Derivation of equations of motion of an unwinding cable from a cylindrical spool package

In this study, the transient analysis of a cable unwinding from a cylindrical spool package is studied. Cables are withdrawn with a constant velocity through a fixed point which is located along the axis of the package. When the cable is flown out of the package, several dynamic forces, such as inertial force, Coriolis force, centrifugal force, tensile force, and fluid-resistance force are acting on the cable. Consequently, the cable becomes to undergo very nonlinear and complex unwinding behavior which is called unwinding balloon. In this paper, to prevent the problems during unwinding such as tangling or cutting, unwinding behaviors of cables in transient state were derived and analyzed. First of all, the governing equations of motion of cables unwinding from a cylindrical spool package were systematically derived using the extended Hamilton's principles of an open system in which mass is transported at each boundary. And the modified finite difference methods are suggested to solve the derived nonlinear partial differential equations. Time responses of unwinding cables are calculated using Newmark time integration methods.