Stabilization of statically unstable columns by axial vibration of arbitrary frequency

The classical Chelomeiʹs problem of stabilization of a statically unstable elastic column by axial harmonic vibration is reconsidered. The excitation frequency is assumed to be arbitrary. Two types of boundary conditions of the column, simply supported and clamped–hinged ends, are considered. Both cases are studied analytically with check by numerical analysis. For undamped columns with the axial force close to the critical stability value, a finite number of triangular stabilization zones appears. This implies that stabilization is attained at medium frequency range (compared to the second eigenfrequency of the unexcited column). The high-frequency stabilization is impossible, as the stabilization regions vanish with the increasing excitation frequency. With addition of internal damping, a continuous stabilization region appears for small excitation amplitudes, which starts at medium frequencies and extends to high-frequency range. The influence of external damping on stabilization regions is shown to be small.