圆形薄膜自由振动的理论解  

Analytical solution of the free vibration of circular membrane

作　　者：林文静[1] 陈树辉[1] 李森[1] 

机构地区：[1]中山大学应用力学与工程系,广州510275

出　　处：《振动与冲击》2009年第5期84-86,共3页Journal of Vibration and Shock

基　　金：国家自然科学基金项目(10672193)

摘　　要：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θ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。The free vibration of circular membrane was studied.The governing differential equation of membrane was set up by Hamilton's principle.Then,by employing the method of separated variables,two second order differential equations regarding the time t and the radial coordinate r and one Bessel equation regarding the circumferential coordinate θ were derived and solved respectively.The analytical solution of natural frequencies and modes of circular membrane and fan-shaped membrane were obtained.The numerical calculations were also carried out by using the finite element software ANSYS.Two examples show that the theoretical results are in good agreement with those numerical results obtained by ANSYS.The theoretical results are lower limits of the numerical results.

关 键 词：圆形薄膜 自由振动 哈密顿原理 分离变量法 贝塞尔函数 

分 类 号：TU311.3[建筑科学—结构工程]