This study investigates the stability of an axially moving free-free beam as a function of moving speed, modulus of elasticity, and axial tensile force. The mode shapes of the beam under different axial tensile forces are also discussed. In this study, we derive the equations of motion of a flexible free-free beam with a uniformly circular cross section that moves axially at a constant speed. Using Galerkin’s technique, we dicretize and reduce the nonlinear partial differential equation of motion to a set of ordinary differential equations by choosing the shape functions to be the eigenfunctions of a free beam. The state-space method is used to investigate the stability of an axially moving free beam for different cases. Numerically, the stability of the system is influenced by different cases in various ways and degrees. The transverse vibration modes with different tensile forces are also considered in this study.

这项研究调查了轴向移动自由梁的稳定性与移动速度，弹性模量和轴向拉力的关系。还讨论了梁在不同轴向拉力下的模态形状。在这项研究中，我们导出了具有均匀圆形横截面，以恒定速度轴向运动的柔性自由自由梁的运动方程。通过使用形状函数作为自由光束的本征函数，我们使用Galerkin的技术将运动的非线性偏微分方程离散化为一组常微分方程。状态空间方法用于研究轴向移动自由光束在不同情况下的稳定性。在数值上，系统的稳定性会受到不同情况的不同程度和程度的影响。