磁场下黏弹性基体中输流纳米管稳定性分析

Stability Analysis of Fluid-conveying Nanotubes Embedded in Viscoelastic Medium Subjected to Magnetic Field

基于非局部连续介质理论,应用哈密顿原理建立轴向磁场作用下黏弹性基体中固支输流单层碳纳米管(Single-walled Carbon Nanotube,SWCNT)系统振动微分方程,应用微分变换法(Differential Transformation Method,DTM)求解该振动方程,着重研究黏弹性基体、轴向磁场、小尺度效应耦合作用时该系统的振动稳定性问题。数值计算结果表明:输流碳纳米管无论是否嵌入黏弹性基体中,磁场的作用均能提高系统的稳定性,而小尺度系数的增加则降低系统稳定性。黏弹性基体的阻尼系数加大系统的振动耗能,当阻尼系数处于较大数值时,系统振动能量迅速耗散,在管内流体流速还处于较低数值时系统即产生发散失稳现象。进一步研究表明在考虑小尺度效应、轴向磁场与基体耦合作用时,较强的轴向磁场可以降低小尺度效应、基体阻尼系数对系统的影响;即使存在小尺度效应,对于弹性系数较大的基体,其阻尼系数对振动系统的影响程度仍大大降低。

Based on the non-local Euler-Bernoulli beam theory and Hamiltonian principle,the system vibration differential equations of fixed-supported single-walled carbon nanotubes(SWCNTs)for fluid transportation embedded in viscoelastic matrix under the action of longitudinal magnetic field is established.The differential transformation method is used to solve the vibration equations.The stability of the SWCNTs under the coupling of viscoelastic matrix,longitudinal magnetic field and small-scale effect is studied.The numerical results indicate that the magnetic field increases the critical velocity of divergent instability and improves the stability of the system whether SWCNTs are embedded in the viscoelastic matrix or not.The viscoelastic matrix stiffness has a similar stabilizing effect on the natural frequency and critical flow velocity of the SWCNTs.Increasing of small-scale coefficient will reduce the stability of the system,making the system more prone to instability as the fluid velocity increases.Increasing of the damping coefficient of the viscoelastic matrix can enlarge the vibration energy dissipation of the system.For larger damping coefficient of the viscoelastic matrix,the vibration energy of the system dissipates rapidly,and the dissipation instability phenomenon of the system will occur when the fluid flow velocity in the nanotube is still low.Further research shows that the small-scale coefficient has noticeable effects on the critical velocity of the fluid flow in the nanotube embedded in the viscoelastic matrix with higher stiffness,and stronger longitudinal magnetic field can reduce the influence of small-scale effect and viscoelastic matrix damping coefficient on the vibration system.This study may be useful for future study of multi-physics problems in a nanoscale system,design of new nanostructures and vibration control devices in a magnetic field.