DYNAMICAL STABILITY OF AN ELASTIC COLUMN AND THE PHENOMENON OF BIFURCATION

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 d irect method with respect to certain integral characteristics of the t ype 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 stabilit y 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 si ngularities described in [1], for example. In this paper a change of v ariables is made which, in fact, is the first step of the procedure me ntioned. 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 stab ility of adjacent shapes of equilibrium when the parameter has supercr itical values (post-buckling).