弹性基体对磁场中输流纳米管的振动影响

李明，邓乾，吕刘飞; Li Ming; Deng Qian; Lv Liufei

引 en

弹性基体对磁场中输流纳米管的振动影响^*

李明
机构：
武汉科技大学理学院,武汉 430065
×
，邓乾
机构：
武汉科技大学理学院,武汉 430065
×
，吕刘飞
机构：
武汉科技大学理学院,武汉 430065
×

武汉科技大学理学院,武汉 430065；

Elastic Matrix Effect on Vibration Characteristics of Fluid-conveyed SWCNT Under A Longitudinal Magnetic Field^*

Li Ming
Affiliation：
Department of Mechanics, Wuhan University of Science and Technology, Wuhan 430065 , China
×
，Deng Qian
Affiliation：
Department of Mechanics, Wuhan University of Science and Technology, Wuhan 430065 , China
×
，Lv Liufei
Affiliation：
Department of Mechanics, Wuhan University of Science and Technology, Wuhan 430065 , China
×

Department of Mechanics, Wuhan University of Science and Technology, Wuhan 430065 , China；

通讯作者:

李明,E-mail:mingli121212@126.com

中图分类号:O353.1

文献标识码:A

文章编号:1672-6553-2023-21(3)-063-006

DOI:10.6052/1672-6553-2022-021

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参考文献
出版信息

参考文献 1

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RAFIEI M,MOHEBPOUR S R,Daneshmand F.Small-scale effect on the vibration of non-uniform carbon nanotubes conveying fluid and embedded in viscoelastic medium [J].Physica E:Low-dimensional Systems and Nanostructures,2012,44(7-8):1372-1379.

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参考文献 12

李明,吕刘飞,郑华升等.磁场对不同温度场中输流悬臂碳纳米管颤振稳定性的影响 [J].固体力学学报,2021,42(1):87-93.LI M,LV L F,ZHENG H S,et al.Magnetic field effect on flutter stability of a fluid-conveying cantilevered carbon nanotube under different temperature fields [J].Chinese Journal of Solid Mechanics,2021,42(1):87-93.(in Chinese)

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目录contents

摘要 Abstract
关键词 Keywords
1 振动控制方程
1.1 振动方程推导
1.2 振动方程求解
2 数值计算结果分析
3 结论
参考文献

摘要

基于非局部连续介质理论,对处于轴向磁场作用下嵌入弹性基体中的输流单层碳纳米管(SWCNT),应用哈密顿原理,采用Euler-Bernoulli梁模型,建立了固支边界条件下该系统的横向振动微分方程.方程中计及磁场力与小尺度效应,弹性基体等效为Pasternak弹性模型.应用微分变换法(DTM)求解方程,着重研究了弹性基体、轴向磁场及小尺度效应耦合作用时该纳米输流管道系统振动稳定性问题.数值计算结果表明:弹性基体与轴向磁场均能提升系统的稳定性;而小尺度效应则降低系统稳定性.进一步的研究表明:随着弹性基体的增强,磁场对系统稳定性的影响受到一定程度的抑制;而对于小尺度效应而言,弹性基体的剪切参数抑制小尺度效应对系统稳定性的影响,但弹性参数放大了这一影响作用.

Abstract

The nonlocal Euler-Bernoulli beam model is used to study the vibration instability characteristics of fluid transporting fixed-fixed supported single-layer carbon nanotubes (SWCNTs) embedded in elastic matrix under longitudinal magnetic field. Considering the magnetic field force and small-scale effect, the elastic matrix is equivalent to Pasternak's property model, and the higher-order vibration differential equation and boundary condition of the fluid-structure coupling system are established by using Hamilton principle. The differential transformation method (DTM) is used to solve the equation, and the stability of SWCNTs under the coupling of elastic matrix, longitudinal magnetic field and small-scale effect is studied. The numerical results indicate that the magnetic field and the elastic matrix improve the stability of the system. The increase of small-scale coefficient will reduce the stability of the system to a certain extent. Further research shows that the influence of longitudinal magnetic field on the stability of flow transporting SWCNTs embedded in elastic matrix will be restrained to varying degrees with the enhancement of the elastic matrix. For the small-scale coefficient, the two parameters of the elastic foundation have the opposite effect, that is, the shear parameter suppresses the influence of small-scale on the stability of the fluid-conveyed carbon nanotube system, while the elastic parameter amplifies the influence.

