Nonlinear vibrations of a functionally graded material microbeam with geometric nonlinearity

Abstract

In this paper, nonlinear vibration analysis of micro scale functionally graded material (FGM) beams with geometric nonlinearity due to large deflection is studied using modified couple stress theory (MCST). MCST is a nonlocal elasticity theory which includes a material length scale parameter since the size of an atomic microstructure becomes comparable to the length of the microbeam. Equations of motion of the micro scale FGM beam are obtained by using Hamilton’s principle. Nonlinear free vibrations of the FGM microbeam with simply supported boundary conditions is investigated where the effect of the length scale parameter on the nonlinear natural frequencies of the microbeam is studied. The nonlinear partial differential equations of motion are converted into nonlinear ordinary differential equations by using Galerkin’s Method. By using describing function method (DFM), a set of nonlinear algebraic equations are obtained which are solved by an iterative eigenvalue solver.