Vibration and instability analysis of tubular nano- and micro-beams conveying fluid using nonlocal elastic theory

A nonlocal Euler–Bernoulli elastic beam model is developed for the vibration and instability of tubular micro- and nano-beams conveying fluid using the theory of nonlocal elasticity. Based on the Newtonian method, the equation of motion is derived, in which the effect of small length scale is incorporated. With this nonlocal beam model, the natural frequencies and critical flow velocities for the case of simply supported system and for the case of cantilevered system are obtained. The effect of small length scale (i.e., the nonlocal parameter) on the properties of vibrations is discussed. It is demonstrated that the natural frequencies are generally decreased with increasing values of nonlocal parameter, both for the supported and cantilevered systems. More significantly, the effect of small length scale on the critical flow velocities is visible for fluid-conveying beams with nano-scale length; however, this effect may be neglected for micro-beams conveying fluid.     

A modified nonlocal beam model for vibration and stability of nanotubes conveying fluid

In this paper, a new, modified nonlocal beam model is developed for analyzing the vibration and stability of nanotubes conveying fluid, in which one single nonlocal nanoscale parameter is included. Using Hamilton’s principle, a new higher-order differential equation of motion and the corresponding higher-order, non-classical boundary conditions are obtained for nanotubes conveying fluid. Based on this modified nonlocal model, effect of nonlocal nanoscale parameter on natural frequencies and critical flow velocities is presented and discussed through numerical calculations. It is found that this factor has great influence on the vibration and stability of nanotubes conveying fluid. In particular, the nonlocal effect tends to induce higher natural frequencies and higher critical flow velocities as compared to the results obtained from the classical and partial nonlocal beam models.     

Longitudinal vibration and stability analysis of carbon nanotubes conveying viscous fluid

Nowadays, carbon nanotubes (CNT) play an important role in practical applications in fluidic devices. To this end, researchers have studied various aspects of vibration analysis of a behavior of CNT conveying fluid. In this paper, based on nonlocal elasticity theory, single-walled carbon nanotube (SWCNT) is simulated. To investigate and analyze the effect of internal fluid flow on the longitudinal vibration and stability of SWCNT, the equation of motion for longitudinal vibration is obtained by using Navier-Stokes equations. In the governing equation of motion, the interaction of fluid-structure, dynamic and fluid flow velocity along the axial coordinate of the nanotube and the nano-scale effect of the structure are considered. To solve the nonlocal longitudinal vibration equation, the approximate Galerkin method is employed and appropriate simply supported boundary conditions are applied. The results show that the axial vibrations of the nanotubesstrongly depend on the small-size effect. In addition, the fluid flowing in nanotube causes a decrease in the natural frequency of the system. It is obvious that the system natural frequencies reach zero at lower critical flow velocities as the wave number increases. Moreover, the critical flow velocity decreases as the nonlocal parameter increases.     

Coupled effects of nano-size,stretching, and slip boundary conditions on nonlinear vibrations of nano-tube conveying fluid by the homotopy analysis method

In this paper, natural frequency and nonlinear response of carbon nano-tube (CNT) conveying fluid based on the coupling of nonlocal theory and von Karman's stretching have been obtained. The homotopy analysis method (HAM) has been used for solving nonlinear differential equation of system and convergence region of approach presented. Effects of mid-plane stretching, nonlocal parameter and their coupling in the model have been investigated. It has been concluded that stretching effect is significant only for higher-amplitude initial excitations and lower beam aspect ratios. Moreover, by including the slip boundary condition, the effect of nano-size flow has been revealed in the nonlinear vibration model. We have concluded that small-size effects of nano-tube and nano-flow have impressed critical velocity of fluid significantly specially for gas fluid. Analytical results obtained from HAM solution show satisfactory agreement with numerical solutions such as Runge–Kutta. Having an analytical approach, we have been able to investigate the unbounded growth of amplitude of vibrations for flow velocities near the critical value. Moreover, by employing the second-order approximation of Galerkin's method, the estimated natural frequency of the first mode is verified. The obtained results would indicate that the effects of higher mode on the first natural frequency are negligible for the doubly-clamped CNT.     

