On the failure of static stability analyses of nonconservative systems in regions of divergence instability

Abstract

The stability of perfect bifurcational discrete dissipative systems under follower loads in regions of existence/non-existence of adjacent equilibria is thoroughly re-examined in the light of recent progress in nonlinear dynamics. A general theory for such nongradient systems described by autonomous ordinary differential equations is developed. Conditions for the existence of adjacent equilibria, the stability of precritical, critical and postcritical states, as well as for different types of local bifurcations are established. Focusing attention on the interaction of geometric nonlinearities and vanishing damping, new findings contradicting widely accepted results of the classical (linear) analysis are discovered. In a small region of adjacent equilibria near a compound branching point, which is explicitly determined, an interaction of two consecutive postbuckling modes occurs related to the following phenomena : in case of vanishing damping, loss of stability may occur via a Hopf (dynamic) bifurcation prior to static (divergence) buckling. Moreover, the critical states of divergence instability may be associated with a double zero Jacobian eigenvalue satisfying also the conditions of a Hopf (local) bifurcation. Besides local (dynamic) bifurcations, global bifurcations are also found. An example is used to illustrate the qualitative findings.