Longitudinal and transverse vibration control of an axially moving string

An active control scheme of an axially moving string system for suppressing longitudinal and transverse vibrations simultaneously is investigated. The control scheme incorporates three inputs: one control force to the right boundary (roller) and two control torques applied to the left and right rollers. Two nonlinear partial differential equations derived using Hamilton's principle govern the longitudinal and transverse motions, where the spatially varying tension due to the transverse and longitudinal vibrations is considered. The asymptotic stability of the closed-loop system is proved using the Lyapunov method. Numerical simulations demonstrate the effectiveness of the proposed control scheme.