A nonlocal meshless solution for flexural vibrations of double-walled carbon nanotubes

•Free lateral vibrations of DWCNTs with arbitrary boundary conditions are studied.•DWCNTs are modeled based on the nonlocal Rayleigh beam theory.•An efficient meshfree approach is developed for solving the governing equations.•The role of influential factors on the flexural frequencies of DWCNTs is addressed.•Vibration mode shapes of DWCNTs under different conditions are extracted.

The true understanding of free vibration of double-walled carbon nanotubes (DWCNTs) plays a vital role in optimal design and dynamic control of such nanostructures. This paper is aimed to examine free flexural vibration of lengthy DWCNTs with arbitrary boundary conditions in the framework of nonlocal elasticity theory. The DWCNTs are embedded in an elastic medium and are subjected to initially axial forces. Equivalent continuum structures associated with the innermost and outermost tubes of the DWCNT are considered. The transverse and rotational interactions of the DWCNT with the surrounding elastic medium are also taken into account. The generalized equations of motion of lengthy DWCNTs are established based on the nonlocal Rayleigh beam theory. Seeking an analytical solution to the developed equations, particularly in their general form, is a very problematic task. As an alternative solution, an efficient numerical scheme is proposed. The effects of slenderness ratio, small-scale parameter, lateral and rotational stiffness of the surrounding matrix, and initially axial force on the first five natural frequencies of DWCNTs under different boundary conditions are comprehensively scrutinized.