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

• The problem of Nicolai on dynamic stability of an elastic non-uniform rod is analyzed.
• New linear equations are derived from nonlinear governing equations.
• The pre-twisting and compressibility effects of the rod are taken into account.
• The influence of small geometrical imperfections on the stability region is obtained.
• Numerical examples are presented.

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.