Inertia forces in rotating shaft problems

The equations of motion presented in the literature for a circular shaft rotating about its centroidal axis assume a constant angular velocity and include a single coupling effect resulting from the Coriolis inertia force component of the gyroscopic moment. The objective of this investigation is to address the limitations of the classical equations of motion and provide a comprehensive model for a shaft rotating about its centroidal axis at an arbitrary angular velocity. The general equations of motion for a flexible body are derived through the application of the principle of virtual work in dynamics and are tailored to the specific case of the rotating shaft problem. The equations are shown to include both the Coriolis and centrifugal inertia forces, and the effect of the inertia terms on the system dynamic stability is demonstrated. The effect of the rotary inertia on the axial and transverse deformations is formulated for the rotating shaft and the coupling terms introduced in the effective mass and stress matrices are obtained. In the absence of the Coriolis coupling, it is demonstrated that the centrifugal force causes the rotating shaft to become unstable when the angular velocity exceeds a certain value defined by the static stress of the stationary shaft. By including the in-plane and out-of-plane Coriolis coupling, the effective damping of the system increases and the instabilities noted at the higher angular velocities are eliminated. The case of a shaft rotating with non-constant angular velocity is also examined and the effect of the angular acceleration on the stability of the shaft is discussed. The generality of the approach presented in this study, as compared to the classical formulation, is further demonstrated by considering the dynamics of a rotating shaft subject to a base excitation. The coupling between the base motion and the deformation of the shaft is examined numerically and the effect of the support motion on the dynamics of the shaft is discussed for both a low level and high level disturbance. The results presented in this investigation clearly demonstrate that many of the detailed formulations previously developed for rotating shafts can be considered as special cases of the general flexible body formulation. As a consequence, general purpose flexible multibody computer algorithms can be used to efficiently and systematically solve more general rotating shaft problems.