Swinging Atwood machine. Far- and near-resonance region (Q1119036)

The swinging Atwood machine, a prototype nonlinear dynamical system, is analyzed following an idea of \textit{N. N. Bogolyubov} and \textit{Y. A. Mitropolsky} [Asymptotic methods in the theory of nonlinear oscillations (1961; Zbl 0151.122); (1958; Zbl 0083.081)]. 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 ratio \(\mu\) to be equal to \(2j^ 2-1\), where j is an integer not equal to one.