Reduction of residual vibrations in a rotating flexible beam with a moving payload mass

The reduction of residual vibrations for the position control of a flexible rotating beam carrying a payload mass is investigated. The common practice used to find the position control of a flexible multi-link arm is to assign a torque actuator to each joint while the payload mass is kept fixed relative to the end-link during the time of manoeuvre. This paper examines the stability of the system if either the payload is freed accidentally to move along the beam during the time of manoeuvre or is allowed to span the beam in a desired path for control purposes. A candidate Lyapunov function is constructed and its time rate of change is examined. It is shown that the use of a PD (proportional plus derivative) torque control yields a convergence of residual vibration to zero, an attainment of the rigid-body rotation to a prespecified desired angle of manoeuvre and a constant velocity of the payload mass as it moves relative to the beam. For manipulation purposes, an additional control force is added to the moving actuator in order to regulate its axial motion. It is shown that allowing the axial motion of the payload mass in a prescribed manner leads to a considerable reduction of its residual vibrations as compared to the case where the payload mass is fixed to the beam tip during the time of manoeuvre. Stability is also verified through simulations of rigid-body rotation and payload axial motion track prespecified reference trajectories.