关键词

输流纳米管；轴向磁场；弹性基体；小尺度系数；微分变换法

Keywords

fluid-conveyed carbon nanotube ； longitudinal magnetic field ； Pasternak foundation ； nonlocal parameter ； DTM

引言
微纳米管道结构系统在生物医药领域、纳米纤维复合材料领域以及分子药物输送、生物传感器等领域已有了较为广泛的应用^[1-2]，而这类输流管道力学问题的应用研究涉及流体力学、固体力学及动力学等多学科^[3].对微纳尺度下力学特性的实验研究以及分子动力学模拟都表明^[4-5]:小尺度效应对纳米材料的各项性质有着相当重要的影响，但经典的连续介质理论由于缺乏尺度相关性而无法准确描述这类小尺度结构的力学行为，为此大量反映尺度相关性的连续介质理论相应提出，其中Eringen的非局部弹性理论^[6]成功应用于研究纳米管道的各类静、动态力学行为，因此在纳米材料力学问题的研究中得到了广泛的应用^[7].
在纳米流体储存、运输以及纳米复合材料等纳米技术领域，碳纳米管常嵌于某一基体中.而模拟这类基体一般可以选用单参数的Winkler弹性模型、双参数的Pasternak弹性模型以及其他一些黏弹性模型等.Mustapha 和Zhong^[8]基于非局部欧拉梁模型研究了温度场中Winkler基体上单壁碳纳米管（SWCNT）的振动性质.Rafiei^[9]等基于相同模型研究了黏弹性Kelvin基体上SWCNT输流时振动响应问题.
近段时期以来，碳纳米管的磁性质以及其在外加磁场中所表现出来的力学特征显示出其在微纳机电系统、纳米传感器以及自旋电子科技等领域潜在的应用前景，基于此，大量相关研究随之出现.Dini等^[10]基于Eringen弹性理论研究了双参数弹性基体上不同边界条件下，弯曲输流双壁碳纳米管在外部磁场作用下的振动特性，结果表明，流体流速、范德华力以及尺度参数对纳米管的振动均有相应的影响.Ghane等^[11]则采用了非局部Timoshenko梁模型，针对输流碳纳米管在外加磁场中的颤振问题，得到了磁场、流速、稀薄效应以及小尺度效应对系统颤振失稳的影响结果.
现有文献主要针对不同参数耦合作用时各参数对纳米输流管道振动特性的研究，但各参数的作用效果受其他参数的影响情况却鲜少提及.本文依据非局部Euler-Bernouli梁模型，针对轴向磁场作用下，嵌入Pasternak弹性基体中的固支输流单层碳纳米管，应用哈密顿原理建立该输流系统的振动控制方程，通过数值计算结果，研究双参数Pasternak弹性基体与纵向磁场、小尺度效应耦合作用时该输流碳纳米管振动特性以及耦合作用时各参数间的相互作用，以期为纳米管在各领域中的应用提供一定的理论依据.
1 振动控制方程
1.1 振动方程推导
图1为弹性基体内固支输流碳纳米管在轴向磁场中的示意图.
图1 轴向磁场中弹性基体上输流单壁碳纳米管
Fig.1 An fluid-conveyed SWCNT in Pasternak medium under the longitudinal magnetic field
建立如图1所示的坐标系，X为纳米管的轴向坐标，碳纳米管长度记为L，外径为D. 系统仅发生面内横向小幅振动，重力及管道外部拉、压力不计，管道内部流体为理想流体的定常流动，流速不变且记为U.
则由Euler-Bernoulli梁应变-位移关系有:

ε_{X X} = - Z \frac{\partial^{2} W}{\partial X^{2}}

(1)

其中W（X，t）为Z向位移，t为时间，ε_XX为X方向的应变.
由非局部弹性理论^[6]，含有小尺度效应的应力-应变关系为:

σ_{X X} - {(e_{0} a)}^{2} \frac{\partial^{2} σ_{X X}}{\partial X^{2}} = E ε_{X X}

