Vibration characteristics of zigzag FGM single-walled carbon nanotubes based on Ritz method with ring-stiffeners

In this study, Rayleigh–Ritz method is used for free vibrations of functionally graded carbon nanotubes based on Sander’s shell theory. The effects of various parameters for length-to-diameter ratio and height-to-diameter ratio for zigzag single-walled carbon nanotubes have been fully inspected. The frequency pattern with mass density per unit lateral area and in-plane rigidity has been calculated and found reverse order. The tube material is taken as functionally graded material and their material quantity is located by the polynomial volume fraction law. It is examined that the frequencies increase on increasing the ratio of height-to-diameter. Moreover, the impact of ring supports around the tube circumferential is studied for their various positions along tube axial length. These rings are restrained the radial displacement in the transverse direction.