SINGULARITIES OF STABILITY BOUNDARIES AND PARADOX OF NICOLAI
Alexei A. Mailybaev, Alexander Seyranian
We present a general approach to the paradox of Nicolai and related effects analyzed as a singularity of the stability boundary. We study potential systems with arbitrary degrees of freedom and two coincident eigenfrequencies disturbed by small non-conservative positional and damping forces. The instability region is obtained in the form of a cone having a finite discontinuous increase in the general case when arbitrarily small damping is introduced. This is a new destabilization phenomenon, which is similar to the effect of the discontinuous increase of the combination resonance region due to addition of infinitesimal damping. Then we
consider the paradox of Nicolai: the instability of a uniform axisymmetric elastic column loaded by axial force and a tangential torque. It is shown that the paradox of Nicolai is related to the conical singularity of the stability boundary which transforms to a hyperboloid with the addition of small dissipation.