Axial free vibration analysis of a tapered nanorod using Adomian decomposition method

This study aimed to conduct an analysis of the axial free vibration of tapered nanorods based on nonlocal elasticity theory. The small-scale effect on the free axial vibration of a tapered nanorod was studied employing the Adomian decomposition method (ADM) and the finite difference method (FDM) as a checking tool where a contradiction existed between the results of this study and available results in one highly cited work in the literature, which was used for comparison purposes in this work. Different boundary conditions for the nanorod were considered: fixed-fixed nanorod, fixed-free nanorod, and fixed-linear spring nanorod. The governing equation of the problem is a variable coefficient differential equation for which analytical solutions are strictly limited. For this type of problem, analytical approximate methods are effective, and there are many studies available in the literature on the application of these methods to solve linear/nonlinear ordinary/partial differential equations. ADM is one of the methods and was successfully used in this study to analyze the free vibration of nanorods. The results obtained in this study have shown that the presented technique is so powerful and has potential for applications in nanomechanics based on nonlocal elasticity theory.