Vibration Control of a Timoshenko Cantilever Beam with Varying Length

This paper addresses the vibration control of a Cartesian palletizer consisting of a trolley and a robotic arm, wherein the robotic arm is modeled as a thick cantilever beam of varying length. The Timoshenko beam theory, which describes the behavior of thick beams, is used to model the robotic arm's dynamics. A mathematical model describing the trolley's motion and the robotic arm's vibration is established based on the extended Hamilton principle. According to this dynamic model, a boundary control law is proposed to suppress the undesired transverse vibration of the robotic arm. The uniform stability of the closed-loop system is proven via the Lyapunov method. The simulation results show that the proposed control law can simultaneously control the trolley's position and the robotic arm's vibration.