The problem of the stability of circular rings connected to each other

The paper considers the problems on the stability of a system 
of circular rings interconnected in such a way that the 
displacements of these rings and the rotation angles of their 
sections coincide at some points. This problem is reduced 
to some variational problem with restrictions on the desired 
functions in the form of linear equations. The Fourier series 
are used for a finite-dimensional approximation. The paper 
also presents the problem on the stability of a system of circular 
rings reinforced with inextensible threads that do not 
withstand compressive forces. In this case, there are constraints 
in the form of inequalities, and after a finite-dimensional 
approximation, the problem is reduced to finding bifurcation 
points for nonlinear programming problems in the 
presence of constraints in the form of inequalities.