TRANSIENT VIBRATION OF GYROSCOPIC SYSTEMS WITH UNSTEADY SUPERPOSED MOTION

The equation of motion for a gyroscopic system with unsteady superpose d motion is derived for the prototypical problem in which motion of an oscillating particle is measured relative to a non-inertial frame. Th e resulting coefficient matrices are time-dependent, and skew-symmetri c acceleration terms are present both as Coriolis acceleration and as a component of net stiffness. Such mathematical structure is also demo nstrated in the context of other unsteady gyroscopic systems, includin g flexible media that translate with time-dependent speed. Following t he asymptotic approach of Krylov, Bogoliubov and Mitropolsky, a pertur bation method is developed for the case in which the superposed motion varies slowly when viewed on the time scale of the system's natural p eriods of oscillation. First-order approximations for the modal amplit ude and phase are obtained in closed form. The method is illustrated t hrough two examples of technical interest: a two-degree-of-freedom mod el of a rotating shaft, and a distributed parameter model of a moving tape or web. (C) 1996 Academic Press Limited