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On the Relation of Static to Dynamic Bifurcation in Nonlinear Autonomous Dissipative or Nondissipative Structural Systems
Author(s) -
Mahrenholtz O.,
Kounadis A. N.
Publication year - 1993
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.19930730302
Subject(s) - jacobian matrix and determinant , mathematics , hurwitz matrix , dissipative system , nonlinear system , eigenvalues and eigenvectors , bifurcation , mathematical analysis , dynamical systems theory , ordinary differential equation , ode , saddle node bifurcation , bifurcation theory , classical mechanics , differential equation , physics , quantum mechanics
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.