Vibration Analysis Of Single-walled Carbon Nanotube Contains Heterojunctions

As a new material, carbon nanotubes, its unique mechanical, electrical, mechanicalproperties, show a very good application prospect in using of composite materials andnano-particle. Non-uniform carbon nanotubes which is caused by the defect of molecularstructure, is called heterojunction carbon nanotubes. Domestic and foreign scholars havedone a lot of research on the uniform type carbon nanotubes, but the study of non-uniformcarbon nanotubes are relatively lack. Pentagon and heptagon defects appear in pairs withina single-walled carbon nanotube. According to its different position, the tube wall minghtkeep straight and the variation of diameter can changes in a stepped way or a continiuroustype. This paper selects one type of heterojunction single-walled carbon nanotubes whichcontains a Pentagon-heptagon intramolecular defect: Single-walled straight carbonnanotubes which contains heterojunctions, analyze its natural vibration subject.Using Euler-Bernoulli beam as the basic model, Hamilton principle and the non-localtheory were invited to establish the vibration differential equation of carbon nanotubes.The stepped carbon nanotubes, using singularity function to express the stepped changesof diameter. Vibration equation of single-walled carbon nanotubes and the correspondingfrequency equation with heterojunctions of equivalent beam model are derived. Bysolving the frequency equation, obtained the natural frequency of different non-localcoefficient and the ratio of length to diameter. Analyzed the effect of different aspect ratioand non-local coefficient on the natural frequency of carbon nanotubes.According to the stepped carbon nanotubes contains heterojunctions, using the modelof Winkler elastic foundation, a non-local free vibration differential equation of embeddedsingle-walled carbon nanotubes with junctions is derived. Furthermore, the vibrationequation of equivalent stepped embedded single-walled carbon nanotubes with junctionsis derived, so does the corresponding frequency equation. Solved the frequency equation,according to the different matrix elasticity coefficient, obtained the natural frequency ofdifferent ratio of length to diameter, and non-local coefficient and its variation.According to the graded cylindrical carbon nanotubes contains junctions, non-localvibration equations of a graded carbon nanotubes is derived using the Hamilton principle.By using the Galerkin method, solved the governing equation, and obtained naturalfrequency with different taper ratio, slenderness ratio and non-local coefficient of carbon nanotubes and its variation.