Abstract
The nonlinear free vibration and frequency veering of a single wall carbon nanotube (SWCNT) based on nonlocal elasticity theory is studied and investigated in this paper. The carbon nanotubes (CNT) is assumed to have an imperfection modeled as half sine and clamped at both ends. The Euler-Bernoulli Beam and Hamilton’s principle were used to derive the nonlinear equation of motion of the SWCNT. The effect of; nonlocal elasticity, geometric initial rise/imperfection, and the effect of the axial force induced by mid-plane stretching are accounted for in the derivation of the nonlinear mathematical model of the CNT. The governing partial differential equation includes quadratic and cubic nonlinearities due to the initial imperfection and the mid-plane stretching. The derived equation of motion is discretized using the assumed mode method by inserting the exact linear eigen mode shape. The resulting nonlinear temporal equation was solved using the method of multiple scales (MMS) to obtain results for the nonlinear natural frequencies of the first three modes of vibrations, for different values of rise/imperfection amplitude, and for different values of the nonlocal parameter. The results are presented in non-dimensional characteristic curves to show the effect of variation of rise/imperfection amplitude and nonlocal parameter on the vibrational behavior of the CNT.
Highlights
Since the discovery of carbon nanotubes (CNTs) in (1991) by Lijima (Lijima, 1991), they have drawn the attention of scientists/researchers and much research and studies have been devoted to analyze and understand their behavior
The objective of this study is to derive a mathematical model based on nonlocal elasticity theory along with continuum mechanics for a CNT with geometric imperfection/initial rise and using the exact mode shape/eigen function that takes into account the waviness of the CNT to discretize the nonlinear integro-partial differential equation of motion
To simplify the numerical solutions of the dynamic behavior of the CNT, all linear and nonlinear results are presented in non-dimensional forms
Summary
Since the discovery of carbon nanotubes (CNTs) in (1991) by Lijima (Lijima, 1991), they have drawn the attention of scientists/researchers and much research and studies have been devoted to analyze and understand their behavior. Studies of micro and nano beams based on the Eringen's nonlocal elasticity theory given (Eringen & Edelen,1972), have been an active research area and many studies and investigations have been published and reported using this theory due to its importance to account for the small size/scale effect on static and dynamic behavior of micro- and nano-beams.
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