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On the effect of damping on the stabilization of mechanical systems via parametric excitation

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Abstract

The effect of damping on the re-stabilization of statically unstable linear Hamiltonian systems, performed via parametric excitation, is studied. A general multi-degree-of-freedom mechanical system is considered, close to a divergence point, at which a mode is incipiently stable and n − 1 modes are (marginally) stable. The asymptotic dynamics of system is studied via the Multiple Scale Method, which supplies amplitude modulation equations ruling the slow flow. Several resonances between the excitation and the natural frequencies, of direct 1:1, 1:2, 2:1, or sum and difference combination types, are studied. The algorithm calls for using integer or fractional asymptotic power expansions and performing nonstandard steps. It is found that a slight damping is able to increase the performances of the control system, but only far from resonance. Results relevant to a sample system are compared with numerical findings based on the Floquet theory.

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Authors and Affiliations

  1. Saint Petersburg State University, St. Petersburg, Russia

    Inga M. Arkhipova

  2. M&MoCS, University of L’Aquila, Via Giovanni Gronchi 18, 67100, L’Aquila, Italy

    Angelo Luongo

Authors
  1. Inga M. Arkhipova
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  2. Angelo Luongo
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Correspondence to Angelo Luongo.

Additional information

This work was partially granted by the Italian Ministry of University and Research (MIUR), under the PRIN10-11 program, project No. 2010MBJK5B.

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Arkhipova, I.M., Luongo, A. On the effect of damping on the stabilization of mechanical systems via parametric excitation. Z. Angew. Math. Phys. 67, 69 (2016). https://doi.org/10.1007/s00033-016-0659-6

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  • DOI: https://doi.org/10.1007/s00033-016-0659-6

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