Approximate Analytical Solution of Nonlinear Natural Frequencies of a Functionally Graded Material Microbeam by using Multiple Harmonic Balance Method

Functionally graded materials (FGMs) provide spatial change of mechanical properties and high performance in thickness direction compared to homogeneous materials which promotes their use in micro/nano scale systems and devices. In micro scale, atomic and molecular interactions due to strong atomic forces affect deformation and stiffness of the beam. Therefore, in this paper nonlinear free vibrations of FGM microbeams are studied by using Modified couple stress theory in order to include the small size effects. Equations of motions of the FGM microbeam are derived by using Euler-Bernoulli beam theory and Hamilton's principle. Power law variation of material properties is taken into account to introduce fractional composition of metallic and ceramic phases in the FGM material. Von Karman geometric nonlinearity as a result of large deformation of the beam is considered and nonlinear ordinary differential equations are obtained by applying Galerkin's method. Multiple harmonic balance method is used to obtain a set of nonlinear algebraic equations which are solved by Newton's method. Employing developed model, effects of higher harmonics, material property index and length scale parameter on the nonlinear natural frequency as a function of vibration amplitude is studied for the case of a simply supported microbeam.