Chaotic Vibration and Internal Resonance Phenomena in Rotor Systems: Part I — Theoretical Analysis

Naturally, the gyroscopic moment is small for the many practical rotating machineries. In addition, some mechanical elements of a rotor system make various types of nonlinearity such as clearance in a ball bearing (Yamamoto, 1955)(Yamamoto, 1977), oil film in a journal bearing (Tondl, 1965), geometrical nonlinearity due to the shaft elongation (Shaw, 1988),(Ishida, 1996), etc. In such rotor systems, the natural frequencies of a forward whirling mode p_f and a backward whirling mode p_b almost satisfy the relation of internal resonance p_f : p_b = 1 : (−1). And then, the critical speeds of a backward harmonic oscillation and a supercombination oscillation are near from the major critical speed. Similarly, in the vicinity of two times of the major critical speed, the critical speeds of the forward and the backward subharmonic resonances of order 1/2 and the combination resonance are close to each other. Therefore, the internal resonance phenomena may occur at the major critical speed and two times of the major critical speed. However there are few studies on the nonlinear phenomena of the rotor systems due to the influence of internal resonance. In this study, we use a 2DOF rotor model and investigate the dynamic characteristics of nonlinear phenomena, especially the chaotic vibration, due to the internal resonance at the major critical speed and the critical speed of two times of the major critical speed. The following are clarified theoretically: (a) the Hopf bifurcation and consecutive period doubling bifurcations possible route to chaos occur at the major critical speed and at two times of the major critical speed, (b) another chaotic vibration from the combination resonance occur at two times of the major critical speed. The results demonstrate that the chaotic vibration is common nonlinear phenomena in the nonlinear rotor system when the effect of the gyroscopic moment is small.