Primary and secondary resonance analysis of porous functionally graded nanobeam resting on a nonlinear foundation subjected to mechanical and electrical loads

Nonlinear free and forced vibration behavior of a porous fiinctionally graded Euler-Bernoulli nanobeam subjected to mechanical and electrical loads is studied based on the nonlocal strain gradient elasticity theory. It is assumed that the porous functionally graded (FG) nanobeam is resting on a nonlinear foundation. Also, material properties of the nanobeam are assumed to vary in the thickness direction. Equations of motion are derived using Hamilton's principle. Galerkin method along with variation iteration method (VIM), Homotopy perturbation method (HPM), Hamiltonian approach method (HAM) and multiple scale method are employed to solve the governing equations based on clamped-clamped, simply-simply and clamped-simply boundary conditions. For verification purposes, the results of this study are compared with those of other studies. The effects of different parameters such as type of porosity distribution, nonlinear foundation, boundary conditions, electrical voltage and size effect parameters on the primary and secondary resonances were investigated. It was found that length scale parameters have a crucial role on the nonlinear vibration behavior of such structures.