Stability Analysis of a Single-Degree-of Freedom Mechanical Model with Distinct Critical Points: I. Bifurcation Theory Approach

Dimitrios S. Sophianopoulos

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DOI: 10.4236/wjm.2013.31005                

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.

Keywords

Mechanical Models; Nonlinear Stability; Distinct Critical Points; Bifurcation Theory; Singularities

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	G. W. Hunt, “Discrete Branching Points in the General Theory of Elastic Stability,” Journal of the Mechanics and Physics of Solids, Vol. 13, No. 5, 1965, pp. 295-310. doi:10.1016/0022-5096(65)90033-5
[2]	J. M. T. Thompson and G. W. Hunt, “Elastic Instability Phenomena,” John Wiley & Sons Ltd., Chichester, 1984.
[3]	L. Godoy, “Theory of Elastic Stability: Analysis and Sensitivity,” Taylor & Francis, Philadelphia, 1999.
[4]	G. W. Hunt, “The Discrete Branching Point as a One Degree of Freedom Phenomenon,” International Journal of Solids and Structures, Vol. 7, No. 5, 1971, pp. 495-503. doi:10.1016/0020-7683(71)90101-6
[5]	H. Troger and A. Steindl, “Nonlinear Stability and Bifurcation Theory,” Springer, New York, 1991. doi:10.1007/978-3-7091-9168-2
[6]	R. Gilmore, “Catastrophe Theory for Scientists and Engineers,” Dover Publ. Co., New York, 1981.
[7]	D. S. Sophianopoulos, “Bifurcations and Catastrophes of a Two-Degrees-of-Freedom Nonlinear Model Simulating the Buckling and Postbuckling of Rectangular Plates,” Journal of The Franklin Institute, Vol. 344, No. 5, 2007, pp. 463-488. doi:10.1016/j.jfranklin.2006.02.012
[8]	V. Gioncu and M. Ivan, “Theory of Critical and Postcritical Behavior of Elastic Structures,” Editure Academieie Republicii Socialiste, Romania, 1984.
[9]	D. S. Sophianopoulos, “Static and Dynamic Stability of a Single-Degree-of-Freedom Autonomous System with Distinct Critical Points,” Structural Engineering & Mechanics, Vol. 4, No. 5, 1996, pp.529-540.
[10]	A. E. R. Woodcock and T. Poston, “A Geometrical Study of the Elementary Catastrophes,” Springer, New York, 1974. doi:10.1016/j.jfranklin.2006.02.012