Subharmonic oscillation in third order non-linear systems

Abstract

In this paper the subharmonic oscillation with frequency n times smaller than that of exciting force is considered. The first paragraph is devoted to the formulation of problem. Here an example of third order equation in which there exists pure subharmonic solution is given. In the second paragraph the condition (2.5), (2.7) for the existence of pure odd subharmonic oscillation in the dynamical system governed by equation (2.1) are derived. The mixed subharmonic oscillation is studied in the third paragraph. The family of tow parameters solutions (3.2) of equation (3.1) with subharmonic term is found by means of asymptotic method. The stationary oscillations are given by formulae (3.14)-(3.16). The stability of subharmonic is oscillation is studied by Liapunov-Routh-Hurwitz criterium in the fourth paragraph. In general, the stability conditions (4.5) lead to the results which are different from those in second order system. To illustrate the presented method a concrete example has been analysed in the fifth paragraph.