中文摘要：

对功能梯度材料制成的环形截面梁,假设材料的物性参数沿壁厚方向按幂率变化,基于Lagrange函数和Hamilton原理,建立了该梁横向自由振动的Hamilton对偶方程组.采用辛方法求解了Hamilton矩阵的辛本征问题,得到了简支、两端固定、悬臂和左端固定右端铰支4种约束的FGM（functionally graded materials）环形截面梁的固有频率和振型函数.算例给出了这4种约束的FGM环形截面梁前8阶无量纲固有频率随材料体积分数的变化规律,分析了材料体积分数对FGM环形截面梁固有频率的影响.

英文摘要：

For a beam of annular cross-section made of functionally graded materials（FGM）, we assume that the physical parameters of the materials along the direction of the wall thickness varying in a simple power law. Based on the Lagrange＇s function and the Hamilton＇s principle, the Hamilton＇s canonical equations for the transverse free vibration of the beam are established. A symplectic eigenvalue problem of the Hamilton matrix is solved by using the symplectic method. Then, the natural frequencies and the vibration mode functions of the beam are obtained, with conditions on the two ends as the simply supported, the fixed, the cantilever and the fixed-simply supported. Numerical examples are given for the first eight order dimensionless natural frequencies against the material volume fraction, and the effect of the material volume fraction on the natural frequency is analyzed.