Koen Verheyden

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

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.

Computation and Stability Analysis of Solutions of Periodic Delay Differential Algebraic Equations

This paper is concerned with the numerical computation, continuation and stability analysis of periodic solutions of periodic delay differential algebraic equations. We consider systems with a time-periodic right hand side function and time-periodic delays. We introduce numerical algorithms based on collocation to compute periodic solutions and their stability. The presented methods combine knowledge from numerical methods for delay equations and differential algebraic equations. Our algorithms are illustrated with numerical results for two models.Copyright