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Vibratory energy exchange between a linear and a nonsmooth system in the presence of the gravity

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Abstract

The concept of the vibratory energy transfer between a linear master DOF and a nonsmooth nonlinear energy sink (NES) in the presence of gravity forces is studied. Different invariant manifolds of the system at different time scales are revealed and necessary conditions for leading the behavior of the system to strongly modulated response (SMR) are enlightened.

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Acknowledgements

The authors would like to thank the “PSA Peugeot Citroën Automobiles” for supporting this research work.

Author information

Authors and Affiliations

  1. École Nationale des Travaux Publics de l’État, Université de Lyon, DGCB and LTDS, UMR CNRS 5513, 3 rue Maurice Audin, 69518, Vaulx-en-Velin Cedex, France

    Alireza Ture Savadkoohi & Claude-Henri Lamarque

  2. PSA Peugeot Citroën Automobiles, DRIA/DSTF/MSMX, route de Gisy, 78943, Vélizy-Villacoublay Cedex, France

    Zoran Dimitrijevic

Authors
  1. Alireza Ture Savadkoohi
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  2. Claude-Henri Lamarque
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  3. Zoran Dimitrijevic
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Correspondence to Alireza Ture Savadkoohi.

Appendix

Appendix

Let us assume the illustrated nonsmooth NES behavior in Fig. 10. In the general case, we have:

(52)

We are interested to evaluate the zero and the first Fourier coefficients of the system which are represented in (12) and (13). If \({\frac{b_{2}}{\omega}}+{\frac{|\varphi_{2}|}{\omega}} \leq\delta\), then f f =0 and f z =0. We have:

(53)
(54)
Fig. 10
figure 10

Schematics of the behavior of the nonsmooth NES with respect to the time in a loop

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Where,

(55)
(56)
(57)
(58)

Integrals of Eqs. (54) and (53) give systems of (15) and (16) (\(f_{f}=-{\frac{i \varphi_{2}}{2}}G_{f} (|\varphi_{2}|^{2} )\)).

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Ture Savadkoohi, A., Lamarque, CH. & Dimitrijevic, Z. Vibratory energy exchange between a linear and a nonsmooth system in the presence of the gravity. Nonlinear Dyn 70, 1473–1483 (2012). https://doi.org/10.1007/s11071-012-0548-2

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  • DOI: https://doi.org/10.1007/s11071-012-0548-2

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