Chaotic behaviour in the dynamical response of coaxial magnetic gears during acceleration

Abstract

Slippage during acceleration has been a significant issue in coaxial magnetic and has been thoroughly investigated in the recent years. Depending on the applied outer load it is important to know the maximum acceleration that can be induced so that the system will not diverge. In the present work a non-dimensional analysis has been conducted in order to develop a stability criterion for the maximum acceleration that can be applied to the system as a function of the outer load. A closed-form relation for the period of the oscillations of the system that will be caused by the acceleration, has also been derived. Furthermore, the case of acceleration with ripple has been investigated. It was demonstrated that during acceleration with ripple under constant applied outer load, the system showcases a chaotic behaviour similar to the driven pendulum. In particular, a thorough analysis on the frequency of the ripple has been conducted. It was observed that when the ripple frequency was slightly lower than the frequency of the oscillation under steady acceleration, the system could potentially diverge even if the acceleration of the system was lower than the critical value. With the present work, a detailed non-dimensional model has been derived that could be a useful tool for determining the stability of coaxial magnetic gears without the requirement of numerical solution of the governing equations. In addition, some interesting observations are made regarding the chaotic behaviour of the dynamical response of coaxial magnetic gears during acceleration with ripple.