Chaotic and Regular Motion in Nonlinear Vibrating Systems
Studies of phenomena arising in nonlinear oscillators are often modelled by an equation of the form where and ψ(x) are approximated by finite Taylor series, and represents a dissipative term. Such a system has an extensive literature. A now classical appr
- PDF / 5,653,232 Bytes
- 86 Pages / 481.89 x 691.654 pts Page_size
- 54 Downloads / 192 Views
W. Szemplinska-Stupnicka Institute of Fundamental Technological Research, Warsaw, Poland
1. Introduction Studies of phenomena arising in nonlinear oscillators are often modelled by an equation of the form
x+
g(i) + 1/J(x) = f cos
Wt
where g(i) and 1/l(x) are approximated by finite Taylor series, and g(i) represents a dissipative term. Such a system has an extensive literature. A ,now classical approach to the study of the system behaviour, such as that presented in the popular book by Hayashi [8 J, is the theoretical analysis based on approximate analytical methods with experimental verification employing electric circuits or electronic computers. In these studies the system is assumed to tend to steady-state oscillation when started with any initial conditions and steady-state solutions are often the main point of interest. Approximate analytical solutions describing various types of resonances and analysis of local stability of the solutions and their domains of attraction provided us with a great deal of knowledge about the system behaviour. The results show a variety of nonlinear phenomena such as: principal, sub, ultra and subultra harmonic resonances and jump phenomena associated with stability limits on resonance curves, which ·seem to leave no room for any irregular, random-like and unpredictable solutions in the deterministic systems. Although chaotic motion in simple deterministic dynamic systems have attracted a great deal of attention in the last decade, results showing "strange attractors" in as classical a vibrating system as that governed by Duffing equation,were a great surprise [ 26-28].
W. Szemplińska-Stupnicka et al., Chaotic Motions in Nonlinear Dynamical Systems © Springer-Verlag Wien 1988
52
W. Szemplinska-Stupnicka
It is pretty obvious that direct applications of the approximate theory of nonlinear vibrations to theoretical study of chaotic motion is impossible and so the return to qualitative topological methods seemed to be the only alternative. Nevertheless one might be tempted to seek a link between the phenomena of nonlinear resonances determined by low order approximate solutions and irregular solutions obtained by computer simulation in order to see the chaotic zones against the background of the classical concepts of resonance curves, stability limits, and jump phenomena. Results on chaotic behaviour obtained by computer simulation allow us to make observ~tions ,which make such an idea an appealing one: one can readily notice that: chaotic motion appears as a transition zone between sub or subultra mT-periodic resonance and the principal T-periodic resonance, chaotic motion often borders and coexists with periodic motion, the motion which can be described by low order approximate solutions. The idea of interpreting and studying regions of chaotic motion from the point of view of the approximate theory of nonlinear vibrations brings a great number of interesting questions: where are the chaotic motion zones located relative to the known phenomena of principal and subharmonic
Data Loading...
