Investigating nonlinear forced vibration of functionally graded nanobeam based on the nonlocal strain gradient theory considering mechanical impact, electromagnetic actuator, thickness effect and nonlinear foundation

Abstract

The present study proposes an investigation to nonlinear vibration response of a functionally graded (FG) nanobeam based on nonlocal strain gradient theory (NSGT) taking into consideration the longitudinal magnetic field and the thickness effect. The nonlinear terms from the governing equation are associated with the curvature of the nanobeam, the nonlinear foundation, the mean axial extension of the nanobeam and electromagnetic actuator for pinned-pinned nanobeam and the effect of discontinuities marked by the presence of the Dirac function (mechanical impact).The governing differential equation of motion and the boundary conditions are modeled within the framework of a simple supported Euler-Bernoulli nanobeam which accounts for the nonlinear von-Kármán strain and using so-called Hamilton's principle. To truncate the continuous system having an infinite degree of freedom, the Galerkin-Bubnov procedure is applied. The resulting nonlinear differential equation is studied by Optimal Auxiliary Functions Method. An explicit analytical solution is proposed for a complex problem near the primary resonance. Our proper procedure is a powerful tool to solve a nonlinear problem without presence of any small parameters. The main characteristic of our technique consists in the existence of some auxiliary functions derived from the expressions of the solution formed for initial linear equation and the form of nonlinear terms calculated for the solution of the linear equation. Also, the presence of some convergence-control parameters assures the rapid convergence of the solutions after only a single iteration. These parameters can be evaluated by means of some rigorous mathematical procedures. We have a large freedom to select the auxiliary functions and the number of convergence-control parameters. The proposed approach is proved to be very accurate, simple, and easy to implement for complicated nonlinear problems. The potential energy is analyzed in detail for different values of the parameters. The local stability of motion near the primary resonance is established using homotopy perturbation method.