The von-Kármán effect on the free torsional vibrations of carbon nanotube made of functionally graded materials (FGMs) is inspected in this research using the theory of nonlocal elasticity. The material properties are assumed to change continuously through the radius of the FGM carbon nanotube (FGM-CNT) according to a power-law distribution. Using Hamilton’s principle, the equations of nonlinear torsional vibrations is obtained. The torsional frequencies and mode shapes of FGM-CNT are derived for two boundary conditions, namely, clamped–clamped and clamped–free. The approximate-analytical solution for the nonlocal nonlinear natural frequencies is extracted using multiple scales method (MSM). Next, the effects of the FG index, the length, the outer radius, the amplitude of vibration and the mode number on the vibrational characteristics of the FGM-CNT are studied. The obtained results in this paper can be used as a key tool for efficient and exact design of devices comprising of torsional nanotubes.

在这项研究中，使用非局部弹性理论研究了冯-卡尔曼对功能梯度材料（FGM）制成的碳纳米管的自由扭转振动的影响。假定材料特性根据幂律分布在FGM碳纳米管（FGM-CNT）的半径范围内连续变化。利用汉密尔顿原理，得到了非线性扭转振动方程。FGM-CNT的扭转频率和模态形状是在两个边界条件下得出的，即被夹紧的和未被夹紧的。使用多尺度方法（MSM）提取了非局部非线性固有频率的近似解析解。接下来，FG指数，长度，外半径的影响 研究了FGM-CNT的振动幅度和模数对振动特性的影响。本文获得的结果可用作有效和准确设计包含扭转纳米管的器件的关键工具。