The deformation field of the flexible beam was described by using the assumed mode method and the finite element method and the dynamic characteristics of a hub-functionally graded material beam undergoing large overall motions were studied. Assuming that the physical parameters of functionally graded materials followed certain kind of power law gradient distribution and varied along the thickness direction, considering both the longitudinal deformation and transversal deformation of the beam, and taking the nonlinear coupling term known as the longitudinal shortening caused by transversal deformation into account, we derived the rigid-flexible coupling dynamics equations of the system described by two different discrete methods with a uniform form via employing Lagrange’s equations of the second kind. The validity of the finite element method established in this paper was 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 was 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 was discussed. The results show that the assumed mode method cannot deal with large deformation problem. When other physical parameters of functionally graded materials beam remain unchanged, the maximum displacement of the beam increases with the increase of functionally graded materials index while the natural frequency of transverse bending of beam increases with the increase of rotational speed, and when rotational speed is constant, the natural frequency will decrease with the increase of functional gradient index.