Dynamic Buckling of 1-DOF Autonomous Potential Systems Under Tilted Cusp Catastrophe

Abstract

Nonlinear dynamic buckling of one-degree-of freedom (1-DOF) undamped systems under step loading (autonomous systems) of constant direction and infinite duration is discussed in detail using Catastrophe Theory. Attention is focused on the relation of static cuspoind catastrophes to the corresponding dynamic catastrophes for 1-DOF autonomous undamped systems by determining properly the dynamic singularity and bifurcational sets for such systems. Using local analysis one has to classify first the total potential energy (TPE) function of the system into one of the elementary Thom’s catastrophes by defining the corresponding control (unfolding) parameters. Subsequently, using global analyses one can readily obtain exact results for the dynamic buckling loads (DBLs) and their imperfection sensitivity of systems subjected to dynamic dual cusp and tilted cusp catastrophes. It was found that the maximum DBL of the dynamic tilted cusp catastrophe corresponds to a limit point lying in the vicinity of the hysteresis point (related to the static tilted cusp catastrophe). Numerical examples illustrate the methodology proposed herein.