Extreme Instability Phenomena in Autonomous Weakly Damped Systems: Hopf Bifurcations, Double Pure Imaginary Eigenvalues, Load Discontinuity

Abstract

The dynamic asymptotic instability of autonomous multi-parameter discrete systems under step compressive loading either of constant direction (conservative load) or of varying direction (follower or nonconservative loading) is thoroughly reconsidered using the efficient - and rather forgotten - Liénard-Chipart stability criterion. Attention is focused on the interaction of nonuniform mass and stiffness distribution with infinitesimal damping. Such parameters alone or combined with others may have a tremendous effect on the Jacobian eigenvalues and thereafter on the local asymptotic dynamic instability which – strangely enough –may occur before static (divergence) instability, even in the case of a positive definite damping matrix. It was also found that such systems when unloaded, although being statically stable, under certain conditions may become dynamically locally unstable to any small disturbance. Hopf bifurcations, double zero eigenvalues, double pure imaginary eigenvalues, loading discontinuity and other phenomena are properly established.