An analytical solution for vibration of a parallel robot where its end-effector is flexible and has a passive prismatic joint(s) has not been presented before. In this research vibration analysis of a PR-PRP parallel robot using an analytical method is investigated. The PR-PRP parallel robot has two planar degrees of freedom and moves by means of two active prismatic joints. The robot moving platform is a flexible link with one passive revolute and one passive prismatic joint. First, the motion equation for a flexible link with the passive prismatic joint is developed. The motion equation is solved by employing an approximate analytical method called ‘‘constrained assumed modes method’’. A time-variant constraint is written for the passive prismatic joint. The developed model allows for inclusion of the effect of physical length for the passive prismatic joint in contact with the moving platform on the vibration response of the system. For verification of the presented model and solution, three case studies are presented and results of the analytical solution are compared with results of a commercial finite element method software. For each case study, two different lengths for the passive prismatic joint are considered. Finally, active vibration control is performed for an applicable motion of the robot using the proportional-integral-derivative controller.

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