Amirkabir University of Technology AUT Journal of Mechanical Engineering 2588-2937 9 4 2025 10 01 A Unified Gram-Schmidt–Ritz Solution for Vibration Analysis of Nanoplates with Elastic Boundary Conditions 413 430 5793 10.22060/ajme.2025.24153.6178 EN Hossein Pakdaman Department of Civil Engineering, K.N. Toosi University of Technology, Valiasr Ave., Tehran, Iran Journal Article 2025 05 16 A novel and unified approach is presented for analyzing the free vibration of rectangular nanoplates with elastic boundary conditions. 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 boundary conditions, including classical boundary conditions, elastic boundary conditions, 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 boundary conditions. https://ajme.aut.ac.ir/article_5793_fe256faf97c200de0f7486ddf56c02f6.pdf