On the behaviour of a
parametric vibrating system with a limited power supply defined on
V.O.Kononenko
The study of non-ideal dynamical systems when the excitation is
influenced by the response of the system has been considered a major challenge
in theoretical and practical engineering science research.
When the excitation is not influenced by the response, it is said to be
an ideal excitation or an ideal source of energy. On the other hand, when the
excitation is influenced by the response of the system, it is said to be
non-ideal. Then, depending of the excitation, one refers to dynamical systems
as ideal or non-ideal.
The behaviour of the ideal dynamical systems is well known in the
current literature, but there are few results on non-ideal dynamical systems.
Generally, non-ideal dynamical systems are those for which the powers supply are limited.
The behaviour of the dynamical systems departs from the ideal, as power
supply becomes more limited. For a non-ideal dynamical system, one must add an
equation that describes how the energy source “supplies the energy” to the equations that governs the
corresponding ideal dynamical system and the response is unknown. Thus, as a
first characteristic, the non-ideal dynamical system has one more degree of
freedom than its ideal counterpart.
In this paper we
discuss some topological properties of parametric non-ideal vibrations of a dynamical system, using Phase Portraits and associated
Poincaré Maps.
This non-ideal dynamical model was mentioned by Kononenko [Kononenko,
1969] but these topological properties were not included in this classical
book.
Kononenko, V. O.
Vibrating Systems with a Limited
Power Supply
London Illife Books LTD, pp. 236,
1969 (In Russian :1959)