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Transformations from Variable Delays to Constant Delays with Applications in Engineering and Biology

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Time Delay Systems

Part of the book series: Advances in Delays and Dynamics ((ADVSDD,volume 7))

Abstract

The transformation from systems with time-varying delays or state-dependent delays to systems with constant delays is studied. The transformation exists if the delay is defined by a transport with a variable velocity over a constant distance. In fact, time-varying or state-dependent delays, which are generated by such a mechanism, are common in engineering and biology. We study two paradigmatic time delay systems in more detail. For metal cutting processes, or more precisely turning with spindle speed variation, we show that the analysis of the machine tool vibrations via constant delays in terms of the spindle rotation angle is more advantageous than the conventional analysis of the vibrations with time-varying delays in the time domain. In a second example, motivated by the McKendrick equation modeling structured populations, we show that systems with variable delays and the equivalent systems with constant delays are, in addition, equivalent to partial differential equations with moving and constant boundaries.

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Notes

  1. 1.

    There is a specific variable transport delay block in the MATLAB/Simulink software environment for such type of delays.

  2. 2.

    No delayed values of \(x_\tau (t)\) appear in the DDE system, and therefore, only an initial value \(x_\tau (0)\) and no initial function must be specified for the additional component \(x_\tau (t)\).

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Authors and Affiliations

  1. Institute of Physics, Chemnitz University of Technology, 09107, Chemnitz, Germany

    Andreas Otto & Günter Radons

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  1. Andreas Otto
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  2. Günter Radons
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Correspondence to Andreas Otto .

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Editors and Affiliations

  1. Department of Applied Mechanics, Budapest University of Technology and Economics, Budapest, Hungary

    Tamás Insperger

  2. Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan, USA

    Tulga Ersal

  3. Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan, USA

    Gábor Orosz

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Otto, A., Radons, G. (2017). Transformations from Variable Delays to Constant Delays with Applications in Engineering and Biology. In: Insperger, T., Ersal, T., Orosz, G. (eds) Time Delay Systems. Advances in Delays and Dynamics, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-319-53426-8_12

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  • DOI: https://doi.org/10.1007/978-3-319-53426-8_12

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  • Print ISBN: 978-3-319-53425-1

  • Online ISBN: 978-3-319-53426-8

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