Stability of synchronous regimes in unbalanced rotors on elastic base

Abstract

The stationary dynamics of an unbalanced rotor (vibrator) on a movable base (linear oscillator) under excitation by a driving torque is studied with focusing on the stability of 1:1 stationary regimes of rotation and oscillation. This problem was well studied previously in the first approximation, dealing, in fact, with averaged regimes, mainly in the framework of asymptotic procedures. We use an efficient analytical procedure, proposed in our previous works for another problem, which sequentially separates the averaged regimes and deviations from them. Describing in the first approximation the known features of the synchronous stationary regimes under consideration, this approach in the second approximation results in the analytical solution for nonuniform rotation whose exactness is confirmed in the numerical simulation. The solution enables us to reveal possibility of parametric instability for oscillations of the rotor angular velocity and to describe two possible mechanisms of this instability. It is shown that the known condition of stability of stationary synchronous rotation-oscillation regimes is only necessary but not sufficient criterion, and two additional necessary conditions of stability are obtained and confirmed by the numerical simulation.