R-函数理论及准Green函数方法在正交各向异性板振动问题中的应用  

R-function theory and Quasi-Green function method for free vibration of clamped orthotropic thin plates

作　　者：李善倾[1] 袁鸿[1] 

机构地区：[1]暨南大学力学与土木工程系,广州510632

出　　处：《应用力学学报》2014年第2期176-181,305,共6页Chinese Journal of Applied Mechanics

基　　金：国家自然科学基金重点项目(11032005);中央高校基本科研业务费专项资金资助(11613328);广东省科技计划项目(2012A030200003);广东省自然科学基金博士启动项目(32213013)

摘　　要：首次将R-函数理论及准Green函数方法应用于求解固支正交各向异性薄板的自由振动问题。首先引入参数变换,将正交各向异性薄板的自由振动微分方程转化为双调和算子的边值问题,并应用R-函数理论,以解析函数形式描述复杂边界形状;利用问题的基本解和边界方程构造了一个准Green函数,该函数满足了问题的齐次边界条件;通过R-函数理论构造适当的边界方程,消除了积分方程核的奇异性;再采用Green公式将其化为第二类Fredholm积分方程。数值算例表明:该方法减少了理论计算量,精度较高。本文还证明了其优越性和正确性,是一种新型的数学方法。The R-function theory and quasi-Green's function approach are employed to solve the free vibration of clamped orthotropic thin plates. Firstly, the governing differential equations are reduced to the boundary value problem of the biharmonic operator by introducing the parameters transformation. The R-function theory is used to describe the complicated boundary shape in analytical function form. A quasi-Green's function is established by using the fundamental solution and boundary equations of the problem. This function satisfies the homogeneous boundary condition of the problem. Irregularity of the kernel of integral equation is overcome by choosing a suitable normalized boundary equation form. Then the differential equation is reduced to Fredholm integral equation of the second kind by Green formula. Numerical example shows that the method is much less theoretical calculation and the accuracy is better. The superiority and correctness of the method is demonstrated, and it is actually a novel mathematical method.

关 键 词：GREEN函数 积分方程 R-函数 正交各向异性板 自由振动 

分 类 号：O326[理学—一般力学与力学基础]