Abstract
The buckling and post-buckling response of a single-degree-of-freedom mechanical model is re-examined in this work, within the context of nonlinear stability and bifurcation theory. This system has been reported in pioneer as well as in more recent literature to exhibit all kinds of distinct critical points. Its response is thoroughly discussed, the effect of all parameters involved is extensively examined, including imperfection sensitivity, and the results obtained lead to the important conclusion that the model is possibly associated with the butterfly singularity, a fact which will be validated by the contents of a companion paper, based on catastrophe theory.
Highlights
The role of distinct critical points, namely symmetric branching points, asymmetric branching points and limit points, has been recognized to be of paramount importance in the General Theory of Elastic Stability [1,2,3]
The buckling and post-buckling response of a single-degree-of-freedom mechanical model is re-examined in this work, within the context of nonlinear stability and bifurcation theory
Its response is thoroughly discussed, the effect of all parameters involved is extensively examined, including imperfection sensitivity, and the results obtained lead to the important conclusion that the model is possibly associated with the butterfly singularity, a fact which will be validated by the contents of a companion paper, based on catastrophe theory
Summary
The role of distinct critical points, namely symmetric (stable or unstable) branching points, asymmetric branching points and limit points, has been recognized to be of paramount importance in the General Theory of Elastic Stability [1,2,3]. Their study, as well as the effect of initial imperfections on their evolution and on the overall system’s response can be performed by applying either the Nonlinear Stability and Bifurcation Theory [5] or the Catastrophe Theory [6] Both theories start from the formulation of the total potential energy function of the system, but proceed afterwards to different directions. In the context of the above remark, the present work deals with the nonlinear stability analysis of a mechanical model with a single active variable (degree of freedom) and with four control parameters, which has been reported in the literature to exhibit all kinds of distinct critical points [8,9]. It should be noted that Cases 1 - 3 involve two control parameters, Cases 4 - 6 three and Case 7 four control parameters, while for all Cases there exists only a single active variable, i.e. rotation θ
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