Nonlinear Vibration Analysis of Axially Functionally Graded Microbeams Based on Nonlinear Elastic Foundation Using Modified Couple Stress Theory

In this study, a non-classical approach was developed to analyze nonlinear free and forced vibration of an Axially Functionally Graded (AFG) microbeam by means of modified couple stress theory. The beam is considered as Euler-Bernoulli type supported on a three-layered elastic foundation with Von-Karman geometric nonlinearity. Small size effects included in the analysis by considering the length scale parameter. It is assumed that the mass density and elasticity modulus varies continuously in the axial direction according to the power law form. Hamilton's principle was implemented to derive the nonlinear governing partial differential equation concerning associated boundary conditions. The nonlinear partial differential equation was reduced to some nonlinear ordinary differential equations via Galerkin's discretization technique. He's Variational iteration methods were implemented to obtain approximate analytical expressions for the frequency response as well as the forced vibration response of the microbeam with doubly-clamped end conditions. In this study, some factors influencing the forced vibration response were investigated. Specifically, the influence of the length scale parameter, the length of the microbeam, the power index, the Winkler parameter, the Pasternak parameter, and the nonlinear parameter on the nonlinear natural frequency as well as the amplitude of forced response have been investigated

Nonlinear Dynamic Response of an Axially Functionally Graded (AFG) Beam Resting on Nonlinear Elastic Foundation Subjected to Moving Load

In recent years, structures made of Functionally Graded materials (FGMs) are used in industries due to the continuously compositional variation of the constituents in FGMs along different directions. In order to develop FGMs, nonlinear vibration analysis to study dynamic behavior is needed. This study proposes nonlinear vibration analysis of a simply supported axially functionally graded (AFG) beam subjected to a moving harmonic load as an Euler-Bernoulli beam utilizing Green’s strain tensor. Axial variation of material properties of the beam is based on the power law. The governing equations of motion are derived via Hamilton’s principle. The Galerkin’s method is implemented to reduce the nonlinear partial differential equations of the system to a number of nonlinear ordinary differential equations. He’s variational method is applied to obtain approximate analytical expressions for the nonlinear frequency and the nonlinear dynamic response of the AFG beam. The effect of some parameters such as the power index and stiffness coefficients, among others, on the nonlinear natural frequency has been investigated. The influence of above mentioned parameters as well as the velocity of the moving harmonic load on the nonlinear dynamic response has been studied. The results indicate that these parameters have a considerable effect on both nonlinear natural frequency and response amplitude