Nonlinear forced vibration analysis of functionally graded non-uniform cylindrical microbeams applying the semi-analytical solution

Abstract

The linear and nonlinear forced vibration response of axially functionally graded (AFG) cylindrical truncated conical and imperfect microbeam subjected to the dynamic harmonically load carried out in the presented research. Based on a couple of modified couple stress theory, the Euler-Bernoulli beam theory and von-Kármán theory, the linear and nonlinear governing equations and related boundary conditions for dynamic response of micro-size tubes are derived employing the Hamilton principle. We considered the uniform and nonuniform functions for the cross-section, in which the convex, linear and exponential functions are the nonuniform sections, and the porosity is regarded as an imperfection. The generalized differential quadrature method (GDQM) is used to prepare the initial conditions for homotopy perturbation (HP) techniques as the semi-analytical approach to calculate the linear and nonlinear results of dynamic responses. The obtained linear and nonlinear results of the free and forced vibration response show the negative and positive effects of some parameters such as the porosity parameter, the nonlinear amplitude, the small-scale parameter, AFG parameter, and different cross-section impact on the dynamic deflection and natural frequency of micro-scale tube and beams with both clamped and simply-supported boundary conditions.