Abstract
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.
Highlights
Experiments show that ignoring the internal length scale, which is the case in the classical continuum mechanics, in micro and nano structures can result in inaccurate structural predictions
Small size effects included in the analysis by considering the length scale parameter
The nonlinear partial differential equation was reduced to some nonlinear ordinary differential equations via Galerkin's discretization technique
Summary
Experiments show that ignoring the internal length scale, which is the case in the classical continuum mechanics, in micro and nano structures can result in inaccurate structural predictions. The results showed that the static deflection and natural frequencies developed by the modified couple stress theory have a significant difference with those obtained by the classical beam theory when the ratio of the beam characteristic size to the internal material length scale parameter is small [29]. Şimşek [42] studied the nonlinear free vibration of AFG microbeams with different boundary conditions based on the modified couple stress theory and Von-Karman's geometric nonlinearity. Free and forced vibration analysis of FG doubly clamped micro-beams was investigated based on the third order shear deformation and modified couple stress theories [48]. The present paper seems to be the first attempt to address the nonlinear free and forced vibration of an AFG microbeam on a nonlinear elastic foundation with doubly-clamped boundary condition, based on the modified couple stress theory and Von-Karman's geometric nonlinearity. Galerkin's method, He's variational method, and He's variational iteration method were employed to determine the nonlinear natural frequency and forced vibration response of the AFG microbeam
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