采用假设模态法和有限元法两种离散方法描述柔性梁的变形场，对作大范围运动的中心刚体-功能梯度材料梁的动力学特征进行研究。假设功能梯度材料的物理参数为沿着梁厚度方向变化的幂函数，考虑梁的纵向拉伸变形和横向弯曲变形，同时计及横向弯曲变形引起的纵向缩短，即非线性耦合项。运用第二类Lagrange方程推导得到两种不同离散方法描述的具有统一形式的系统刚柔耦合动力学方程。通过与假设模态法的数值仿真结果对比，验证本文建立有限元模型的正确性。通过大变形算例，说明基于小变形假设的假设模态法计算上的局限性。在此基础上讨论功能梯度指数对作大范围转动柔性梁动力学特性的影响。结果表明基于小变形假设的假设模态法并不能处理大变形问题；在功能梯度材料梁其他物理参数不变的条件下，梁的最大位移随着功能梯度指数增大而增大；横向弯曲固有频率会随着转速的增加而变大；当转速一定时，固有频率会随着功能梯度指数增大而减小。

The dynamic characteristics of a hub-functionally graded material beam undergoing large overall motions are studied. The deformation field of the flexible beam is described by using the assumed mode method and the finite element method. Assuming that the physical parameters of functionally graded materials follow certain kind of power law gradient distribution and vary along the thickness direction. The longitudinal deformation and transversal deformation of the beam are both considered, and the nonlinear coupling term which is known as the longitudinal shortening caused by transversal deformation is also taken into account. The rigid-flexible coupling dynamics equations of the system described by two different discrete methods which have a uniform form are derived via employing Lagrange’s equations of the second kind. The validity of the finite element method established in this paper is verified by comparison with the numerical simulation results of the assumed mode method. The limitation of the assumed mode method based on small deformation assumption is illustrated by the example of large deformation. On this basis, the influence of functional gradient distribution rules on the dynamic characteristics of flexible beams undergoing large overall motions is discussed. The results show that the assumed mode method cannot deal with large deformation problem. Remaining other physical parameters of functionally graded materials beam unchanged, the maximum displacement of the beam increases with the increase of functionally graded materials index. The natural frequency of transverse bending of beam increases with the increase of rotational speed, when rotational speed is constant, the natural frequency will decrease with the increase of functional gradient index.

国家自然科学基金项目（11772158,11802263）；江苏省自然科学基金青年基金项目（BK20180895）;中央高校项目（30917011103）