In this thesis, strain gradient elasticity theory containing three material length scale parameters is applied to analyze the in-plane and out of plane frequencies of free vibration behavior of a simply supported functionally graded nanoplate. The material properties of FGM nanoplate are assumed to vary continuously and across the thickness direction on the basis of simple power law function and technique. To describe deformation of FGM nanoplate, a general displacement filed will be used which includes several kinematic theories such as principle, the equations of motion and boundary conditions are derived. Galerkin’s procedure is implemented to solve the governing equations and achieve mass and stiffness matrices of the system. For validity and accuracy of the present solution, obtained numerical results are compared with the previous literature. Finally, selected numerical results are presented to investigate the influences of material length scale parameters, power law index, length to thickness ratio, and transverse shear deformation on the in-plane and out of plane frequencies of functionally graded nanoplate with the simply supported boundary condition at all edges. Considering the obtained results, it was clear that due to increasing stiffness of the system, the frequency of the size-dependent plate model is more than when the plate thickness becomes comparable to material length scale parameters. these size effects decrease or even disappear as the thickness of the plate is far greater than the material length scale parameters. In addition, increasing the power law index and decreasing plate thickness would decrease natural frequencies in both Keywords Free vibration, Functionally graded nanoplate, Strain gradient elasticity theory, Galerkin’s procedure