(2)

其中σ_XX分别为X方向的应力，E为纳米管弹性模量，e₀为材料常数，a为材料内部特征长度，对于碳纳米管而言，C-C键长一般为0.142nm.
碳纳米管内部流体作用于纳米管的力可表示为^[5]:

F_{f} = m_{f} (\frac{\partial^{2} W}{\partial t^{2}} + 2 U \frac{\partial^{2} W}{\partial X \partial t} + U^{2} \frac{\partial^{2} W}{\partial X^{2}})

(3)

其中m_f为每单位长度上纳米管内部流体的质量.
轴向磁场中，由磁场引起的作用于纳米管横向方向上单位长度的洛伦兹力为^[12]:

F_{Z} = η A H_{X}^{2} \frac{\partial^{2} W}{\partial X^{2}}

(4)

其中A为管道横截面积，η为磁导率，H_X为轴向磁场强度.
Pasternak弹性基体对纳米管的作用力表示为^[13]:

F = K W - G \frac{\partial^{2} W}{\partial X^{2}}

(5)

其中K为弹性系数，G为剪切系数.
基于上述各式，由弹性基体、磁场以及管内流体共同作用于碳纳米管的功为:

W_{E x t} = \frac{1}{2} \int_{0}^{L} - (K W - G \frac{\partial^{2} W}{\partial X^{2}} + F_{z} + F_{f}) W d X

(6)

管道系统的总动能为:

T = \frac{1}{2} (m_{c} + m_{f}) \int_{0}^{L} {(\frac{\partial W}{\partial t})}^{2} d X

(7)

其中m_c为每单位长度上纳米管的质量.
管道系统的应变能为:

U = \frac{1}{2} \int_{0}^{L} \int_{A} ε_{X X} σ_{X X} d A d X

(8)

由弯矩M与应力σ_XX的关系式:

M = \iint σ_{X X} Z d A

(9)

应用哈密顿原理:

\int_{t_{1}}^{t_{2}} δ (U - T - W_{Ext}) d t = 0

(10)

得系统振动微分方程为:
$E I \frac{\partial^{4} W}{\partial X^{4}} + (m_{f} U^{2} - η A H_{X}^{2} - G) \frac{\partial^{2} W}{\partial X^{2}} +$

\begin{matrix} 2 m_{f} U \frac{\partial^{2} W}{\partial X \partial t} + K W + (m_{c} + m_{f}) \frac{\partial^{2} W}{\partial t^{2}} - \\ {(e_{0} a)}^{2} [(m_{f} U^{2} - η A H_{X}^{2} - G) \frac{\partial^{4} W}{\partial X^{4}} + \\ 2 m_{f} U \frac{\partial^{4} W}{\partial X^{3} \partial t} + K \frac{\partial^{2} W}{\partial X^{2}} + \\ (m_{c} + m_{f}) \frac{\partial^{4} W}{\partial X^{2} \partial t^{2}}] = 0 \end{matrix}

(11)

其中EI为纳米管弯曲刚度.
边界条件为:

X = 0, L : W = \frac{\partial W}{\partial X} = 0

(12)

1.2 振动方程求解
引入无量纲变量及参数:
$\begin{matrix} w = \frac{W}{L}, x = \frac{X}{L}, τ = \sqrt{\frac{E I}{m_{c} + m_{f}}} \frac{t}{L^{2}}, \\ β = \frac{m_{f}}{m_{c} + m_{f}}, u = U L \sqrt{\frac{m_{f}}{E I}}, μ = {(\frac{e_{0} a}{L})}^{2}, \\ ψ = η A H_{X}^{2} \frac{L^{2}}{E I}, g = G \frac{L^{2}}{E I}, k = K \frac{L^{4}}{E I} \end{matrix}$
则上述方程（11）及边界条件（12）可改写为无量纲方程

\begin{matrix} \frac{\partial^{4} w}{\partial x^{4}} + (u^{2} - ψ - g) \frac{\partial^{2} w}{\partial x^{2}} + 2 u \sqrt{β} \frac{\partial^{2} w}{\partial x \partial τ} + \\ k w + \frac{\partial^{2} w}{\partial τ^{2}} - μ [(u^{2} - ψ - g) \frac{\partial^{4} w}{\partial x^{4}} + \\ 2 u \sqrt{β} \frac{\partial^{4} w}{\partial x^{3} \partial τ} + k \frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{4} w}{\partial x^{2} \partial τ^{2}}] = 0 \end{matrix}

