Determination of fundamental coupled torsional—radial frequency of single-walled carbon nanotubes

Research indicates that single-walled carbon nanotubes have a unique coupled torsional–radial vibration as one of their fundamental modes. Determination of their vibration frequency is required for efficient use of single-walled carbon nanotube in nano-electromechanical systems. However, there is no mathematical expression for these frequencies and their dependence on single-walled carbon nanotube geometry is unknown. This article examines the effect of diameter, length, and chirality on the fundamental coupled torsional–radial vibration frequency of single-walled carbon nanotube using molecular–structural–mechanics–approach, finite element analysis, and regression analyses. Consequently, a first-ever mathematical form of this frequency is derived. The form quickly and accurately predicts these frequencies at 1.5% in-sample, and 7.2% out-sample mean absolute percentage error. single-walled carbon nanotubes’ fundamental coupled torsional–radial vibration frequency is found independent of diameter and inversely proportional to length where the proportionality constant depends on chirality. The coupling of modes and the similarity of the frequency form with cylindrical shell suggest that single-walled carbon nanotube behave like thin shells in these vibrations. A form for effective circumferential shear modulus of single-walled carbon nanotube is also derived. This modulus is found to depend only on the chirality where achiral single-walled carbon nanotubes have higher values than chiral single-walled carbon nanotubes. Proposed mathematical forms can be used for characterization of single-walled carbon nanotubes, determination of single-walled carbon nanotubes’ effective shear modulus, and tuning operational frequency of single-walled carbon nanotube-based nano-electromechanical systems.