Instability of cantilever beams with non-linear elements: Butterfly catastrophe

Theocaris, P. S.

doi:10.1016/0020-7403(84)90047-x

Cited by 2 publications

(4 citation statements)

References 14 publications

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“…• The set of critical points of the U-unfolding is the so-called four-dimensional Catastrophe Manifold CM U , given by [5] …”

Section: Theoretical Backgroundmentioning

confidence: 99%

“…• The set of the singular (degenerate) critical points S U is a four-dimensional Sub-Manifold of the previous one, satisfying [5]:…”

Section: Theoretical Backgroundmentioning

confidence: 99%

“…From the One-Dimensional Elementary Catastrophes (Cuspoids), the Butterfly (A 5 according to the classification of Thom [4]), possesses the highest degree of complexity and moreover it rarely appears in the Theory of Elastic Stability [5]; however, it has been found to occur in other engineering disciplines, as for instance in the mesophase stability response of external magnetic or electric fields [7]. In order to show that the one-dimensional system studied herein exhibits the Butterfly Singularity, one may either use Equations (6) and (8) or employ and extend the alternative elimination procedure of Deng [8].…”

Section: Computation Of Catastrophe Manifolds and Bifurcation Sets VImentioning

confidence: 99%

“…Among the former type of systems, mechanical models (original ones or products of adequate discretization of 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 however is not an easy task, and this remark is fully justified from the admittedly limited relevant publications [5]- [13].…”

Section: Introductionmentioning

confidence: 99%

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Stability Analysis of a SDOF Mechanical Model with Distinct Critical Points: II. Catastrophe Theory Approach

Sophianopoulos¹,

Pantazi²

2015

WJM

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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.

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“…• The set of critical points of the U-unfolding is the so-called four-dimensional Catastrophe Manifold CM U , given by [5] …”

Section: Theoretical Backgroundmentioning

confidence: 99%

“…• The set of the singular (degenerate) critical points S U is a four-dimensional Sub-Manifold of the previous one, satisfying [5]:…”

Section: Theoretical Backgroundmentioning

confidence: 99%