Analytical assessment of nonlinear forced vibration of functionally graded porous higher order hinged beams

Abstract

This study focuses on the nonlinear free and forced vibration behavior of functionally graded porous beams considering high-order bidirectional porosity distributions. It is assumed that the closed-cell voids are distributed non-uniformly along the thickness and length of the beam. A nonlinear formulation is derived based on the Reddy beam theory, von Karman geometrical nonlinearity using the Hamilton principle. Subsequently, Galerkin technique is implemented to reduce the partial differential equations to a set of nonlinear ordinary differential equations in time. The primary resonance of the system is inspected when the beam is under a combination of a harmonic transverse load and a static compressive axial load. Hence, the harmonic balance method and method of multiple scales are used to develop closed form approximate solutions. It is shown, the method of multiple scales underestimates the amplitude of vibration when either amplitude of vibration or axial load, or both are high. Effects of beam’s aspect ratio, porosity distributions, beam’s shear deformation and porosity volume fraction are studied. Results revealed, the proposed porosity distributions are more effective than the conventional porosity distributions. Furthermore, comparing beams with the same mass reveals that beams with more voids at the center have lowest amplitude of vibration.