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 (a, b, and c) 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 nanoand 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. Keywords Single-walled carbon nanotube Á Nonlocal Rayleigh beam theory Á Moving load Á Moving mass Á Conversion coefficient

Free vibration and wave propagation of axially moving functionally graded Timoshenko microbeams

“…Furthermore, the relations for the bending moment and shear force in framework of the nonlocal theory can be obtained by integrating Eqs. (22) and (23) according to the Timoshenko beam theory, shown as…”

“…With regard to thick beams, such a model needs some modifications to include the effect of shear strain. As we know, the Timoshenko beam model [12][13][14][15][16][17][18][19][20][21][22][23], which takes the influences of transverse shear deformation into consideration, is more appropriate for the analyses of thick beams. Accordingly, the nonlocal Timoshenko microbeam model can be established by combining the nonlocal theory with the Timoshenko beam model.…”

Vibrational control scrutiny of physically affected SWCNT acted upon by a moving nanoparticle in the framework of nonlocal–strain gradient theory

“…The radius of the SWCNT is R , the length is L and the thickness is t . In order to express the equation of equilibrium in terms of the mechanical and electrical components of displacement, the stress-strain relation for piezoelectric materials is given by [46][47][48][49][50][51][52][53]:…”

“…Likewise [C] and [ ] denote the elastic stiffness and stress-temperature coefficient matrices, respectively. The nonzero strain and displacement field appropriate to Rayleigh beam takes the form as [46][47][48][49][50][51][52][53]: According to the high-order nonlocal-strain gradient theory developed in Lim et al [28], the stress can be defined as:…”

“…(6) has been recently used widely for the study of micro-and nano-structure elements. Therefore, the bending moments within the SWCNT structure are peculiar to their local counterparts which are proposed as [28,[46][47][48][49][50][51][52][53]:…”