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

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.