Abstract
In a recent publication [1], the fully nonlinear stability analysis of a Single-Degree-of Freedom (SDOF) model with distinct critical points was dealt with on the basis of bifurcation theory, and it was demonstrated that this system is associated with the butterfly singularity. The present work is the companion one, tackling the problem via the Theory of Catastrophes. After Taylor expanding the original potential energy function and introducing Padè approximants of the trigonometric expression involved, the resulting truncated potential is a universal unfolding of the original one and an extended canonical form of the butterfly catastrophe potential energy function. Results in terms of equilibrium paths, bifurcation sets and manifold hyper-surface projections fully validate the whole analysis, being in excellent agreement with the findings obtained via bifurcation theory.
Highlights
Problems in Non-Linear Buckling, where various phenomena may arise, such as discontinuities, singularities and instabilities, can be efficiently treated by using Catastrophe Theory [2]-[4], and a better insight on them can be readily gained
Due to its topological character, this particular Theory allows for the establishment of universal solutions and the direct production of qualitative results, since it deals with various forms of potential energy functions incorporating the behavior of classes of complex systems
There are three distinguished differences/obstacles, which do not allow us to fully classify it among the seven elementary Catastrophes, namely: (a) the “constant” term Z, which has a negligible influence [4] [6], since Catastrophe Manifolds and Bifurcation Sets, associated with 1st and 2nd order derivatives of Vtr with respect to the active variable θ, are not affected by its presence, (b) the 5th order term related to coefficient H, which does not depend on the external load, remaining constant as λ is varied, and (c) the effect of the presence of coefficient A on the truncated potential. We initially address the latter of the aforementioned obstacles
Summary
Problems in Non-Linear Buckling, where various phenomena may arise, such as discontinuities, singularities and instabilities, can be efficiently treated by using Catastrophe Theory [2]-[4], and a better insight on them can be readily gained. The Theory of Catastrophes supplies the tools for comprehension and detection of numerous types of bifurcations and it has been proven highly efficient as far as potential discrete or continuous systems are concerned Among the former type of systems, mechanical models Pantazi the latter systems) with a few degrees of freedom are included, which, despite their geometrical and overall simplicity, may exhibit a quite complicated post-buckling response, associated with all kinds of distinct critical points. Their treatment via the foregoing Theory is not an easy task, and this remark is fully justified from the admittedly limited relevant publications [5]-[13]. After Taylor expansions and Padé approximations of trigonometric functions, the universal unfolding of the system’s original potential energy yields the butterfly singularity, in full accordance with the findings of the Theory of Bifurcations [1]
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