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Na Qie, Wei-Fan Houa, Ji-Huan He (2021)
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New analytical approach for fractal Kmodel
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. Keywords Nonlinear vibration, Hamiltonian-based method, variational principle, carbon nanotube, residual equation Introduction The carbon nanotube (CNT) has attracted wide attention since it is first discovered by Japanese scientist Iijima in 1991. The vibration of the CNT is usually used to directly or indirectly measure the elastic modulus or other mechanical behaviors of 2–5 the CNTs. Therefore, the study of the vibration characteristics of the CNT is important. In this study, we will investigate 6,7 the vibration of CNT embedded in the elastic medium, which can be governed as 2 4 4 d μ π EI k π E þ þ μ þ μ ¼ 0 (1.1) 2 4 4 dt l ρA ρA 4l ρ where l is the length of the CNT, ρ represents the density, E indicates Young’s modulus, and A and I mean the cross- sectional area and cross-sectional inertia moment, respectively. There are many methods available for obtaining the 6 8 frequency property of equation (1.1) such as the incremental harmonic balanced method, variational iteration method, 9 10,11 12 Fourier series and Stokes’ transformation, homotopy perturbation method, and He’s frequency formulation. In this study, we will use a new method called the Hamiltonian-based method to determine the frequency property. The overall structure of this study is arranged as follows. The variational principle of the problem is constructed and its Hamiltonian is obtained in The variational principle and the Hamiltonian.In Hamiltonian-based method, the Hamiltonian-based method is adopted to solve the problem. In Results and discussion, the correctness of the Hamiltonian-based method is verified via a comparison with He’s frequency formulation through the numerical results. And the conclusion is presented in Conclusion and future recommendation. The variational principle and the Hamiltonian 13–22 The variational principle of equation (1.1) can be easily established via the semi-inverse method, which reads as School of Physics and Electronic Information Engineering, Henan Polytechnic University, Jiaozuo, China Corresponding author: Guo-Dong Wang, School of Physics and Electronic Information Engineering, Henan Polytechnic University, 2001 Century Avenue, Jiaozuo 454003, China. Email: wgd@hpu.edu.cn; my200920@outlook.com Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (https:// creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). Wang and Wang 113 4 4 1 ∂μ 1 π EI k π E 2 4 JðμÞ¼ þ μ þ μ dt: (2.1) 4 4 2 ∂t 2 l ρA ρA 16l ρ Where He–Weierstrass function can be obtained as 4 4 1 1 π EI k π E 0 2 2 4 Eðt,Ξ,Ξ ,ζÞ¼ ζ þ μ þ μ 4 4 2 2 l ρA ρA 16l ρ 4 4 1 dμ 1 π EI k π E dμ dμ 2 4 þ μ þ μ ζ (2.2) 4 4 2 dt 2 ρA dt dt l ρA 16l ρ 1 1 dμ dμ dμ ¼ ζ ζ 2 2 dt dt dt where the variable ζ that defined as dμ ζ ¼ (2.3) dt From equation (2.2), it can be proved that ∂ E Eðt, μ, μ , ζÞ¼ 0 and > 0 (2.4) ∂ζ which shows that equation (2.1) is a minimal variational principle. 4 4 2 4 4 4 In equation (2.1), K ¼ 1=2ð∂μ=∂tÞ is the kinetic energy, and P ¼ 1=2ðπ EI =l ρA þ k=ρAÞμ þ π E=16l ρμ repre- sents the potential energy. It is well known that the total energy is conserved in the whole process of the vibration. So the Hamiltonian can be obtained as 4 4 1 ∂μ 1 π EI k π μ 2 4 Η ¼ K þ P ¼ þ þ μ þ μ ¼ constant ¼ Η (2.5) 4 4 2 ∂t 2 l ρA ρA 16l ρ Equation (2.5) can be simplified as 1 dμ 2 4 þ αμ þ βμ ¼ Η (2.6) 2 dt 4 4 4 4 where α ¼ 1=2ðπ EI =l ρA þ k=ρAÞ, β ¼ π E=16l ρ. Hamiltonian-based method In this section, we will apply the Hamiltonian-based method to obtain the solution of equation (1.1). As is known to all, the kinetic energy and the potential energy are changed during the oscillation process, but the total energy will keep unchanged for a conservative oscillator. 