基于辛方法的功能梯度圆柱壳振动特性分析

Analysis of vibration characteristics of functionally graded cylindrical shells based on Symplectic method

基于辛方法分析了功能梯度圆柱壳的自由振动特性。从薄壳理论和功能梯度材料特性出发,得到了功能梯度圆柱壳自由振动时的拉格朗日密度函数。引入对偶变量,经哈密顿正则变换,导出了功能梯度圆柱壳自由振动的哈密顿正则方程,将问题转化为求解哈密顿矩阵的辛本征值问题,得到了两端固支和两端简支两种边界条件下功能梯度圆柱壳的量纲为一的固有频率。数值结果表明:简支和固支两种边界条件下功能梯度圆柱壳的量纲为一的固有频率随体积分数、厚径比、环向波数的变化规律基本相同,但在数值上略有差别;量纲为一的固有频率随环向波数的增大呈现先减小后增大的现象,随厚径比的增大而增大,随材料体积分数的增大而逐渐减小。

Free vibration characteristics of functionally graded cylindrical shells are investigated based on Symplectic method.According to the thin shell theory and the properties of functionally graded materials,Lagrange’s density function for free vibration of functionally graded cylindrical shells is given.Introducing the dual variable and using Hamilton canonical transformation,the Hamilton canonical equations for free vibration of functionally graded cylindrical shells are derived,and the original problem is transformed into solving the Symplectic eigenvalue problem of Hamilton matrix.Then the dimensionless natural frequencies of functionallygraded cylindrical shells with the simply supported and the fixed at both ends are obtained.For boundary conditions of simply supported and fixed supported at both ends,respectively,the numerical results show that the change rules of dimensionless natural frequencies for functionally graded cylindrical shell versus the material volume fraction,ratio of the shell thick to average radius and circumferential wave number are basically the same,but there are slight differences in value.The dimensionless natural frequencies firstly decrease and then increase with the increase of the circumferential wave number,increase with the increase of the ratio of the shell thick to average radius,and decrease gradually with the increase of the material volume fraction.