Non-linear vibration analysis of circular nano-plate based on nonlocal strain gradient method

The non-linear vibration analysis of a clamped circular nano-plate is performed using the nonlocal strain gradient model. For this purpose, a combination of the differential form of the nonlocal elasticity theory and strain gradient model, along with Hamilton’s principle, are used in conjunction with the von Kármán nonlinear strain-displacement relationships and Galerkin weighted residual method to discretize the governing equations. The analysis has focused on the effect of nonlocal and material parameter, initial conditions, and frequency number, on the overall behavior of the nano-plate. Using Galerkin method, the system of non-linear differential equations is obtained and the natural non-linear frequencies, as well as the mode shapes, are determined. Results indicate that increasing the parameters of nonlocal and material has a decreasing and increasing effect (respectively) in the frequency ratio in all modes. This indicates that these parameters have a softening and hardening behavior on the plate vibrational behavior respectively. Also, increasing the initial value of deflection produces a rising trend in the frequency ratio for all modes, nonlocal parameter λ and material parameter Ls.