Nonlinear Vibrations of Axially Functionally Graded Timoshenko Tapered Beams

This paper presents the coupled axial-transverse-rotational nonlinear forced vibrations of Timoshenko tapered beams made of an axially functionally graded (AFG) material subjected to an external harmonic excitation. Two sources of nonlinearities are considered in modeling and numerical simulations: (i) the geometric nonlinearities arising from induced nonlinear tension due to the clamped–clamped boundary conditions and large deformations, and (ii) nonlinear expressions to address the nonuniform geometry and mechanical properties of the beam along the length. More specifically, a nonlinear model is developed based on the Timoshenko beam theory accounting for shear deformation and rotational inertia. Exponential distributions are presumed for the cross-sectional area, moduli of elasticity, mass density, and Poisson's ratio of the AFG tapered Timoshenko beam. The kinetic and potential energies, the virtual work of the external harmonic distributed load, and the one done by damping are implemented into Hamilton's energy principle. The resultant nonuniform nonlinearly coupled partial differential equations are discretized into a set of nonlinear ordinary differential equations utilizing Galerkin's technique. In the discretization scheme, a large number of modes, both symmetric and asymmetric, are employed due to the asymmetric characteristic of the nonuniform beam with respect to its length. The effect of different parameters, including the gradient index and different taper ratios, on the force-vibration-amplitude and frequency-vibration-amplitude diagrams is examined; the effect of these parameters on the natural frequencies, internal resonances, and asymmetric characteristics of the AFG system is investigated as well.