Motion of a pendulum with vibrating suspension axis at unconventional values of parameters

Abstract

In the classical papers [1,2] motion of a pendulum with vibrating suspen- sion axis was considered in the case, when frequency of external loading is much higher than the natural frequency of the pendulum in the absence of this loading. The present paper is concerned with the analysis of pendulums motion at unconventional values of parameters. Case, when frequency of ex- ternal loading and the natural frequency of the pendulum in the absence of this loading are of the same order, is studied. Vibration intensity is assumed to be relatively low. A new modi cation of the method of direct separation of motions (MDSM) [3,4] is proposed to study corresponding equations, which in the considered case dont contain a small parameter explicitly. A condition of pendulums upper position stabilization is determined for this case by its means. It is noted that in the considered range of parameters not only the e ective sti ness of the system changes due to the external loading, but also its e ective mass. It is shown that application of the classical asymptotic methods in this case leads to erroneous results. So, the applicability range of the MDSM turns out to be broader than the one of these methods.