An oscillator-rotator system: Vibrational maintenance of rotation, stationary synchronous regimes, stability, vibration mitigation

The steady-state dynamics of an oscillator-rotator system excited by a vibrating base is studied with account of reciprocal influence of the rotation on the oscillations. This problem, associated with many engineering applications, in previous works was considered either within framework of 1DoF model neglecting the feedback effect of the rotator on the main body oscillation or by means of rather complicated asymptotic procedures. The proposed straightforward analytical procedure, which sequentially separates the averaged regime and deviations from it without assumption of smallness of inertia forces in the rotator, has been proved to be very exact and efficient. It allows already in the first approximation to reveal previously unnoticed peculiar features of the synchronous stationary regimes. In the second approximation the analytical solution, based on the Fourier series, without introducing small parameters and corresponding asymptotic procedures, results in the approximate analytical description of non-uniform rotation, which is excellently confirmed by the numerical simulation. It has been shown that the synchronous regimes with rotational response of the pendulum may become unstable due to parametric resonances for fluctuations of the angular velocity. These resonances are not associated with a certain ratio of natural frequencies for pendulum swinging and the body oscillation, and can be generated by two different sources. The domain of existence and stability of the stationary synchronous regimes in the parameter space has been determined. As an example of practical application of the analysis, the problem of mitigation of the forced harmonic oscillations by the rotator is considered and some recommendations are given.