Swinging Atwood machine. Far- and near-resonance region

The swinging Atwood machine, a prototype nonlinear dynamical system, is analyzed following an idea of Bogoliubov and Mitropolsky. A series solution is found for the radial and angular displacement as functions of time. The analysis is repeated in the resonance case, when the frequency of the driving force maintains a fixed ratio to that of the free motion. The condition of resonance requires the mass ration μ to be equal to 2j2 - 1, where j is an integer not equal to one