Quantum mechanics

Galerkin method

Beam (structure)

Vibration

Acoustics

Classical mechanics

Hamilton's principle

Physics

Nonlinear system

Multiple-scale analysis

Mathematics

Optics

Equations of motion

Axial symmetry

Mathematical analysis

Mechanics

Applied Mathematical Modelling

• Nonlinear behaviour is of a hardening type. • Both symmetric and asymmetric modes are present. • Larger gradient index results in shift in resonance. This paper aims at investigating the nonlinear vibration characteristics of axially functionally graded (AFG) shear deformable tapered beams subjected to external harmonic excitations. A coupled nonlinear longitudinal-transverse-rotational set of equations governing the motion of the AFG system is derived utilising the third-order shear deformation beam theory via Hamilton's energy principle. The beam under consideration is tapered; i.e. the width of the beam varies along the length. The tapered geometry along with the nonuniform material properties arising from the AFG nature of the beam increases the complexity in the modelling and numerical simulations. The expressions for the kinetic and potential energies of the AFG shear-deformable tapered beam together with the expressions for the work of damping and external excitation are derived and implemented in Hamilton's principle. The nonlinear partial differential equations are discretised making use of the Galerkin technique and solved with the aid of a continuation scheme. The effect of different parameters such as the gradient index and the tapered ratios on the force- and frequency-amplitude diagrams of the AFG system is examined.

Nonlinear vibration analysis of axially functionally graded shear-deformable tapered beams