Frequency Entrainment in a Self-Oscillatory System with Ex-ternal Force

This paper deals with forced oscillations in a self-oscillatory system of the negative-resistance type. When no external force is applied, the system produces a self-excited oscillation. Under the impression of a periodic force, the frequency of the self-excited oscillation falls in synchronism with the driving frequency within a certain band of frequencies. This phenomenon of frequency entrainment also occurs when the ratio of the two frequencies is in the neighborhood of an integer (different from unity) or a fraction. Under this condition, the natural frequency of the system is entrained by a frequency which is an integral multiple or submultiple of the driving frequency. In this paper, special attention is directed toward the study of periodic oscillations as caused by frequency entrainment. The amplitude characteristics of the entrained oscillations are obtained by the method of harmonic balance, and the stability of these oscillations is investigated by making use of Hill's equation as a stability criterion. The regions in which different types of entrained oscillation, as well as beat oscillation, occur are sought by varying the amplitude and frequency of the external force. The theoretical results are compared with the solutions obtained by analog-computer analysis and found to be in satisfactory agreement with them.