Nonlocal effects on the longitudinal vibration of a complex multi-nanorod system subjected to the transverse magnetic field

Abstract

In this communication we examine the free longitudinal vibration of a complex multi-nanorod system (CMNRS) using the nonlocal elasticity theory. Discussion is limited to the cases of two types of boundary conditions, namely, clamped–clamped (C–C) and clamped–free (C–F), where nanorods are coupled in the “Free-Chain” system by an elastic medium. Each nanorod in CMNRS is subjected to the influence of transversal magnetic field. The longitudinal vibration of the system are described by a set of m partial differential equations, derived by using D’Alembert’s principle and classical Maxwell’s relation, which includes Lorentz magnetic force. Analytical expressions for the nonlocal natural frequencies are obtained in closed-form by using the method of separations of variables and trigonometric method. Results for the nonlocal natural frequencies are compared for the special cases of a single and double-nanorod system with the existing results in the literature. Numerical examples are given in order to examine the effects of nonlocal parameter, stiffness coefficient and transversal magnetic field on nonlocal natural frequencies of axially vibrating CMNRS.