On the Relation of Static to Dynamic Bifurcation in Nonlinear Autonomous Dissipative or Nondissipative Structural Systems

Abstract
Dynamical dissipative or nondissipative discrete systems under constant directional (conservative) step loading described by nonlinear autonomous Ordinary Differential Equations (ODEs) are considered. Attention is restricted on gradient structural systems which under the same loading applied statically exhibit a branching point critical response bifurcating from a trivial prebuckling path. The connection of static bifurcations and stability with the corresponding dynamic bifurcations and stability are comprehensively discussed after a thorough investigation of the structural form of the Jacobian matrix and its characteristic equation. The relation of the eigenvalues of the statical system with those of the dynamical system associated with the Jacobian matrix is examined at the precritical, critical and postcritical stage. This enables us to establish very simple stability conditions (related basically to the sign of second variation of the total potential energy) compared to those of Routh‐Hurwitz.