Sd. Furta, DYNAMICAL STABILITY OF AN ELASTIC COLUMN AND THE PHENOMENON OF BIFURCATION, Mathematical methods in the applied sciences, 17(11), 1994, pp. 855-875
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).