▎ 摘　　要

This paper presents a simple exact solution for coupled nonlinear free vibration analysis of are used in nonlinear dynamics, they are employed to find the exact solution of a coupled system for the first time. The nonlinear dynamic equation including geometric nonlinearity and Eringen nonlocal theory is uncoupled by elliptical functions, and exact solutions for simply supported boundary conditions are obtained. The results are compared with the harmonic balance method. The nonlinear frequency of MLGS is studied for its effects with a small scale parameter, and the linear and nonlinear van der Waals force.