Geometrically Nonlinear Coupled Modeling and Frequency Analysis of Rotating AFG Beams Incorporating Complete Gradient and Strain Formulations

This paper develops a geometrically nonlinear coupled finite element model for high-speed rotating AFG beams using the complete Green–Lagrange strain tensor. The model extends the complete Green–Lagrange formulation, previously used for homogeneous rotating beams, to AFG beams and retains all higher-order axial nonlinear terms. Instead of directly adding a density-dependent centrifugal stiffening term, it solves the steady-state axial deformation from dynamic equilibrium and uses it as an intermediate variable carrying the joint distribution of E(x) and ρ(x), so that both properties enter the geometric stiffening terms. The gyroscopic coupling matrix G is fully retained. Numerical results show that (i) under the present Green–Lagrange formulation, the model predicts a monotonic stiffening trend consistent with complete stiffening in the limiting case at high rotational speeds, whereas the conventional centrifugal-force model can exhibit nonphysical softening under the fully coupled formulation; (ii) the coupled model captures frequency veering and mode exchange; and (iii) the steady-state tip deformation and first bending frequency show opposite trends with k. These results provide a reference for gradient design of rotating AFG beams within the considered parameter range.