Stability Analysis of Periodic Motion of the Swinging Atwood Machine

Abstract

AbstractThe swinging Atwood machine is a conservative Hamiltonian system with two degrees of freedom that is essentially nonlinear. A general solution of its equations of motion cannot be written in symbolic form, only in some special case it is integrable. A very interesting peculiarity of the system is an existence of a state of dynamical equilibrium when the oscillating body of smaller mass balances a body of larger mass. This state is described by periodic solution of the equations of motion that is constructed in the form of power series in a small parameter. In this paper, we investigate the system dynamics in the neighbourhood of the periodic solution. Its perturbed motion is described in linear approximation by the fourth order system of differential equations with periodic coefficients. We computed a fundamental matrix for this system and found its characteristic exponents in the form of power series in a small parameter. We have shown that owing to oscillations the state of dynamical equilibrium of the swinging Atwood machine is stable in linear approximation. All the relevant symbolic calculations are performed with the aid of the computer algebra system Wolfram Mathematica.KeywordsSwinging Atwood’s machinePeriodic solutionCharacteristic exponentsStabilityComputer algebraMathematica