Necessity of transient-state unwinding equation of motion for analyzing unwinding motion of a thin cable

When a thin cable is unwound from a spool dispenser, factors that influence balloon shapes are initial tensile force, unwinding velocity, balloon height, package radius, and fluid condition. Historically, a thin cable’s unwinding motion has been analyzed using transient- and steady-state unwinding equations of motion. The transient-state unwinding equation of motion can be derived by using Hamilton’s principle for an open system in which mass can change within a control volume. In the process of solving the transient-state unwinding equation of motion, two-point boundary conditions are exactly adopted for each time step. The steady-state unwinding equation of motion can be derived by using a perturbation scheme from the transient-state unwinding equation of motion. To find a solution for the steady-state unwinding equation of motion, the shooting method is utilized, but due to insufficient boundary conditions at the lift-off point, finding a unique solution is impossible. Therefore, analysis of the unwinding motion of a thin cable should be performed using the transient-state unwinding equation of motion.