Wave propagation in fluid-conveying viscoelastic single-walled carbon nanotubes with surface and nonlocal effects

In this paper, the transverse wave propagation in fluid-conveying viscoelastic single-walled carbon nanotubes is investigated based on nonlocal elasticity theory with consideration of surface effect. The governing equation is formulated utilizing nonlocal Euler-Bernoulli beam theory and Kelvin-Voigt model. Explicit wave dispersion relation is developed and wave phase velocities and frequencies are obtained. The effect of the fluid flow velocity, structural damping, surface effect, small scale effects and tube diameter on the wave propagation properties are discussed with different wave numbers. The wave frequency increases with the increase of fluid flow velocity, but decreases with the increases of tube diameter and wave number. The effect of surface elasticity and residual surface tension is more significant for small wave number and tube diameter. For larger values of wave number and nonlocal parameters, the real part of frequency ratio raises.     

Analytical approaches for vibration analysis of multi-walled carbon nanotubes modeled as multiple nonlocal Euler beams

This paper concerns with the effect of small scale on the vibrational characteristics of multi-walled carbon nanotubes (MWCNTs) modeled as multiple nonlocal Euler beams. In this model, each nanotube interacts with its neighbors through the van der Waals force. Analytical approaches are expressed to solve coupled governing equations of the motion. Results for double- and five-walled carbon nanotubes (DWCNTs and FWCNTs), as two specific examples of MWCNTs, are presented for various boundary conditions. Then, effect of small scale on the natural and intertube resonant frequencies and their associated amplitude ratios are discussed. Besides the effect of small scale, the effect of end conditions on the vibrational properties and a comparison between the methods are provided. Natural and intertube frequencies reduce with the introduction of nonlocal parameter. However, reduction of intertube frequencies is less than the natural frequencies. Moreover, it is provided that the effect of small scale stiffens the van der Waals force and causes MWCNTs to behave similar to a single beam in high values of nonlocal parameter. Also, this study reveals that in high mode numbers, natural frequencies of a multiple classical Euler beams system tend to frequencies of its constituent beams.     

Flow-induced oscillations of a cantilevered pipe conveying fluid with base excitation

It is known that a plain cantilevered pipe conveying fluid loses its stability by a Hopf bifurcation, leading to either planar or non-planar flutter for flow velocities beyond the critical flow velocity for Hopf bifurcation. If an external mass is attached to the end of the pipe (an end-mass), the resulting dynamics become much richer, showing 2D and 3D quasiperiodic and chaotic oscillations at high flow velocities. In this paper, a cantilevered pipe, with and without an end-mass, subjected to a small-amplitude periodic base excitation is considered. A set of three-dimensional nonlinear equations is used to analyze the pipe?s response at various flow velocities and with different amplitudes and frequencies of base excitation. The nonlinear equations are discretized using the Galerkin technique and the resulting set of equations is solved using Houbolt?s finite difference method. It is shown that for a plain pipe (with no end-mass), non-planar post-instability oscillations can be reduced to planar periodic oscillations for a range of base excitation frequencies and amplitudes. For a pipe with an end-mass, similarly to a plain pipe, three-dimensional period oscillations can be reduced to planar ones. At flow velocities beyond the critical flow velocity for torus instability, the three-dimensional quasiperiodic oscillations can be reduced to two-dimensional quasiperiodic or periodic oscillations, depending on the frequency of base excitation. In all these cases, a low-amplitude base excitation results in reducing the three-dimensional oscillations of the pipe to purely two-dimensional oscillations, over a range of excitation frequencies. These numerical results are in agreement with the previous experimental work.     

In-plane vibration analysis of curved carbon nanotubes conveying fluid embedded in viscoelastic medium

The effect of the induced vibrations in the carbon nanotubes (CNTs) arising from the internal fluid flow is a critical issue in the design of CNT-based fluidic devices. In this study, in-plane vibration analysis of curved CNTs conveying fluid embedded in viscoelastic medium is investigated. The CNT is modeled as a linear elastic cylindrical tube where the internal moving fluid is characterized by steady flow velocity and mass density of fluid. A modified-inextensible theory is used in formulation and the steady-state initial forces due to the centrifugal and pressure forces of the internal fluid are also taken into account. The finite element method is used to discretize the equation of motion and the frequencies are obtained by solving a quadratic eigenvalue problem. The effects of CNT opening angle, the elastic modulus and the damping factor of the viscoelastic surrounded medium and fluid velocity on the resonance frequencies are elucidated. It is shown that curved CNTs are unconditionally stable even for a system with sufficiently high flow velocity. The most results presented in this investigation have been absent from the literature for fluid-induced vibration of curved CNTs embedded in viscoelastic foundations.     

