Harmonic motions of a weakly forced autoparametric vibrating system

Abstract

Abstract. Based on the method of averaging, in this paper we investigate the continuation of harmonic motions for a weakly forced autoparametric vibrating system which models the dynamics of a forced pendulum positioned on a vertically excited mass. The result shows that when the very weakly force is imposed on the pendulum, a state of dynamic balance (a harmonic motion) is preserving.

Periodic solutions of nonlinear differential systems by the method of averaging

“…There are lots of papers related to forced autoparametric vibrating systems, see [8], [2], [1]. The following system is based on the forced autoparametric vibrating system in [14] and the appropriate frequency modulation is carried out:…”

“…R e m a r k 3.1. In [14], we described a system which is the same as (3.1) (up to a scaling factor (p √ 1 + R) −1 ), and the existence of a periodic solution for it is obtained by a direct application of the local average theorem in [4] because the form of the fundamental matrix for the linearized system is known. However, in the present case, the second equation (3.8) of the linearized system for the average system (3.5) is a non-autonomous Hill equation and the fundamental matrix is unknown.…”