Geometrically nonlinear free vibration analysis of axially functionally graded taper beams

Large amplitude free vibration analysis is carried out on axially functionally graded (AFG) tapered slender beams under different boundary conditions. The problem is addressed in two parts. First the static problem corresponding to a uniform transverse loading is solved through an iterative scheme using a relaxation parameter and later on the subsequent dynamic problem is solved as a standard eigenvalue problem on the basis of known static displacement field. The mathematical formulation of the static problem is based on the principle of minimum total potential energy, whereas Hamilton's principle has been applied for the dynamic analysis. To account for the geometric non-linearity arising due to large deflection, nonlinear strain displacement relations are considered. The dynamic behaviour has been presented in the form of backbone curves in a dimensionless frequency–amplitude plane. The results are successfully validated with the previously published results.