(13)

及相应的边界条件

x = 0,1 : w = \frac{\partial w}{\partial x} = 0

(14)

设方程（13）的解为，代入方程（13）有

\begin{matrix} \frac{d^{4} φ}{d x^{4}} + (u^{2} - ψ - g) \frac{d^{2} φ}{d x^{2}} + 2 u \sqrt{β} Ω \frac{d φ}{d x} + \\ (Ω^{2} + k) φ - μ [(u^{2} - ψ - g) \frac{d^{4} φ}{d x^{4}} + \\ 2 u \sqrt{β} Ω \frac{d^{3} φ}{d x^{3}} + (Ω^{2} + k) \frac{d^{2} φ}{d x^{2}}] = 0 \end{matrix}

(15)

采用微分变换法（DTM）求解方程（15），DTM法应用于纳米级别输流管道系统振动稳定性分析已在文献^[12]得到证实.
则有方程（15）的微分变换形式:
$[1 - μ (u^{2} - ψ - g)] (n + 4)! Φ (n + 4) -$

\begin{matrix} 2 μ u \sqrt{β} Ω (n + 3)! Φ (n + 3) + [(u^{2} - ψ - g) - \\ μ (Ω^{2} + k)] (n + 2)! Φ (n + 2) + \\ 2 u \sqrt{β} Ω (n + 1)! Φ (n + 1) + \\ (Ω^{2} + k) n! Φ (n) = 0 \end{matrix}

(16)

边界条件的微分变换形式:

Φ (0) = Φ (1) = 0

(17)

\begin{matrix} \sum_{n = 0}^{\infty} Φ (n) = 0, \\ \sum_{n = 0}^{\infty} n Φ (n) = 0 \end{matrix}

(18)

令Φ（2）=C₁，Φ（3）=C₂，进而与式（17）一起代入式（16），迭代求得Φ（n），n=4，5，···，N. 然后将Φ（n），n=0，1，2，···，N代入式（18），可得到以下两个方程:

[\begin{matrix} a_{11} a_{12} \\ a_{21} a_{22} \end{matrix}] \{\begin{matrix} C_{1} \\ C_{2} \end{matrix}\} = 0

(19)

