On the stability problem of Nicolai with variable cross-section and compressibility effect

We consider the problem of Nicolai on dynamic stability of an elastic cantilever rod loaded by an axial compressive force and tangential twisting torque in continuous formulation. The rod is assumed to be non-uniform, i.e. having variable cross-section with non-equal principal moments of inertia. New linear equations and boundary conditions are derived from nonlinear governing equations. These equations form the basis for analytical and numerical studies. The important new details of this formulation include the pre-twisting effect due to the torque and compressibility of the rod. General formulae for the influence of small geometrical imperfections to the stability region are derived and numerical examples are presented.