A Unified Gram-Schmidt–Ritz Solution for Vibration Analysis of Nanoplates with Elastic Boundary Conditions

A novel and unified approach is presented for analyzing the free vibration of rectangular nanoplates with elastic boundary conditions (BCs). The theoretical modeling is achieved using the nonlocal Mindlin plate theory, which accounts for the size-dependent behavior of nanoplates, while the artificial spring technique is employed to accommodate a wide range of BCs, including classical BCs, elastic BCs, and their combinations. The governing equations of motion are derived using the virtual displacement principle, followed by the application of the weighted residual method to obtain the nonlocal quadratic functional. The Rayleigh-Ritz method, employing Gram-Schmidt polynomial series as the admissible displacement functions, is then utilized to solve the eigenvalue problems associated with the free vibration of nanoplates. The present approach is validated through a series of comparison and convergence studies, which demonstrate its high accuracy and low computational cost. Finally, parametric numerical investigations are conducted to elucidate the effects of variations in spring stiffness on the natural frequencies of nanoplates. It is shown that the proposed method can easily compute the natural frequencies of nanoplates with elastic BCs.