Forced Axial Vibration of a Single-Walled Carbon Nanotube Embedded in Elastic Medium under Various Moving Forces

Abstract

The dynamic free and forced axial vibrations subjected to moving exponential and harmonic axial forces of a single-walled carbon nanotube (SWCNT) embedded in an elastic medium, are studied in this paper. Two different boundary conditions of SWCNT, including clamped-clamped and clamped-free, are taken into account. Eringen’s nonlocal elasticity theory is used to show the nonlocality for the model. The constitutive equations and their boundary conditions are derived by Hamilton’s principle. Employing the general solution, the derived equations are analytically solved to obtain two items. Firstly, the axial natural frequencies, secondly, the time-domain axial displacements at the middle of the carbon nanotube (CNT), and then the maximum axial displacements. The responses are validated with previous works, and the results demonstrates good agreement to them to verify the influence of the nonlocal parameter on the nondimensional natural frequencies for three various mode numbers. In the time-domain section, the effects of the nonlocal parameter, length, nondimensional stiffness of the elastic medium, and velocity of the moving load on the axial displacement are investigated. Also, the influences of the excitation frequency to natural frequency for the harmonic moving load, as well as the time constant for the exponential moving load on the axial displacement, are illustrated. Finally, the effect of the nonlocal parameter on the maximum axial deflection versus velocity parameter is schematically indicated.