轴向运动薄板的自由振动分析

首次利用解析法求解了轴向运动薄板的自由振动问题，并对解析结果进行了Galerkin法验证。基于Kirchhoff薄板理论，根据Hamilton原理推导轴向运动薄板自由振动的控制方程，分别采用解析法和Galerkin法求解控制方程，得到了四边简支条件下系统固有频率的解析解和数值解。同时，得到了第一阶临界速度的解析表达式。轴向速度为零时，对比了解析解、Galerkin数值解和ANSYS软件解，三种方法所得结果高度吻合。随后对比了不同速度条件下的解析解与Galerkin解，分析了预应力与临界速度的关系。发现在低速条件下离心力是影响系统振动的主要因素，科氏力影响可忽略；第一阶固有频率的解析解仅适用于低速条件，高阶固有频率的解析解适用的速度范围大。

FREE VIBRATION ANALYSIS OF THE AXIALLY MOVING THIN PLATES

For the first time, the analytical method is used to solve the free vibration problem of an axially moving thin plate, and the analytical results are verified by Galerkin method. Based on Kirchhoff''s thin plate theory, the governing equation of the free vibration of an axially moving thin plate is derived according to Hamilton''s principle. The governing equation is solved by the analytic method and Galerkin method respectively, and the analytical and numerical solutions of the natural frequency of the system are obtained under the boundary condition of four sides simply supported. At the same time, the analytical expression of the first order critical velocity is obtained. When the axial velocity is zero, the analytical solution, Galerkin solution and ANSYS software solution are compared, and the results are in good agreement. Then the analytical solution and Galerkin solution under different velocities are compared, and the relationship between the prestress and critical velocity is analyzed. It is found that the centrifugal force is the main factor affecting system vibration at low velocity, and the coriolis force can be ignored. The analytical solution of the first order natural frequency is only suitable for the low velocity condition, and the analytical solution of the higher order natural frequency is suitable for a wide range of velocities.