Analysis of motion of inverted pendulum with vibrating suspension axis at low-frequency excitation as an illustration of a new approach for solving equations without explicit small parameter

In the classical papers (see, e.g. P.L. Kapitsa, Pendulum with vibrating axis of suspension. Usp. Fiz. Nauk 44 1 (1954) 7-20 (in Russian)) motion of pendulum with vibrating suspension 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 inverted pendulums motion at unconventional values of parameters. Case when frequency of external 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 modification of the method of direct separation of motions (MDSM) is proposed to study the corresponding equation which in the considered case does not contain a small parameter explicitly. The aim is to obtain solutions of this equation in the stability domain. It is revealed that in the considered range of parameters not only the effective stiffness of the system changes due to the external loading, but also its effective mass. Applicability of the proposed approach for solving non-linear equations without small parameter is demonstrated; as an illustration, a damped Duffing equation is considered. © 2014 Elsevier Ltd.