Effect of gravity and tangential air resistance on unwinding cable

In the field of unwinding dynamics, most of the researches so far have only considered the normal component of air resistance. In this research, the transient-state equation of motion that accurately contains all the boundary conditions at the guide-eyelet and lift-off points is derived. The transient-state equation of motion is derived from Hamilton's principle for an open system, because the total mass of an unwinding cable varies continuously. The virtual work in Hamilton's principle includes gravity, normal air resistance, and tangential air resistance. The air resistances are assumed to be proportional to the square of the normal and tangential velocities, and the effects of gravity and tangential air resistance on the unwinding cable are verified on the basis of the maximum balloon radius and the associated error. The results show that the effect of gravity and tangential air resistance on the maximum balloon radius is within 4 %, which is negligible.