Passive and Active Inertia Forces in Flexible Body Dynamics

This investigation is devoted to a discussion on the effect of the coupling between the longitudinal and transverse displacements of kinematically driven rotating beams. To this end, the inertia forces that act on a flexible body as the result of the finite rotation are categorized into two classes. These are the passive and active inertia forces. While the effect of the active inertia forces of kinematically driven systems is recognized in the absence of external disturbances and nonzero initial conditions, the passive inertia forces of such systems are not recognized in the case of zero initial conditions and in the absence of external excitations. Depending on the assumed displacement field, three classes of mechanical systems are defined in this paper. These are the active, partially active, and passive systems. The active system has a mathematical model in which both passive and active inertia forces are fully presented. In a partially active system, a part of the passive inertia forces and the active inertia forces appear in the mathematical model. The vibration of the kinematically driven passive system is governed by homogeneous equations which contain only the passive inertia forces. In the case of zero initial conditions and in the absence of external excitation, the response of the passive kinematically driven system is zero regardless of the value of the angular velocity. The effect of the inertia forces of the passive system appear as a time varying modification of the system parameters. It is shown in this investigation that a rotating beam model in which the axial deformation is neglected is a partially active or passive system. It is also demonstrated that the neglect of the effect of the longitudinal displacement has two significant effects. It decouples the modes of vibration and makes the form of the complementary solution independent of the sense of rotation. The behavior of the active, partially active, and passive systems when they are subjected to driving constraints (specified motion) is examined and it is shown that the response of the passive system converges to the partially active system if the effect of the initial conditions becomes dominant as compared to the effect of the active inertia forces of the partially active system.