Eur. Phys. J. Plus (2022) 137: 420
https://doi.org/10.1140/epjp/s13360-022-02594-x

Technical Report

Nonlinear nano-rod-type analysis of internal resonances and geometrically considering nonlocal and inertial effects in terms of Rayleigh axial vibrations

¹ School of Mechanical Engineering, Iran University of Science and Technology, 16846-13114, Narmak, Tehran, Iran
² Center of Excellence in Railway Transportation, Iran University of Science and Technology, 16846-13114, Narmak, Tehran, Iran
³ School of Engineering, Damghan University, Damghan, Iran

^a So_jamali@mecheng.iust.ac.ir

Received: 1 August 2021
Accepted: 13 March 2022
Published online: 4 April 2022

Abstract

In the present study, a comprehensive modeling of the geometrically nonlinear free axial vibration of nanorod in presence of nonlocal and inertial of lateral motions effects is presented to analyze internal modals interactions. The nonlinear axial vibration of the nanoscale rod is studied based on Eringen’s nonlocal theory by considering Method of Multiple Scale (MS). In order to verify the accuracy of this method, the results are compared with the results of fourth-order Runge–Kutta numerical method, which has a good precision. Comparison of this method and He’s variational method shows that the convergence of it is faster than the He’s variational method in results. The governing equation and boundary conditions are derived by Hamilton's principle. The amplitude-frequency relationship and interaction between the modes are studied in terms of the first five frequency modes in detail. Also, comparison of the effects of nonlocal and nonlinear factors for Simple and Rayleigh theories versus variations of aspect ratio (L/d) in terms of clamped–clamped and clamped-free boundary conditions is investigated. Finally, the stringent values of the nonlinear amplitudes corresponding to the internal resonances of the nanoscale rod are computed, and also a comparison between Simple and Rayleigh theories based on nonlinear amplitudes in both boundary conditions is plotted. The results show that nonlocal and inertial of lateral motions effects lead to extra internal modals interactions for nanorod vibrations, and also, with increasing nonlocal coefficient, the nonlinear amplitude of the internal resonances increases. The presented nonlocal solutions can be beneficial for those who are interested in designing micro/nano electromechanical systems. Moreover, the solutions via results can be utilized as a trustworthy reference for evaluating the validity of future researches in free and force vibrations.