令上式系数矩阵行列式为零，即可获得输流碳纳米管系统无量纲复频率Ω，其中虚部Im（Ω）是系统的无量纲频率，研究^[4]表明，在Im（Ω）=0时，固支管道系统出现屈曲失稳，此时流速称为系统发散失稳的临界流速u_cr.
2 数值计算结果分析
本文采用的分析参数为^[12]:管内流体的密度ρ_f=1000kg/m³，碳纳米管密度ρ_c=2300kg/m³，其外层半径R₀=3nm，壁厚t_d=0.1nm，弹性模量E=3.4TPa. 泊松比ν=0.3.振动中为不计剪切变形与转动惯量，长径比取L/2R₀=40，磁导率η=4π×10^-7，其余参数选择在算例中将给予说明，选取DTM算法收敛截取数60以确保各项数值解足够精确^[12].
图2为不考虑磁场作用及尺度效应时，Pasternak弹性基体的弹性系数k与剪切系数g对管道频率、管道发散失稳无量纲临界流速的影响.
由图2可以看出有无弹性基体的作用，随着管内流体流速的增加，输流固支碳纳米管均将经历模态频率逐渐降低，进而出现系统发散失稳（一阶模态频率为零），再到第一、二阶模态耦合颤振这一复杂振动过程.不同之处在于，嵌入弹性基体的系统，其临界流速将提高，管道的稳定性将提升，这一结论现有文献获取的结论一致.进一步比较可以发现剪切系数g的提高对系统稳定性的提升优于弹性系数k.
图2 Pasternak弹性基体中无量纲固有频率与无量纲流速变化关系（β=0.5，H_X=μ=0）
Fig.2 The imaginary component of the eigenfrequency, Ω as functions of the flow velocity, embedded in Pasternak foundation ( β=0.5, H_X=μ=0)
对于外加轴向磁场以及碳纳米管小尺度效应对管道稳定性的影响，文献^[12]已有了相应结论，即轴向磁场增强将提高管道稳定性，而小尺度效应则降低管道的稳定性.
图3 不同弹性模量k时磁场对临界流速的影响（β=0.5，g=μ=0）
Fig.3 Critical flow velocities with longitudinal magnetic field in a fluid-conveyed SWCNT for different values k (β=0.5, g=μ=0)
将碳纳米管输流系统嵌入弹性基体后，图3、图4给出了弹性系数k与剪切系数g对磁场作用下管道发散失稳临界流速的影响.
分析图3可以看到，对于不同的弹性系数k，磁场的增强都能提高纳米管道系统临界流速进而提升其稳定性.但详细比较增加相同磁场强度临界流速的提升值时，可以发现，不同弹性系数k下磁场对稳定性的提升程度不同.例如，H_X均由9×10⁷A/m增加到10×10⁷A/m，当k=0时，无量纲临界流速u_cr由8.83增至9.32，增量Δu_cr为0.49; 当k=100，200，···，500时，Δu_cr分别为0.48，0.46，···，0.41，尽管差别不大，但仍可说明:随着弹性系数k的增大，磁场对管道稳定性的提升作用受到了一定程度的抑制.
图4 不同剪切模量g时磁场对临界流速的影响（β=0.5，k=μ=0）
Fig.4 Critical flow velocities with longitudinal magnetic field in a fluid-conveyed SWCNT for different values g (β=0.5, k=μ=0)
分析图4同样可以得到类似结论:不同剪切系数g的基体中，输流碳纳米管的稳定性会随着磁场的增强而提高，但剪切系数g仍然一定程度地抑制了磁场对管道稳定性的提升作用，尽管这一抑制作用不明显，且磁场越弱，这一作用越不明显.
图5 不同弹性模量k时小尺度系数对临界流速的影响（β=0.5，H_X=g=0）
Fig.5 Critical flow velocities with nonlocal parameter in a fluid-conveyed SWCNT for different values k (β=0.5, H_X=g=0)
图5、图6给出了考虑小尺度效应时弹性系数k与剪切系数g对管道发散失稳临界流速的影响.
由图5、图6可以看到，有无弹性基体，小尺度系数μ的增加都会降低管道系统稳定性.详细比较两个基体参数在不同数值下，增加相同小尺度系数μ，无量纲临界流速的降低值Δu_cr可以发现，弹性基体的剪切系数g抑制了小尺度参数μ对系统稳定性的影响，例如:μ由0.18增至0.2，当g=0时，Δu_cr=-0.25; 当g=50时，Δu_cr=-0.12. 但弹性系数k却放大了小尺度系数对管道稳定的影响，如:μ由0.18增至0.2，当k=0时，Δu_cr=-0.25; 当k=500时，Δu_cr=-0.4，说明弹性基体的两个参数对尺度效应的影响出现了相反的结果.
图6 不同剪切模量g时小尺度系数对临界流速的影响（β=0.5，H_X=k=0）
Fig.6 Critical flow velocities with longitudinal magnetic field in a fluid-conveyed SWCNT for different values g (β=0.5, H_X=k=0)
3 结论
本文应用非局部Euler-Bernouli梁模型，着重研究了在轴向磁场、尺度效应与弹性基体耦合作用时，嵌入弹性基体中的输流固支SWCNT的振动失稳特性，弹性基体采用Pasternak弹性模型.研究结果表明:随着磁场的增强，Pasternak弹性基体的两个参数不同程度地抑制了磁场对系统稳定性的影响; 但对小尺度系数的影响，两个参数却表现为相反的作用，即剪切参数抑制了小尺度对纳米管道系统的影响而弹性参数则放大了这一影响效应.
依据所得结论可以发现，通过外加轴向磁场及增加弹性基体可以提高纳米输流管道稳定性，但是一味增大二者强度从经济性考虑并不是一个明智的策略; 另外可以选取剪切系数高、弹性系数低的基体来改善小尺度效应对系统稳定性的影响.
参考文献
- [1] SAWANO S,ARIE T,AKITA S.Carbon nanotube resonator in liquid [J].Nano Letters,2010,10(9):3395-3398.
