Dynamics and vibrations of flexible robot arms have received considerable attention in recent years. The flexibility of the arm affects the function of the robot and complicates its dynamics as well. Generally, the base of the robot arm has some elasticity, which also affects the precision of its function. We model the robot arm as a flexible beam moving in a vertical plane and resting on two springs: one is in the vertical direction and the other one is in the rotational direction. A lumped mass, which simulates the pay load, is attached to the tip of the beam. The beam translates and rotates as a rigid body and moreover it deforms in the lateral direction. The extended Hamilton principle is used to derive the governing equations of motion and their corresponding boundary conditions. We obtained three coupled differential equations: two ordinary-differential equations governing the rigid-body motion of the arm and a partial differential equation governing its deformation. An exact solution for the natural frequencies and mode shapes of the vibrations of the arm about an equilibrium position is obtained. The significance of the effect of the flexibility of the link and the base and the ratio of the mass at the tip point to the mass of the beam on the natural frequencies and mode shapes is investigated.