The motion of a mass hanging from an overhead crane

We study the dynamics of a mass hanging from an overhead crane whose pulley support undergoes a small vertical oscillation. Firstly, we use Lagrangian mechanics to derive a differential equation describing the motion of the crane system and then we reduce the equation in three particular cases by making certain assumptions about the motion of the system. In one of these cases, we use Melnikov theory to establish the existence of both chaotic and subharmonic periodic solutions of the crane system, for particular parameter values.