Vibration analysis of shear-deformable AFG imperfect beams

The vibrations of axially functionally graded (AFG) beams is examined incorporating shear deformation and an imperfection. A coupled nonlinear model is developed in the Hamiltonian framework for an AFG imperfect beam. Different sources of displacement nonlinearities are involved in this problem, namely, the geometrical nonlinearities due to the large deformations, and the nonlinearities due to the presence of a geometrical imperfection along the length of the beam. Expressions for the virtual work of external excitation and damping as well as formulas for the motion and elastic energies of the system are inserted into Hamilton's energy/work principle. The equations are truncated employing Galerkin's technique while incorporating symmetric as well as asymmetric modes. It is shown that the nonuniform material properties along with the tapered geometry of the beam as well as the presence of geometric imperfections results in the contribution of both asymmetric and symmetric vibration modes. Hence, a well-optimised scheme on the basis of a continuation method is developed to solve the high DOF nonlinear problem. Numerical simulations are performed to investigate the influences of the gradient index, the imperfection and the taper values.