Abstract
The equations of motion for a gyroscopic system with unsteady superposed motion are derived for the prototypical problem in which motion of an oscillating particle is measured relative to a non-inertial frame. The resulting coefficient matrices are time-dependent, and skew-symmetric acceleration terms are present both as Coriolis acceleration and as a component of net stiffness. Such mathematical structure is also demonstrated in the context of other unsteady gyroscopic systems, including flexible media that translate with time-dependent speed. Following the asymptotic approach of Krylov, Bogoliubov and Mitropolsky, a perturbation method is developed for the case in which the superposed motion varies slowly when viewed on the time scale of the natural periods of oscillation. First-order approximations for the modal amplitude and phase are obtained in closed form. The method is illustrated through two examples of technical interest: a two degree-of-freedom model of a rotating shaft, and a distributed parameter model of a moving tape.
