A New Analytical Approach for Free Vibration, Buckling and Forced Vibration of Rectangular Nanoplates Based on Nonlocal Elasticity Theory

In this paper, an analytical Hamiltonian-based model for the dynamic analysis of rectangular nanoplates is proposed using the Kirchhoff plate theory and Eringen’s nonlocal theory. In a symplectic space, the dynamic problem is reduced to solving a unified Hamiltonian dual equation formed by a total unknown vector consisting of displacements, rotation angles, bending moments and generalized shear forces. The exact solutions for free vibration, buckling and steady state forced vibration are established by the eigenvalue analysis and expansion of eigenfunction without any trial functions. In addition, the explicit expressions of the characteristic equations, mode functions and steady state response of the nanoplate with two opposite edges that are simply supported or guided supported are obtained. To verify the accuracy and reliability of the present method, numerical results are compared with published solutions and excellent agreement is obtained. Comprehensive benchmark results that consider the nonlocal effect on the dynamic behaviors of rectangular nanoplates are also presented in dimensionless tabular and graphical forms.