中文摘要：

轴向运动梁的横向振动是具有实际工程背景的动力学问题．该文应用Cosserat弹性杆模型讨论圆截面轴向运动梁的动力学建模及其运动稳定性．以沿梁中心线的弧坐标代替方向固定的坐标轴，根据梁截面的姿态随弧坐标和时间的变化确定梁的变形过程．从欧拉的速度场概念出发，考虑梁截面转动的惯性效应和剪切变形，建立大变形轴向运动梁的动力学方程．其小变形特例为轴向运动的三维Timoshenko梁．基于该模型分析了轴向运动梁准稳态运动的静态和动态稳定性，导出可导致失稳的临界轴向速度．证明空间域内的欧拉稳定性条件是时间域内的Lyapunov稳定性的必要条件．

英文摘要：

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-dimensionM 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.