Dynamic and Stability of Harmonic Driving Flexible Cartesian Robotic Arm with Bolted Joints Based on the Sensitivity and Multiple Scales Method

Flexible Cartesian robotic arms (CRAs) are typical multicoupling systems. Considering the elastic effects of bolted joints and the motion disturbances, this paper investigates the dynamic and stability of the flexible CRA. With the kinetic energy and potential energy of the comprising components, Hamilton’s variational principle and Duhamel integral are utilized to derive the dynamic equation and vibration differential equation. Based on the proposed elastic restraint model of the bolted joints, boundary conditions and mode equations of the flexible CRA are determined with using the principle of virtual work. According to the mode frequencies and sensitivities analysis, it reveals that the connecting stiffness of the bolted joints has significant influences, and the mode frequencies are more sensitive to the tensional stiffness. Moreover, describing the motion displacement of the driving base as combination of an average motion displacement and a harmonic disturbance, the vibration responses of the system are studied. The result indicates that the motion disturbance has obvious influence on the vibration responses, and the influence enhances under larger accelerating operations. The multiple scales method is introduced to analyze the parametric stability of the system, as well as the influences of the tensional stiffness and the end-effector on the stability.