Vibration characteristics of zigzag FGM single-walled carbon nanotubes based on Ritz method with ring-stiffeners
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ORIGINAL PAPER
Vibration characteristics of zigzag FGM single-walled carbon nanotubes based on Ritz method with ring-stiffeners M Hussain*
and M N Naeem
Department of Mathematics, Govt. College University Faisalabad, Faisalabad 38040, Pakistan Received: 15 December 2019 / Accepted: 19 June 2020
Abstract: 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. Keywords: Rayleigh–Ritz; Lagrangian functional; Ring-stiffeners; MATLAB; CNTs
1. Introduction The discovery of carbon nanotubes (CNTs) is very important and interest of research due to considerable observation and research publications every year. CNTs have a variety of uses and applications in potential looking fields, some of which are charge detectors, electronics, communication, composite materials, biotechnology, environment, energy storage, chemical and optical [1]. In many practical applications, the use of CNTs such as pressure vessels, silos core-barrels of pressure water reaction, chimney design CNTs are core elements. Usually tubes are stiffened by rings to amplify the stiffness and strength, reduce the weight of structure to be designed. While designing the ring-stiffened tubes, it is important to know their resonant frequencies because fatigue could be caused by excessive vibrations. Moreover, the enemy’s sonar equipment may detect the submarine by the vibration of its hull. When ring-stiffeners of equal strength are narrowly and equally spaced sensitive equipment may be affected in the aircrafts. The ring-stiffened CNTs are attributed to sustain more load and pressure within buckling and vibrations. More than one type of
*Corresponding author, E-mail: [email protected]; [email protected]
materials is used to structure the functionally graded materials (FGMs) by some material manufacturing law and their physical properties vary from one surface to another surface. Some researchers have independently reported the synthesis of SWCNTs [2, 3]. They proved that these nanotubes have extraordinary properties. SWCNTs can be prepared by rolling a single graphene layer with hexagonal cells to form the structure of cylindrical fullerene. Snow et al. [4] examined the imaging stability of SWCNTs with atomic force microscopy as probes. For long a
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