The Chelomei Problem: High or Low Frequency Stabilization?
The possibility to increase the stability of elastic systems by
means of vibrations was originally pointed out by V.N. Chelomei
in 1956. In particular, he arrived at the conclusion that an
elastic rod compressed by a periodic longitudinal force, whose
constant part is greater than the
critical Euler's value, can be stabilized by high-frequency
longitudinal vibrations applied to the rod end. In this study,
formulas for the upper and lower critical frequencies of rod
stabilization are derived and analyzed. It is shown that,
in contrast to the case of high-frequency stabilization of
an inverted pendulum with a vibrating suspension point, the
rod is stabilized at excitation frequencies of the order of
the natural frequency of transverse oscillations belonging to a certain interval.