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Research Papers

Bifurcations of Equilibria in Potential Systems at Bimodal Critical Points

[+] Author and Article Information
Alexei A. Mailybaev

Institute of Mechanics, Moscow State Lomonosov University, Michurinsky prospect 1, 119192 Moscow, Russiamailybaev@imec.msu.ru

Alexander P. Seyranian

Institute of Mechanics, Moscow State Lomonosov University, Michurinsky prospect 1, 119192 Moscow, Russiaseyran@imec.msu.ru

J. Appl. Mech 75(2), 021016 (Feb 26, 2008) (11 pages) doi:10.1115/1.2793136 History: Received December 25, 2006; Revised August 20, 2007; Published February 26, 2008

Bifurcations of equilibria at bimodal branching points in potential systems are investigated. General formulas describing postbuckling paths and conditions for their stability are derived in terms of the original potential energy. Formulas describing unfolding of bimodal branching points due to a change of system parameters are given. A full list of possible cases for postbuckling paths, their stability, and unfolding depending on three system coefficients is presented. In order to calculate these coefficients, one needs the derivatives of the potential energy and eigenvectors of the linearized problem taken at the bifurcation point. The presented theory is illustrated by a mechanical example on stability and postbuckling behavior of an articulated elastic column having four degrees of freedom and depending on three problem parameters (stiffness coefficients at the hinges). For some of the bimodal critical points, numerical results are obtained illustrating influence of parameters on postbuckling paths, their stability, and unfolding. A surprising phenomenon that a symmetric bimodal column loaded by an axial force can buckle with a stable asymmetric mode is recognized. An example with a constrained sum of the stiffnesses of the articulated column shows that the maximum critical load (optimal design) is attained at the bimodal point.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 3

Unfolding of the bimodal bifurcation: Cases 9–16

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Figure 4

Structure of nontrivial equilibria for nearly bimodal critical point

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Figure 5

Elastic articulated column loaded by an axial force

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Figure 6

Stiffness parameters for columns undergoing bimodal buckling

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Figure 7

Stiffnesses and buckling modes of the elastic column (b0=1,b1=0.25,b2=1)

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Figure 8

Stiffnesses and buckling modes of the elastic column (b0=1,b1=0.15,b2=0.76465)

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Figure 9

Critical load depending on stiffness parameters

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Figure 1

Pitchfork bifurcation: (a) supercritical (v1111>0) and (b) subcritical (v1111<0). Stable equilibria are shown by solid lines.

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Figure 2

Unfolding of the bimodal bifurcation: Cases 1–8

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