The thermal effect on vibration and instability of carbon nanotubes conveying fluid

Based on the theory of thermal elasticity mechanics, an elastic Bernoulli–Euler beam model is developed for vibration and instability analysis of fluid-conveying single-walled carbon nanotubes (SWNTs) considering the thermal effect. Results are demonstrated for the dependence of natural frequencies on the flow velocity as well as temperature change. The influence of temperature change on the critical flow velocity at which buckling instability occurs is investigated. It is concluded that the effect of temperature change on the instability of SWNTs conveying fluid is significant.     

Assessment of nanotube structures under a moving nanoparticle using nonlocal beam theories

Dynamic analysis of nanotube structures under excitation of a moving nanoparticle is carried out using nonlocal continuum theory of Eringen. To this end, the nanotube structure is modeled by an equivalent continuum structure (ECS) according to the nonlocal Euler-Bernoulli, Timoshenko and higher order beam theories. The nondimensional equations of motion of the nonlocal beams acted upon by a moving nanoparticle are then established. Analytical solutions of the problem are presented for simply supported boundary conditions. The explicit expressions of the critical velocities of the nonlocal beams are derived. Furthermore, the capabilities of various nonlocal beam models in predicting the dynamic deflection of the ECS are examined through various numerical simulations. The role of the scale effect parameter, the slenderness ratio of the ECS and velocity of the moving nanoparticle on the time history of deflection as well as the dynamic amplitude factor of the nonlocal beams are scrutinized in some detail. The results show the importance of using nonlocal shear deformable beam theories, particularly for very stocky nanotube structures acted upon by a moving nanoparticle with low velocity.     

Flexural sensitivity and resonance of cantilever micro-sensors based on nonlocal elasticity theory

Sensors based on microcantilevers, especially ones with uniform structure, have ultrahigh sensitivities. The normalized natural frequencies and the sensitivity of lateral vibration of an elastic microcantilever sensor in contact with a surface are derived analytically based on the Euler–Bernoulli beam theory by taking into account the small scale effect. The interaction of the sensor with the surface is modeled by linear springs, which restricts the results to experiments involving low-amplitude excitations. The results show that the normalized natural frequencies of nonlocal microcantilever are smaller than those for its local counterpart, especially for higher values of small scale parameters. Also, each mode has a different sensitivity to variations in surface stiffness. Moreover, the most sensitivity is observed at the first mode of vibration. When the nonlocal effect is not taken into account, the natural frequencies and the sensitivity of the microcantilever in contact with the surface are compared with those obtained in previous study without considering the nonlocal effect.     

Longitudinal vibration and instabilities of carbon nanotubes conveying fluid considering size effects of nanoflow and nanostructure

In this study, the effects of small-scale of the both nanoflow and nanostructure on the vibrational response of fluid flowing single-walled carbon nanotubes are investigated. To this purpose, two various flowing fluids, the air-nano-flow and the water nano-flow using Knudsen number, and two different continuum theories, the nonlocal theory and the strain-inertia gradient theory are studied. Nano-rod model is used to model the fluid-structure interaction, and Galerkin method of weighted residual is utilizing to solve and discretize the governing obtained equations. It is found that the critical flow velocity decreases as the wave number increases, excluding the first mode divergence that it has the least value among of the other instabilities if the strain-inertia gradient theory is employed. Moreover, it is observed that Kn effect has considerable impact on the reduction of critical velocities especially for the air-flow flowing through the CNT. In addition, by increasing a nonlocal parameter and Knudsen number the critical flow velocity decreases but it increases as the characteristic length related to the strain-inertia gradient theory increases.     