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. Here the solution of equation (1.1) is assumed with the following form μðtÞ¼ M cosðϖtÞ (3.1) With the initial conditions of equation (3.1), we can determine the Hamiltonian constant as 2 4 Η ¼ αM þ βM (3.2) Substituting equation (3.2) into equation (2.6), there is 1 dμ 2 4 2 4 þ αμ þ βμ αM βM ¼ 0 (3.3) 2 dt 114 Journal of Low Frequency Noise, Vibration and Active Control 41(1) Taking equation (3.1) into equation (3.3) leads to the residual equation, which is 2 2 2 2 2 4 4 2 4 R ¼ ϖ M sin ðϖtÞþ αM cos ðϖtÞþ βM cos ðϖtÞ αM βM (3.4) Now we define two average residuals T =4 R ¼ R cosðϖ tÞdt (3.5) 1 1 1 1 0 T =4 R ¼ R cosðϖ tÞdt (3.6) 2 2 2 So the frequency–amplitude formulation can be obtained as follows 2 2 ~ ~ ϖ R ϖ R 1 2 2 2 1 ϖ ¼ (3.7) ~ ~ R R 1 2 Here we select two arbitrary frequencies as ϖ ¼ 1 and ϖ ¼ 2, and then we get the two residual equations as follows 1 2 2 2 2 2 4 4 2 4 R ¼ M sin ðtÞþ αM cos ðtÞþ βM cos ðtÞ αM βM (3.8) 2 2 2 2 4 4 2 4 R ¼ 2M sin ð2tÞþ αM cos ð2tÞþ βM cos ð2tÞ αM βM (3.9) So the two average residuals are obtained as T =4 2 2 M 10α þ 14M β 5 R ¼ R cos tdt ¼ (3.10) 1 1 T 15π T =4 2 2 2M 5α þ 7M β 10 R ¼ R cosð2tÞdt ¼ (3.11) 2 2 T 15π 2 0 According to equation (3.7), there is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ~ ~ ϖ R ϖ R 14 1 2 2 1 ϖ ¼ ¼ 2α þ βM (3.12) ~ ~ R R 1 2 With this, we get the solution of equation (1.1)as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μðtÞ¼ M cos 2α þ βM t (3.13) That is 0 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 2 4 π EI k 7M π E @ A μðtÞ¼ M cos þ þ t (3.14) 4 4 l ρA ρA 40l ρ Results and discussion Recently, He’s frequency formulation, which is first proposed by Chinese mathematician Ji-Huan He, has been widely used 25 26 27 to solve the nonlinear vibrations arising in three-dimensional printing technology, micro-electromechanical, N/MEMS, 28,29 and so on . By using He’s frequency formulation, we can get the frequency–amplitude formulation of equation (1.1)as 4 2 4 π EI k 3M π E ϖ ¼ þ þ (4.1) 4 4 l ρA ρA 16l ρ Wang and Wang 115 Figure 1. Comparison between the Hamiltonian-based method and He’s frequency formulation. Figure 2. Comparison between the Hamiltonian-based method and He’s frequency formulation. Then the solution of equation (1.1) can be obtained as 0 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 2 4 π EI k 3M π E @ A μðtÞ¼ M cos þ þ t (4.2) 4 4 l ρA ρA 16l ρ which has a good agreement with equation (3.14). For E ¼ 1, ρ ¼ 1, l ¼ π, M ¼ 1, I ¼ 1, A ¼ 1, and k ¼ 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. When we select E ¼ 1, ρ ¼ 1, l ¼ π, M ¼ , I ¼ 1, A ¼ 1, and k ¼ 1, the comparison of the two methods is plotted in Figure 2, which also has a good agreement. 116 Journal of Low Frequency Noise, Vibration and Active Control 41(1) Conclusion and future recommendation In this study, a novel and effective method called the Hamiltonian-based method is used to study the nonlinear vibration of the CNT embedded in elastic medium. The comparison between our proposed method and He’s frequency formulation shows a good agreement, which strongly proves the correctness of the Hamiltonian-based method. The finding in this study is expected to open up new horizons for the study of the nonlinear vibration. Recently, the fractal and fractional calculus have been widely used to model many complex problems arising in 30,31 32–35 36–38 39 40–46 circuit, physics, filter, biomedical science, and so on. The proposed method in this work is also extended to the fractal cases. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by Program of Henan Polytechnic University (No.B2018-40), the Fundamental Research Funds for the Universities of Henan Province (NSFRF210324), Innovative Scientists and Technicians Team of Henan Provincial High Education (21IRTSTHN016), and Key Project of Scientific and Technology Research of Henan Province (212102210224). ORCID iDs Kang-Jia Wang https://orcid.org/0000-0002-3905-0844 Guo-Dong Wang https://orcid.org/0000-0001-6574-1668 References 1. Iijima S. Helical microtubules of graphitic carbon. Nature 1991; 354: 56–58. 2. Qian D, Wagner GJ, LIU WK, et al. Mechanics of carbon nanotubes. Appl Mech Rev 2002; 55(6): 495–533. 3. 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"Journal of Low Frequency Noise, Vibration and Active Control" – SAGE
Published: Jul 26, 2021
Keywords: Nonlinear vibration; Hamiltonian-based method; variational principle; carbon nanotube; residual equation
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