We study the dynamics of a mass hanging from an overhead crane whose p
ulley 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 pa
rticular cases by making certain assumptions about the motion of the s
ystem. In one of these cases, we use Melnikov theory to establish the
existence of both chaotic and subharmonic periodic solutions of the cr
ane system, for particular parameter values.