A mathematical model that accurately represents an overhead crane with flexible cable and load hoisting/lowering is developed. The analysis includes the transverse vibrations of the flexible cable and the trolley motion as well as the load hoisting/lowering motions. A set of highly non-linear partial differential equations and ordinary differential equations that govern the motion of the crane system within time-varying spatial domain is derived via calculus of variation and Hamilton's principle. Variable-time modified Galerkin method has been used to discretize the non-linear system. State space transformation is then used to get a set of first order ordinary differential equation. A proportional derivative control scheme is applied to derive the underlying crane so that the cable and payload swing are damped out. Numerical simulations for the control performance of the considered system are presented for various operating conditions.