aligned} S_{1}= & {} \frac{\alpha }{2}-\frac{\chi \kappa (2\lambda -2\alpha _{2}\cos (\frac{\omega \tau _{2}}{2}))}{(2\lambda -2\alpha _{2}\cos (\frac{\omega \tau _{2}}{2}))^{2}+(\omega +2\alpha _{2}\sin (\frac{\omega \tau }{2}))^{2}}\nonumber \\&-\frac{\alpha _{1}}{\omega }\sin \left( \frac{\omega \tau _{1}}{2}\right) ,~~~~S_{2}=\frac{-\beta }{8}, ~~~~S_{3}=-\frac{h}{2\omega }\\ S_{4}= & {} \frac{\sigma }{\omega }+\frac{\chi \kappa (\omega +2\alpha _{2}\sin (\frac{\omega \tau _{2}}{2}))}{(2\lambda -2\alpha _{2}\cos (\frac{\omega \tau _{2}}{2}))^{2}+(\omega +2\alpha _{2}\sin (\frac{\omega \tau _{2}}{2}))^{2}}\nonumber \\&-\frac{\alpha _{1}}{\omega }\cos \left( \frac{\omega \tau _{1}}{2}\right) ,~~~~S_{5}=\frac{3\gamma }{4\omega }\ \end{aligned}$$

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Belhaq, M., Ghouli, Z. & Hamdi, M. Energy harvesting in a Mathieu–van der Pol–Duffing MEMS device using time delay. Nonlinear Dyn 94, 2537–2546 (2018). https://doi.org/10.1007/s11071-018-4508-3

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  • DOI: https://doi.org/10.1007/s11071-018-4508-3

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