DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

A TWO-DEGREES-OF-FREEDOM HAMILTONIAN MODEL:
AN ANALYTICAL AND NUMERICAL STUDY
RAFFAELLA PAVANI
A well-studied Hamiltonian model is revisited. It is shown that known numerical results
are to be considered unreliable, because they were obtained by means of numerical methods
unsuitable for Hamiltonian systems. Moreover, some analytical results are added.
Copyright © 2006 Raffaella Pavani. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Our aim is to study a well-known structural engineering problem about anomalous
elastic-plastic responses of a two-degrees-of-freedom model of a fixed ended beam with
short pulse loading (e.g., [2] and references therein). In particular, the resulting elastic
vibrations may be chaotic. We will tackle this problem mainly from the point of view of
numerical analysis, but we provide even some new theoretical results.
This system was already extensively studied using Runge-Kutta methods with variable
stepsize and many results can be found in [2] and references therein, but here we want to
show that some other numerical methods can be more effective in order to understand
the qualitative behavior of the orbits, in particular when chaotic behavior is detected.
Moreover, some new analytical results support our conclusions.
In Section 2, the problem is described and equations of the used mathematical model
are provided. In Section 3, we present some theoretical results about the behavior of solutions
close to the equilibrium point. In Section 4, numerical results are reported and
comparisons with already known results are shown. At last, Section 5 is devoted to a final
discussion.
2. Beam mathematical model
The two-degrees-of-freedom model of a fixed ended beam is provided by Carini et al. [2].
This model was deeply studied in the field of structural engineering and enjoys a large literature,
which we do not cite here for the sake of brevity (e.g., see references in [2]). In
particular it is known that a beam, deformed into the plastic range by a short transverse
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 905–913