Study on the nonlinear vibration of embedded carbon nanotube via the Hamiltonian-based method

Kang-Jia Wang,Guo-Dong Wang

doi:10.1177/14613484211032757

Abstract

This article mainly studies the vibration of the carbon nanotubes embedded in elastic medium. A new novel method called the Hamiltonian-based method is applied to determine the frequency property of the nonlinear vibration. Finally, the effectiveness and reliability of the proposed method is verified through the numerical results. The obtained results in this work are expected to be helpful for the study of the nonlinear vibration.

Highlights

The carbon nanotube (CNT) has attracted wide attention since it is first discovered by Japanese scientist Iijima[1] in 1991
He’s frequency formulation, which is first proposed by Chinese mathematician Ji-Huan He, has been widely used to solve the nonlinear vibrations arising in three-dimensional printing technology,[25] micro-electromechanical,[26] N/MEMS,[27] and so on
For E 1⁄4 1, ρ 1⁄4 1, l 1⁄4 π, M 1⁄4 1, I 1⁄4 1, A 1⁄4 1, and k 1⁄4 1, we plot a comparison between the Hamiltonian-based method and He’s frequency formulation in Figure 1, which shows a good match between the two methods

Summary

Introduction

The carbon nanotube (CNT) has attracted wide attention since it is first discovered by Japanese scientist Iijima[1] in 1991. There are many methods available for obtaining the frequency property of equation (1.1) such as the incremental harmonic balanced method,[6] variational iteration method,[8] Fourier series and Stokes’ transformation,[9] homotopy perturbation method,[10,11] and He’s frequency formulation.[12] In this study, we will use a new method called the Hamiltonian-based method to determine the frequency property. In equation (2.1), K 1⁄4 1=2ð∂μ=∂tÞ2 is the kinetic energy, and P 1⁄4 1=2ðπ4EI =l4ρA þ k=ρAÞμ2 þ π4E=16l4ρμ[4] represents the potential energy. We will apply the Hamiltonian-based method to obtain the solution of equation (1.1). The Hamiltonian-based method is based on this and the variational theory, so it can present a more accurate solution compared with the VIM or HPM. We select two arbitrary frequencies as π1 1⁄4 1 and π2 1⁄4 2, and we get the two residual equations as follows

M sin2ðtÞ αM 2

Results and discussion

Conclusion and future recommendation