Dynamical stability of an elastic column and the phenomenon of bifurcation

Abstract

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).