On the Averaging in Strongly Damped Systems: The General Approach and its Application to Asymptotic Analysis of the Sommerfeld Effect

Passage through resonance of an unbalanced rotor mounted on a strongly damped oscillating system that is excited by an asynchronous motor of limited power is investigated using an averaging procedure for a partially strongly damped system. The approach is closely related to a singular perturbation technique. It is demonstrated that the system's dynamics can be reduced on the slow manifold to a single first-order differential equation predicting both the stationary and transient solutions of the original system. Furthermore, the technique provides insight into the system's dynamics, enabling an outline of two simple methods to avoid capture into resonance. The first method is based on an active compensation of the small averaged vibrational torque, and the second method is a passive mechanical device reducing vibration amplitudes. The approximate results obtained were compared with the direct numerical simulations of the full system and demonstrate very good accuracy everywhere except in the vicinity of the bifurcation point. In this vicinity, the asymptotically long time interval is not sufficient for predicting capture into resonance.