Mathematical Problems in Engineering
Volume 2008 (2008), Article ID 471080, 25 pages
The Effect of Infinitesimal Damping on the Dynamic Instability Mechanism of Conservative Systems
1Department of Civil Engineering, University of Thessaly, Pedion Areos, 38 334 Volos, Greece
2Laboratory of Metal Structures, Department of Civil Engineering, National Technical University of Athens, Zografou Campus, 157 80 Athens, Greece
3Research Center Pure and Applied Mathematics, Academy of Athens, Soranou Efessiou 4, 115 27 Athens , Greece
Received 17 December 2007; Accepted 19 February 2008
Academic Editor: Jose Balthazar
Copyright © 2008 Dimitris S. Sophianopoulos et al. 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.
The local instability of 2 degrees of freedom (DOF) weakly damped systems is thoroughly discussed using the Liénard-Chipart stability criterion. The individual and coupling effect of mass and stiffness distribution on the dynamic asymptotic stability due to mainly infinitesimal damping is examined. These systems may be as follows: (a) unloaded (free motion) and (b) subjected to a suddenly applied load of constant magnitude and direction with infinite duration (forced motion). The aforementioned parameters combined with the algebraic structure of the damping matrix (being either positive semidefinite or indefinite) may have under certain conditions a tremendous effect on the Jacobian eigenvalues and then on the local stability of these autonomous systems. It was found that such systems when unloaded may exhibit periodic motions or a divergent motion, while when subjected to the above step load may experience either a degenerate Hopf bifurcation or periodic attractors due to a generic Hopf bifurcation. Conditions for the existence of purely imaginary eigenvalues leading to global asymptotic stability are fully assessed. The validity of the theoretical findings presented herein is verified via a nonlinear dynamic analysis.