This paper deals with a problem of modeling and vibration analysis of a flexible robotic arm under fast motion. Conventional method of modeling does not exactly express the bending mechanism of a flexible beam. Then a new description of deformation on the flexible shaft and Hamilton's principle are used to derive the dynamic model of the flexible robotic arm. A nonlinear dynamic model is proposed to describe the dynamic behavior of the flexible beam and the dynamic equations of the flexible robotic arm are also presented. Derived equations consist of some nonlinear forces and nonlinear stiffness. The present research considers the effect from those nonlinear forces coupling with angular functions (angular position, velocity, acceleration). A forced vibration method is applied to evaluate the characteristics of flexible vibration. In this research, numerical calculations based on two strategies are applied to analyze the flexible vibration. In the first strategy (AIM, angle independent method), angles of joints are assumed to be independent of the external forces. In the second strategy (ADM, angle dependent method), the effect on the angular functions of joints due to the external forces is considered. Especially, the gravity affects significantly on the residual vibration of flexible robotic arm and on the angular functions.