On Sub-Harmonic and"Summed and Differential Harmonic"Oscillations of Rotating Shaft

Abstract

The author has found experimentally that sub-harmonic oscillations and "summed and differential harmonic"oscillations take place in a shaft system with non-linear spring characteristics and multiple degrees of freedom. In the present paper, these vibrations are treated mathematically and analytically. Conclusions arrived at may be summarized as follows : (1) The equation of response curves of sub-harmonic oscillations of order 1/2 has qualitatively the same construction as that of rectilinear vibratory system. The hard or soft spring characteristics result in response curves of both hard and soft spring types. (2) When two vibrations taking place at the critical speed of "summed and differential harmonic"oscillations are both forward or both backward precessional whirling motions, only the summed harmonic type can occur and the differential type does not take place. When "summed and differential harmonic"oscillation consists of forward and backward precessional motions, only differential type has possibility of occurrence.