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

Abstract

This paper is a preliminary work to address the problem of dynamical systems with parameters varying in time. An idea to predict their behaviour is proposed. These systems are called \emph{transient systems}, and are distinguished from \emph{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, which 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.