Vibration analysis of two orthogonal slender single-walled carbon nanotubes with a new insight into continuum-based modeling of van der Waals forces

Transverse vibrations of doubly orthogonal slender single-walled carbon nanotubes (SWCNTs) at the vicinity of each other are of interest. The van der Waals (vdW) forces play an important role in dynamic interactions between two adjacent nanotubes. Using Lennard-Jones potential function, such a phenomenon is appropriately modeled by a newly introduced vdW force density function. By employing Hamilton's principle, the equations of motion are obtained based on the nonlocal Rayleigh beam theory. In fact, these are integro-partial differential equations and seeking an exact or even analytical solution to them is a very difficult job. Therefore, an efficient numerical solution is proposed. The effects of the intertube distance, slenderness ratio, small-scale parameter, aspect ratio, and elastic properties of the surrounding medium on the free vibration of the nanosystem are addressed. The obtained results could be regarded as a pivotal step for better realizing of dynamic behaviors of more complex systems consist of multiple orthogonal networks of nanotubes.