Nonlinear Vibration Analysis of Nano-disks Based on Nonlocal elasticity Theory Using Variational Iteration Method

In this study, the small scale effect on the nonlinear free-field vibration of a Nano-disk has been investigated using Nonlocal elasticity Theory. The formulation is based on the classical theory and the non-linear model of von Kármán strain. To take into account the small scale and the nonlinear geometric effects, the governing differential equation based on the Nonlocal elasticity theory was extracted according to Hamilton principle, while the shear stresses as well as the inertial effect were ignored. Von Karman geometric model was used in this process. Moreover, selecting the appropriate function based on the boundary condition for the Nnao-disk, the equation of motion was discreted using the Galerkin method. Regarding the existence of nonlinear terms, the variational iteration method (VIM) was used to solve the discreted equations. In this method, using a correction function and finding a general Lagrange coefficient, the equation was converted into a recursive sequence, the limit of which is considered as the solution of the equation. For further comparison, the nonlinear equation was solved using the fourth order Runge-Kutta method. According to the results, a very good agreement is observed between the two methods, while the former method makes the solution much easier.

Keywords:

Nonlinear free vibration , Nano-disks , Nonlocal elasticity Theory , Von Karman strain