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Mathematical and Computational Tools for the Stability Analysis of Time-Varying Delay Systems and Applications in Mechanical Engineering

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Applications of Time Delay Systems

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 352))

Summary

An overview of eigenvalue based tools for the stability analysis of linear periodic systems with delays is presented. It is assumed that both the system matrices and the delays are periodically varying. First the situation is considered where the time-variation of the periodic terms is fast compared to the system’s dynamics. Then averaging techniques are used to relate the stability properties of the time-varying system with these of a time-invariant one, which opens the possibility to use frequency domain tools. As a special characteristic the averaged system exhibits distributed delays if the delays in the original system are time-varying. Both analytic and numerical tools for the stability analysis of the averaged system are discussed. Special attention is paid to the characterization of situations where a variation of a delay has a stabilizing effect. Second, the assumption underlying the averaging approach is dropped. It is described how exact stability information of the original, periodic system can be directly computed. The two approaches are briefly compared with respect to generality, applicability and computational efficiency. Finally the results are illustrated by means of two examples from mechanical engineering. The first example concerns a model of a variable speed rotating cutting tool. Based on the developed theory and using the described computational tools, both a theoretical explanation and a quantitative analysis are provided of the beneficial effect of a variation of the machine speed on enhancing stability properties, which was reported in the literature. The second example concerns the stability analysis of an elastic column, subjected to a periodic force.

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Author information

Authors and Affiliations

  1. Department of Computer Science, K.U. Leuven, Celestijnenlaan 200A, B-3001, Heverlee, Belgium

    Wim Michiels & Koen Verheyden

  2. HEUDIASYC (UMR CNRS 6599), Université de Technologie de Compiègne, Centre de Recherche de Royallieu, BP 20529, 60205, Compiègne, cedex, France

    Silviu-Iulian Niculescu

Authors
  1. Wim Michiels
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  2. Koen Verheyden
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  3. Silviu-Iulian Niculescu
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Editor information

Editors and Affiliations

  1. ECE Department, Boise State University, 1910 University Dr., Boise, ID, 83725-2100, USA

    John Chiasson

  2. UMR CNRS 6597, Ecole Centrale de Nantes, IRCCyN, 1 rue de la Noë, BP 92101, 44321, Nantes, France

    Jean Jacques Loiseau

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Michiels, W., Verheyden, K., Niculescu, SI. (2007). Mathematical and Computational Tools for the Stability Analysis of Time-Varying Delay Systems and Applications in Mechanical Engineering. In: Chiasson, J., Loiseau, J.J. (eds) Applications of Time Delay Systems. Lecture Notes in Control and Information Sciences, vol 352. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-49556-7_13

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  • DOI: https://doi.org/10.1007/978-3-540-49556-7_13

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-49555-0

  • Online ISBN: 978-3-540-49556-7

  • eBook Packages: EngineeringEngineering (R0)

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