Bending Vibration Of Carbon Nanotubes

Carbon nanotubes (CNTs) were discovered by lijima, a Japanese scientist, in1991. From then, great interest has been aroused in theoretical and experimental researches on CNTs. CNTs have many unique physical, mechanical properties, and they can be used as reinforced fibers to enhance the performance of some composite materials. CNTs have been widely used in various fields.Vibration is an important dynamic characteristics of CNTs. Use of this feature makes it possible that CNTs are applied as atomic-resolution mass sensors. Due to size effect of CNTs, the classic continuum theory is not adequate to capture this effect. In this paper, bending vibrations of CNTs in various boundary conditions are considered, in particular the influence of nonlocal effects with the framework of nonlocal elasticity theory. Also, a model of nonlocal Euler-Bernoulli beams with varying cross-section is proposed and bending vibration of horn-shaped CNTs is emphasized. Main achievements are as follows:First, free vibration of cantilever CNTs with classical Euler-Bernoulli beam theory is considered, and the effects of mass and its rotary inertia on the end are analyzed. The attached mass on the free end decreases the natural frequency; rotary inertia also decreases the natural frequency.Second, vibration of uniform CNTs using nonlocal Euler-Bernoulli beam theory in various boundary conditions is studied. Emphasis is to analyze the effect of the small scale factor on the vibration frequency and mode shapes. In simply-supported and clamped boundary conditions, the natural frequency decreases as the small scale factor grows. In cantilever boundary conditions, the second order and the third order frequency decrease, but the first order frequency increases; for the simply-supported CNTs, the small scale factor almost has no effect on the mode shapes of vibration, but for the clamped and cantilevered CNTs, the influence is more obvious.Third, a governing differential equation of bending vibration of nonlocal Euler-Bernoulli beams with variable cross-section is derived. Using the integral method, the natural frequencies are obtained for various boundary conditions and the effect of the nonlocal parameter on the natural frequency is discussed.