Abstract: The natural frequencies of nonlinear planar vibration of axially moving beams are numerically investigated in the supercritical speed range. The finite difference schemes are presented for the static equilibrium equation in the coupled plane of the beam in the supercritical range, and the non-trivial solutions are obtained. Based on the non-trivial statically equilibrium configuration, a typical governing equation of continuous gyroscopic systems is established in the supercritical range via introducing a coordinate transform. The natural frequencies are investigated for the planar vibration via the 8-term Galerkin method to truncate the corresponding governing equations of the beam in the supercritical state without nonlinear parts under the fixed boundary conditions. The effect of number of the terms of the Galerkin truncation method on the solution of the natural frequencies is also studied by analyzing the numerical results.

Key words: vibration and wave, axially moving beam, nonlinearity, planar vibration, natural frequency, supercritical

摘要： 通过数值方法研究超临界速度下，两端固定边界的轴向运动梁平面耦合非线性振动固有频率。发展有限差分法，确定在超临界范围轴向运动梁的径向与横向耦合平面内非平凡静平衡位形。基于非平凡静平衡位形，经坐标变换，建立超临界轴向运动梁连续陀螺系统的标准控制方程。运用高阶Galerkin截断，研究超临界运动状态下梁平面振动的固有频率；并研究Galerkin截断阶数对计算结果的影响。

关键词: 振动与波, 轴向运动梁, 非线性, 平面振动, 固有频率, 超临界