Abstract
The paradox of destabilization of a conservative or non-conservative system by small dissipation, or Ziegler’s paradox (1952), has stimulated an interest in the sensitivity of reversible and Hamiltonian systems with respect to dissipative perturbations. We discuss the motion of a particle in Brouwer’s rotating vessel, a typical gyroscopic system, that has an unstable equilibrium caused by internal damping for a wide range of rotation velocities. Using quasi-periodic averaging-normalization by Mathematica, we find that modulation of the rotation frequency in the cases of single-well and saddle equilibria stabilizes the system for a number of combination resonances, thus producing quenching of the unstable motion.
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Acknowledgments
O.N. Kirillov drew my attention to the papers by Brouwer [5] and Bottema [3]. In a pleasant cooperation with Theo Ruijgrok, Mathematica was used to handle the normalization procedure in this paper.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Verhulst, F. Brouwer’s rotating vessel I: stabilization. Z. Angew. Math. Phys. 63, 727–736 (2012). https://doi.org/10.1007/s00033-011-0177-5
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DOI: https://doi.org/10.1007/s00033-011-0177-5