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On the use of the theory of dynamical systems for transient problems

A preliminary work on a simple model

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Abstract

This paper is a preliminary work to address the problem of dynamical systems with parameters varying in time. An idea to predict their behavior is proposed. These systems are called transient systems, and are distinguished from steady systems in which parameters are constant. In particular, in steady systems the excitation is either constant (e.g., nought) or periodic with amplitude, frequency, and phase angle which do not vary in time. We apply our method to systems, which are subjected to a transient excitation that is neither constant nor periodic. The effect of switching-off and full-transient forces is investigated. The former can be representative of switching-off procedures in machines; the latter can represent earthquake vibrations, wind gusts, etc., acting on a mechanical system. This class of transient systems can be seen as the evolution of an ordinary steady system into another ordinary steady system, for both of which the classical theory of dynamical systems holds. The evolution from a steady system to the other is driven by a transient force, which is regarded as a map between the two steady systems.

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Acknowledgements

U.G. would like to acknowledge the financial support from the Italian Ministry of Education, Universities and Research (MIUR) under the PRIN program 2010/11 N. 2010MBJK5B, Dynamics, Stability, and Control of Flexible Structures.

L.M. would like to thank Giulio Ghirardo (Department of Engineering, University of Cambridge) for his help with vector graphics.

Finally, both authors would like to thank the anonymous reviewers for helpful comments.

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Authors and Affiliations

  1. Dipartimento di Ingegneria Industriale, Università di Padova, Via Marzolo 9, 35131, Padova, Italy

    Ugo Galvanetto

  2. Department of Engineering, University of Cambridge, Trumpington Street, CB2 1PZ, Cambridge, UK

    Luca Magri

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  1. Ugo Galvanetto
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  2. Luca Magri
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Correspondence to Luca Magri.

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Galvanetto, U., Magri, L. On the use of the theory of dynamical systems for transient problems. Nonlinear Dyn 74, 373–380 (2013). https://doi.org/10.1007/s11071-013-0976-7

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  • DOI: https://doi.org/10.1007/s11071-013-0976-7

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