Imao Nagasaka

Forced oscillations of a vertical continuous rotor with geometric nonlinearity

Nonlinear forced oscillations of a vertical continuous rotor with distributed mass are discussed. The restoring force of the rotor has geometric stiffening nonlinearity due to the extension of the rotor center line. The possibility of the occurrence of nonlinear forced oscillations at various subcritical speeds and the shapes of resonance curves at the major critical speeds and at some subcritical speeds are investigated theoretically. Consequently, the following is clarified: (a) the shape of resonance curves at the major critical speed becomes a hard spring type, and (b) among various kinds of nonlinear forced oscillations, only some special kinds of combination resonances have possibility of occurrence.

Forced Oscillations of a Continuous Asymmetrical Rotor With Geometric Nonlinearity (Major Critical Speed and Secondary Critical Speed)

Forced oscillations in the vicinities of both the major and the secondary critical speeds of a continuous asymmetrical rotor with the geometric nonlinearity are discussed. When the self-aligning double-row ball bearings support the slender flexible rotor at both ends, the geometric nonlinearity appears due to the stiffening effect in elongation of the shaft if the movements of the bearings in the longitudinal direction are restricted. The nonlinearity is symmetric when the rotor is supported vertically, and is asymmetric when it is supported horizontally. Because the rotor is slender, the natural frequency p fn of a forward whirling mode and p bn of a backward whirling mode have the relation of internal resonance P fn :P bn = 1 : (-1). Due to the influence of the internal resonance, various phenomena occur, such as Hopf bifurcation, an almost periodic motion, the appearance of new branches, and the diminish of unstable region. These phenomena were clarified theoretically and experimentally. Moreover, this paper focuses on the influences of the nonlinearity, the unbalance, the damping, and the lateral force on the vibration characteristics.

Forced Vibrations of a Very Slender Continuous Rotor With Geometrical Nonlinearity (Harmonic and Subharmonic Resonances)

When both ends of an elastic continuous rotor are supported simply by double-row self-aligning ball bearings, the geometrical nonlinearity appears due to the stiffening effect in the elongation of the rotor if the movement of the bearings in the longitudinal direction is restricted. As the rotor becomes more slender, the geometrical nonlinearity becomes stronger. In this paper, we study on unique nonlinear phenomena caused by both of the nonlinear spring characteristics and an initial axial force in the vicinity of the major critical speed ω c and twice ω c in a very slender continuous rotor. When the rotor is supported horizontally, the difference in support stiffness and the asymmetrical nonlinearity appear as a result of shifting from the equilibrium position. By the influences of the internal resonance and the initial axial force, the nonlinear resonance phenomena become very complicated. For example, the peak resonance splits into two peaks, and these two peaks leave each other and then one becomes a hard spring type while the other becomes a soft spring type, respectively. Moreover, almost periodic motions and chaotic vibrations appear. In this paper, we prove the above phenomena theoretically and experimentally.

Forced Oscillations of a Continuous Rotor with Nonlinear Spring Characteristcs. Vertical Shaft with Geometric Nonlinearity.

Nonlinear forced oscillations of a vertical continuous rotating shaft with distributed mass are discussed. The restoring force of the shaft has geometric stiffening nonlinearity due to the extension of the shaft center line. The possibility of the occurrence of nonlinear forced oscillations at various subcritical speeds and the shapes of resonance curves at the major critical speeds and at some subcritical speeds are investigated. Consequently, the following is clarified : (a) the shape of resonance curves at the major critical speed becomes a hard-spring type, and (b) among various kinds of nonlinear forced oscillations, only some special kinds of summed-and-differential harmonic oscillations have possibility of occurrence.