Vibrations of double-nanotube systems with mislocation via a newly developed van der Waals model

• Frequency analysis of doubly parallel SWCNT with arbitrary configuration is studied.
• A new continuum-based vdW force model is developed accounting for mislocation.
• Nonlocal shear deformable beams are employed and their equations are obtained.
• The limitations of the nonlocal Timoshenko beam are explained for various BCs.
• The roles of crucial factors on first five frequencies are explored comprehensively.

This study deals with transverse vibrations of two adjacent-parallel-mislocated single-walled carbon nanotubes (SWCNTs) under various end conditions. These tubes interact with each other and their surrounding medium through the intertube van der Waals (vdW) forces, and existing bonds between their atoms and those of the elastic medium. The elastic energy of such forces due to the deflections of nanotubes is appropriately modeled by defining a vdW force density function. In the previous works, vdW forces between two identical tubes were idealized by a uniform form of this function. The newly introduced function enables us to investigate the influences of both intertube free distance and longitudinal mislocation on the natural transverse frequencies of the nanosystem which consists of two dissimilar tubes. Such crucial issues have not been addressed yet, even for simply supported tubes. Using nonlocal Timoshenko and higher-order beam theories as well as Hamilton's principle, the strong form of the equations of motion is established. Seeking for an explicit solution to these integro-partial differential equations is a very problematic task. Thereby, an energy-based method in conjunction with an efficient meshfree method is proposed and the nonlocal frequencies of the elastically embedded nanosystem are determined. For simply supported nanosystems, the predicted first five frequencies of the proposed model are checked with those of assumed mode method, and a reasonably good agreement is achieved. Through various studies, the roles of the tube's length ratio, intertube free space, mislocation, small-scale effect, slenderness ratio, radius of SWCNTs, and elastic constants of the elastic matrix on the natural frequencies of the nanosystem with various end conditions are explained. The limitations of the nonlocal Timoshenko beam theory are also addressed. This work can be considered as a vital step towards better realizing of a more complex system that consists of vertically aligned SWCNTs of various lengths.

A general configuration of a system of doubly parallel SWCNTs. Frequency analysis of doubly parallel SWCNTs embedded in an elastic matrix is of highly interest. To this end, a powerful meshfree method is implemented to solve the integral-partial differential governing equations resulted from the nonlocal Timoshenko and higher-order beam theories.Figure optionsDownload as PowerPoint slide