Resonant dynamics of axially functionally graded imperfect tapered Timoshenko beams

Abstract

This paper addresses the nonlinear resonant dynamics of axially functionally graded (AFG) tapered beams subjected to initial geometric imperfections, based on the Timoshenko beam theory. A rigorous coupled axial–transverse–rotational nonlinear model is developed taking into account the geometric nonlinearities due to the large deformations coupled with an initial imperfection along the length of the beam as well as nonlinear expressions accounting for nonuniform tapered geometry and mechanical properties. The Hamilton’s energy principle is used to balance the kinetic and potential energies of the AFG imperfect tapered Timoshenko beam with the work done by damping and the external excitation load. This results in a set of strongly nonlinear partial differential equations (PDEs). The Galerkin decomposition scheme involving an adequate number of both symmetric and asymmetric modes is utilized to reduce the PDEs to a set of nonlinear ordinary differential equations. A well-optimized numerical scheme is developed to handle the high-dimensional discretized model.