Nonlinear Vibration Analysis of Nano-Disks Based on Nonlocal Elasticity Theory Using Homotopy Perturbation Method

Abstract

This paper introduces a novel approach for small-scale effects on nonlinear free-field vibration of a nano-disk using nonlocal elasticity theory. The formulation of a nano-disk is based on the nonlinear model of von Kármán strain in polar coordinates and classical plate theory. To analyze the nonlinear geometric and small-scale effects, the differential equation based on nonlocal elasticity theory was extracted from Hamilton principle, while the inertial and shear-stress effects were neglected. The equation of motion was discretized using the Galerkin method on selecting an appropriate function based on the boundary condition used for the nano-disk. Due to presence of nonlinear terms, the homotopy method was used in conjunction with the perturbation method (HPM) to ease up the solution and completely solve the problem. For further comparison, the nonlinear equations were solved by the fourth-order Runge–Kutta method, the solution of which was compared with that of HPM. Excellent agreements in results were observed between the two methods, indicating that the latter method can simplify the solution, and hence, can be applied to nonlinear nano-disk problems to seek their solution with a high accuracy.