圆形薄膜自由振动的理论解

本文研究圆形薄膜的自由振动。首先根据哈密顿原理建立薄膜横向振动的动力学方程，然后采用分离变量法，导出时间t\、径向坐标r和环向坐标 变量分离的2个二阶常微分方程和1个贝塞尔方程并分别求解，求得周边固定圆形薄膜、扇形薄膜自由振动的理论解，从而得到固有频率及其振型的解析表达式。最后，应用ANSYS有限元计算软件计算上述几种类型自由振动的频率及其模态并与理论解比较。ANSYS有限元数值解与理论解二者十分接近，理论解是有限元数值解的下限。

Analytical solution of the free vibration of circular membrane

The free vibration of circle membrane is studied in this paper. Firstly, the governing differential equation of membrane is set up by Hamilton’s principle. Then, by employing the method of separated variables, two second order differential equations regarding the time t, the radial direction coordinate r and one Bessel equation regarding the circumferential direction coordinate are derived. These equations are solved respectively. Therefore, the analytical solution of natural frequencies and modals of circle membrane and fan-shaped membrane are obtained. Finally, the numerical calculations are also taken by using the finite element software ANSYS. Two examples show that the theoretical results are pretty good agreement with those numerical results obtained by ANSYS. The theoretical results are lower limits of the numerical results.