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 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 pfn of a forward whirling mode and pbn of a backward whirling mode have the relation of internal resonance pfn:pbn=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.

Currently, some kinds of helicopters use pendulum absorbers in order to reduce vibrations. Present pendulum absorbers are designed based on the antiresonance concept used in the linear theory. However, since the vibration amplitudes of the pendulum are not small, it is considered that the nonlinearity has influence on the vibration characteristics. Therefore, the best suppression cannot be attained by using the linear theory. In a helicopter, periodic forces act on the blades due to the influences of the air thrust. These periodic forces act on the blades with the frequency which is the integer multiple of the rotational speed of the rotor. Our previous study proposed a 2-degree-of-freedom (2DOF) model composed of a rotor blade and a pendulum absorber. The blade was considered as a rigid body and it was excited by giving a sinusoidal deflection at its end. The present paper proposes a 3DOF model that is more similar to the real helicopter, since the freedom of the fuselage is added and the periodic forces are applied to the blade by aerodynamic force. The vibration is analyzed considering the nonlinear characteristics. The resonance curves of rotor blades with pendulum absorbers are obtained analytically and experimentally. It is clarified that the most efficient condition is obtained when the natural frequency of the pendulum is a little bit different from the frequency of the external force. Various unique nonlinear characteristics, such as bifurcations, are also shown.

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