Abstract: The lateral vibration of an axially moving beam is a dynamics problem with practical engineering background.In this paper the Cosserat's model of elastic rod was applied to discuss the dynamics modeling and stability of an axially moving beam with circular cross section.The arc-coordinate along the center line of the beam was used instead of the fixed coordinate.The deformation process of the beam was expressed by the attitude motion of the cross section with the variation of the arc-coordinate and time.Considering the inertial effect and shear strain of the cross section,the dynamics equations of the beam with large deformation were established from the view point of the concept of velocity field of Euler.The three-dimensional motion of an axially moving Timoshenko's beam can be regarded as a special case of small deformation.The stability problem of quasi-stationary state of the axially moving beam was discussed in the static and dynamic states,and the critical axial velocity before buckling was derived.It was proved that the Euler's stability conditions of the moving beam in the space domain are the necessary conditions of Lyapunov's stability in the time domain.