圆环形薄膜自由振动的理论解  

The Analytical Solution of Free Vibration of Annular Membrane

作　　者：林文静[1] 陈树辉[1] 张启明[1] 

机构地区：[1]中山大学工学院应用力学与工程系,广东广州510275

出　　处：《中山大学学报（自然科学版）》2008年第S2期103-108,共6页Acta Scientiarum Naturalium Universitatis Sunyatseni

基　　金：国家自然科学基金资助项目(10672193)

摘　　要：为研究出圆环形薄膜的自由振动,采用解析方法,求得几何形状简单规则的圆环形平面薄膜、扇环形薄膜的固有频率及其振型的理论解。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。最后给出圆环形薄膜、扇环形薄膜自由振动两个算例,说明固有频率及其模态的理论解和用有限元软件ANSYS求得的数值解十分接近,理论解是有限元数值解的下限。The analytical solution of free vibration of annular membrane is studied in this paper.The natural frequencies and modals of annular membrane and sectorial-annular-shaped membrane are obtained by using analytical method.Firstly,the governing differential equation of membrane is derived 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 θ are obtained.These equations can be solved respectively.Finally,two examples are given to show that the theoretical solutions of the natural frequencies and modals of annular membrane and sectorial-annular-shaped membrane are in pretty good agreement with those of numerical solutions obtained by the finite element software ANSYS.The theoretical solutions are the lower limits of the numerical results.

关 键 词：圆环形薄膜 自由振动 哈密顿原理 分离变量法 贝赛尔函数 

分 类 号：O327[理学—一般力学与力学基础]