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Dynamical stability of an elastic column and the phenomenon of bifurcation
Author(s) -
Furta Stanislav D.
Publication year - 1994
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1670171103
Subject(s) - mathematics , bifurcation , stability (learning theory) , gravitational singularity , instability , mathematical analysis , buckling , sobolev space , lyapunov function , nonlinear system , mechanics , physics , quantum mechanics , machine learning , computer science , thermodynamics
The article studies the stability of rectilinear equilibrium shapes of a non‐linear elastic thin rod (column or Timoshenko's beam), the ends of which are pressed. Stability is studied by means of the Lyapunov direct method with respect to certain integral characteristics of the type of norms in Sobolev spaces. To obtain equations of motion, a model suggested in [16] is used. Furta [6] solved the problem of stability for all values of the parameter except bifurcational ones. When values of the system's parameter become bifurcational, the study of stability is more complicated already in a finite‐dimensional case. To solve a problem like that, one often has to use a procedure of solving the singularities described in [1], for example. In this paper a change of variables is made which, in fact, is the first step of the procedure mentioned. To prove instability, we use a Chetaev function which can be considered as an infinite‐dimensional analogue of functions suggested in [14, 9]. The article also investigates a linear problem on the stability of adjacent shapes of equilibrium when the parameter has supercritical values (post‐buckling).