- [2] WANG Q,ARASH B.A review on applications of carbon nanotubes and graphenes as nano-resonator sensors [J].Computational Materials Science,2014,82(1):350-360.
- [3] 随岁寒,李威.输流管道弯曲和振动的有限元分析 [J].动力学与控制学报,2022,20(4):83-90.SUI S H,LI W.The finite element analysis on bending and vibration of the fluid-conveying pipes [J].Journal of Dynamics and Control,2022,20(4):83-90.(in Chinese)
- [4] AMIRI A,VESAL R,TALEBITOOTI R.Flexoelectric and surface effects on size-dependent flow-induced vibration and instability analysis of fluid-conveying nanotubes based on flexoelectricity beam model [J].International Journal of Mechanical Sciences,2019,156(6):474-485.
- [5] WANG L,NI Q.A reappraisal of the computational modelling of carbon nanotubes conveying viscous fluid [J].Mechanics Research Communications,2009,36(7):833-837.
- [6] ERINGEN A C.On differential equations of nonlocal elasticity and solutions of screw dislocations and surface waves [J].Journal of Applied Physics,1983,54(2):4703-4710.
- [7] MOHAMMAD M.On the plastic buckling of curved carbon nanotubes [J].Theoretical and Applied Mechanics Letters,2020,10(1):46-56.
- [8] MUSTAPHA K B,ZHONG Z W.The thermo-mechanical vibration of a single-walled carbon nanotube studied using the Bubnov-Galerkin method [J].Physica E:Low-dimensional Systems and Nanostructures,2010,43(1):375-381.
- [9] RAFIEI M,MOHEBPOUR S R,Daneshmand F.Small-scale effect on the vibration of non-uniform carbon nanotubes conveying fluid and embedded in viscoelastic medium [J].Physica E:Low-dimensional Systems and Nanostructures,2012,44(7-8):1372-1379.
- [10] DINI A,ABBAS Z B M,MAHMOUD S.Effects of van der Waals forces on hygro-thermal vibration and stability of fluid-conveying curved double-walled carbon nanotubes subjected to external magnetic field [J].Physica E:Low-dimensional Systems and Nanostructures,2019,106(10):156-169.
- [11] GHANE M,SAIDI A,BAHAADINI R.Vibration of fluid-conveying nanotubes subjected to magnetic field based on the thin-walled Timoshenko beam theory [J].Applied Mathematical Modelling,2020,80(11):65-83.
- [12] 李明,吕刘飞,郑华升等.磁场对不同温度场中输流悬臂碳纳米管颤振稳定性的影响 [J].固体力学学报,2021,42(1):87-93.LI M,LV L F,ZHENG H S,et al.Magnetic field effect on flutter stability of a fluid-conveying cantilevered carbon nanotube under different temperature fields [J].Chinese Journal of Solid Mechanics,2021,42(1):87-93.(in Chinese)
- [13] GHAVANLOO E,DANESHMAND F,RAFIEI M.Vibration and instability analysis of carbon nanotubes conveying fluid and resting on a linear viscoelastic winkler foundation [J].Physica E:Low-dimensional Systems and Nanostructures,2010,42(9):2218-2224.

