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.

研究表明，单壁碳纳米管具有独特的扭转-径向振动耦合作为其基本模式之一。为了在纳米机电系统中有效使用单壁碳纳米管，需要确定其振动频率。然而，对于这些频率没有数学表达式，并且它们对单壁碳纳米管几何形状的依赖性是未知的。本文使用分子-结构-力学方法，有限元分析和回归分析，研究了直径，长度和手性对单壁碳纳米管基本耦合扭转-径向振动频率的影响。因此，得出了该频率的第一个数学形式。该表格可快速准确地预测样本中1.5％的频率，以及7。2％的样本外平均绝对百分比误差。发现单壁碳纳米管的基本扭转-径向振动频率与直径无关，与长度成反比，而比例常数取决于手性。模式耦合以及频率形式与圆柱壳的相似性表明，在这些振动中，单壁碳纳米管的行为像薄壳。还推导了单壁碳纳米管的有效周向剪切模量的形式。发现该模量仅取决于手性，其中非手性单壁碳纳米管具有比手性单壁碳纳米管更高的值。拟议的数学形式可用于表征单壁碳纳米管，