A simplified-nonlocal model for transverse vibration of nanotubes acted upon by a moving nanoparticle

Abstract

This study provides a simplified solution for estimating the dynamic response of a single-walled carbon nanotube when excited by a moving nanoparticle. At first, the strong form of the equation of motion for a nonlocal Rayleigh nanotube is deduced, and the inertia effect of a moving nanoparticle along a nanobeam is then considered. For obtaining a weak form of the above nonlocal model, we use the Galerkin method, where the test functions are a set of orthogonal polynomials generated from a polynomial satisfying given boundary conditions. This process leads to a second-order differential equation which for a moving load the matrix coefficients are time dependent. In the state-space formulation, the forced response depends upon a transition matrix that can be locally approximated by the matrix exponential by assuming that the coefficients are locally constant. The normalized frequencies for a moving force are calculated and compared to those obtained in previous studies, and good agreement between them was observed. After acquiring the dynamic responses of a nanotube for a wide range of velocities and weights of moving nanoparticles, as well as for the nonlocal effects on a nanobeam, a nonlinear regression analysis is adapted to estimate the response of a nanobeam according to an analogous classical Rayleigh beam. These equivalent results in three multipliers ( $$\alpha$$ , $$\beta$$ , and $$\gamma$$ ) are functions of kinetic parameters and nonlocal effects. Due to the normalization of the variables, these multipliers can be used for various types of beam-like structures in both the nano- and macro-domains. The accuracy of these coefficients is evaluated using the results gained by the analytical solution. This paper offers a remedy for a time-consuming process by means of some simple substitutions.