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.