基本信息

中图分类号: O353.1
文献标识码: A
DOI: 10.6052/1672-6553-2022-021
文章编号: 1672-6553-2023-21(3)-063-006

基金信息

国家自然科学基金资助项目(51909196)；
National Natural Science Foundation of China(51909196)；

引用信息

稿件历史

收稿日期: 2022-03-28

参考文献

[1] SAWANO S,ARIE T,AKITA S.Carbon nanotube resonator in liquid [J].Nano Letters,2010,10(9):3395-3398.
[2] WANG Q,ARASH B.A review on applications of carbon nanotubes and graphenes as nano-resonator sensors [J].Computational Materials Science,2014,82(1):350-360.
[3] 随岁寒,李威.输流管道弯曲和振动的有限元分析 [J].动力学与控制学报,2022,20(4):83-90.SUI S H,LI W.The finite element analysis on bending and vibration of the fluid-conveying pipes [J].Journal of Dynamics and Control,2022,20(4):83-90.(in Chinese)
[4] AMIRI A,VESAL R,TALEBITOOTI R.Flexoelectric and surface effects on size-dependent flow-induced vibration and instability analysis of fluid-conveying nanotubes based on flexoelectricity beam model [J].International Journal of Mechanical Sciences,2019,156(6):474-485.
[5] WANG L,NI Q.A reappraisal of the computational modelling of carbon nanotubes conveying viscous fluid [J].Mechanics Research Communications,2009,36(7):833-837.
[6] ERINGEN A C.On differential equations of nonlocal elasticity and solutions of screw dislocations and surface waves [J].Journal of Applied Physics,1983,54(2):4703-4710.
[7] MOHAMMAD M.On the plastic buckling of curved carbon nanotubes [J].Theoretical and Applied Mechanics Letters,2020,10(1):46-56.
[8] MUSTAPHA K B,ZHONG Z W.The thermo-mechanical vibration of a single-walled carbon nanotube studied using the Bubnov-Galerkin method [J].Physica E:Low-dimensional Systems and Nanostructures,2010,43(1):375-381.
[9] RAFIEI M,MOHEBPOUR S R,Daneshmand F.Small-scale effect on the vibration of non-uniform carbon nanotubes conveying fluid and embedded in viscoelastic medium [J].Physica E:Low-dimensional Systems and Nanostructures,2012,44(7-8):1372-1379.
[10] DINI A,ABBAS Z B M,MAHMOUD S.Effects of van der Waals forces on hygro-thermal vibration and stability of fluid-conveying curved double-walled carbon nanotubes subjected to external magnetic field [J].Physica E:Low-dimensional Systems and Nanostructures,2019,106(10):156-169.
[11] GHANE M,SAIDI A,BAHAADINI R.Vibration of fluid-conveying nanotubes subjected to magnetic field based on the thin-walled Timoshenko beam theory [J].Applied Mathematical Modelling,2020,80(11):65-83.
[12] 李明,吕刘飞,郑华升等.磁场对不同温度场中输流悬臂碳纳米管颤振稳定性的影响 [J].固体力学学报,2021,42(1):87-93.LI M,LV L F,ZHENG H S,et al.Magnetic field effect on flutter stability of a fluid-conveying cantilevered carbon nanotube under different temperature fields [J].Chinese Journal of Solid Mechanics,2021,42(1):87-93.(in Chinese)
[13] GHAVANLOO E,DANESHMAND F,RAFIEI M.Vibration and instability analysis of carbon nanotubes conveying fluid and resting on a linear viscoelastic winkler foundation [J].Physica E:Low-dimensional Systems and Nanostructures,2010,42(9):2218-2224.

弹性基体对磁场中输流纳米管的振动影响^*

Elastic Matrix Effect on Vibration Characteristics of Fluid-conveyed SWCNT Under A Longitudinal Magnetic Field^*

摘要

Abstract

关键词

Keywords

1 振动控制方程

1.1 振动方程推导

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

1.2 振动方程求解

(13)

(14)

(15)

(16)

(17)

(18)

(19)

2 数值计算结果分析

3 结论

参考文献

基本信息

基金信息

引用信息

稿件历史

参考文献

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弹性基体对磁场中输流纳米管的振动影响*

Elastic Matrix Effect on Vibration Characteristics of Fluid-conveyed SWCNT Under A Longitudinal Magnetic Field*

摘要

Abstract

关键词

Keywords

1 振动控制方程

1.1 振动方程推导

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

1.2 振动方程求解

(13)

(14)

(15)

(16)

(17)

(18)

(19)

2 数值计算结果分析

3 结论

参考文献

基本信息

基金信息

引用信息

稿件历史

参考文献

微信公众号二维码

手机版网站二维码

弹性基体对磁场中输流纳米管的振动影响^*

Elastic Matrix Effect on Vibration Characteristics of Fluid-conveyed SWCNT Under A Longitudinal Magnetic Field^*