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A robot with a long extendible flexible arm which can also undergo both vertical translation and rotary motion is considered. First, a distributed-parameter model for the robot arm dynamics is developed. It is found that the extending motion could enhance the arm vibrations. Then, a Galerkin type approximation based on an appropriate time-dependent basis for the solution space is used to obtain an approximate finite-dimensional model for simulation studies. A feedback control for damping the motion-induced vibrations is derived by considering the time rate-of-change of the total vibrational energy of the flexible arm. The paper concludes with some simulation results for a special case with the proposed control law.