Nonlocal effect on the axisymmetric nonlinear vibrational response of nano-disks using variational iteration method

Abstract

In this study, the nonlinear free vibration of a nano-disk considering small scale effects has been investigated by using the nonlocal elasticity. To take into account the nonlinear geometric effects, the nonlinear model of von Karman strain has been used while the governing differential equation was extracted according to Hamilton principle. The Galerkin weighted residual method in conjunction with the variational iteration method (VIM) was introduced to solve the governing equations for simply supported and clamped edge boundary conditions. For further comparison, the nonlinear equation was solved using the fourth-order Runge-Kutta method. Very good agreements were observed between the results of both methods, while the former method made the solution much easier. Additionally, it was observed that the ratio of thickness to radius, h/R, plays an important role on the nonlinear frequencies. This effect appears to be minute if the local elasticity theory is adopted. However, results indicated that the nonlocal effect may be ignored provided h/R ratio is very small.