On size-dependent nonlinear vibration of porous and imperfect functionally graded tapered microbeams

In current study, for the first time, the size dependent nonlinear vibration behavior of imperfect uniform and non-uniform functionally graded (FG) microbeams is investigated based on modified couple stress and Euler–Bernoulli theories. Due to difficulty of solving governing nonlinear differential equations of uniform and especially non-uniform microbeams, a few number of authors have studied nonlinear vibration of mechanical structures. It is assumed that a microbeam is made of FG material and for investigating the material properties, two types of porous distributions in a microbeam cross section area are considered. The governing differential equations are obtained using Hamilton's principle and considering the Von-Kármán's nonlinear strain. A generalized differential quadrature method (GDQM) and direct iterative method are presented to obtain the numerical results for the microbeam with simply and clamped edges through three boundary conditions. The influence of changes in some parameters such as nonlinear amplitude, material length scale, rate of thickness, FG index and porosity volume fraction on fundamental normalized frequency is studied and validity of the results is studied by several numerical examples.