基于Timoshenko梁模型的周期充液管路弯曲振动带隙特性和传输特性

充液管路的固液耦合振动广泛存在于各种工程领域中,对其弯曲振动控制进行研究具有重要意义.将声子晶体的周期性思想引入到管路结构设计中,将管壁设计成沿轴向交替排列的周期性复合结构.基于Timoshenko梁模型,采用传递矩阵法计算了固液耦合条件下周期管路结构的弯曲振动能带结构及其传输特性,同时分析了材料阻尼系数、周期和非周期支撑对带隙特性的影响.充液周期管路结构的弯曲振动带隙特性为管路的振动控制提供了一条新的技术途径. 关键词： 声子晶体 充液管路 振动带隙     

Dynamic behavior of triple-walled carbon nanotubes conveying fluid

In this study, the instability of triple-walled carbon nanotubes (TWCNTs) conveying fluid is studied based on an Euler–Bernoulli beam model. The van der Waals (vdW) interactions between different carbon nanotubes (CNTs) are taken into account in the analysis, and the Galerkin discretization approach is used to solve the coupled equations of the motions. Numerical simulations show that the interlayer vdW interactions play a significant role in the natural frequencies and the stability of TWCNTs. The critical flow velocities—associated with divergence, restabilization and flutter—are determined. The effects of different inner radius and the value of mode N used in Galerkin discretization on the dynamical behaviors of the fluid-conveyed TWCNTs are also examined in detail. Results reveal that the internal moving fluid plays an important role in the instability of TWCNTs.     

Flow-thermoelastic vibration and instability analysis of viscoelastic carbon nanotubes embedded in viscous fluid

The flexural vibration of viscoelastic carbon nanotubes (CNTs) conveying fluid and embedded in viscous fluid is investigated by the nonlocal Timoshenko beam model. The governing equations are developed by Hamilton's principle, including the effects of structural damping of the CNT, internal moving fluid, external viscous fluid, temperature change and nonlocal parameter. Applying Galerkin’s approach, the resulting equations are transformed into a set of eigenvalue equations. The validity of the present analysis is confirmed by comparing the results with those obtained in literature. The effects of the main parameters on the vibration characteristics of the CNT are also elucidated. Most results presented in the present investigation have been absent from the literature for the vibration and instability of the CNT conveying fluid.     

Buckling of single layer graphene sheet based on nonlocal elasticity and higher order shear deformation theory

Higher order shear deformation theory (HSDT) is reformulated using the nonlocal differential constitutive relations of Eringen. The equations of motion of the nonlocal theories are derived. The developed equations of motion have been applied to study buckling characteristics of nanoplates such as graphene sheets. Navier's approach has been used to solve the governing equations for all edges simply supported boundary conditions. Analytical solutions for critical buckling loads of the graphene sheets are presented. Nonlocal elasticity theories are employed to bring out the small scale effect on the critical buckling load of graphene sheets. Effects of (i) nonlocal parameter, (ii) length, (iii) thickness of the graphene sheets and (iv) higher order shear deformation theory on the critical buckling load have been investigated. The theoretical development as well as numerical solutions presented herein should serve as reference for nonlocal theories as applied to the stability analysis of nanoplates and nanoshells.     

Wave characteristics of single-walled fluid-conveying carbon nanotubes subjected to multi-physical fields

Wave propagation in single-walled carbon nanotubes (SWCNTs) conveying fluids and placed in multi-physical fields (including magnetic and temperature fields) is studied in this paper. The nanotubes are modelled as Timoshenko beams. Based on the nonlocal beam theory, the governing equations of motion are derived using Hamilton's principle, and then solved by Galerkin approach, leading to two second-order ordinary differential equations (ODEs). Numerical simulations are carried out to verify the analytical model proposed in the present study, and determine the influences of the nonlocal parameter, the fluid velocity and flow density, the temperature and magnetic field flux change, and the surrounding elastic medium on the wave behaviour of SWCNTs. The results show that the nonlocal parameter has a considerable influence on dynamic behaviour of the nanotube and the fluid flow inside it. The results also show that the magnetic and temperature fields play an important role on the wave propagation characteristics of SWCNTs.     

Reply to the comments of M.E. Golmakani and J. Rezatalab,Comment on “Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates” (by R. Aghababaei and J.N. Reddy,Journal of Sound and Vibration 326 (2009) 277–289), Journal of Sound Vibration, 333 (2014) 3831–3835

Golmakani and Rezatalab [1] suggested in their paper that the deflection of a simply supported nonlocal elastic plate under uniform load is not affected by the small length scale terms. They based their proof on the use of Navier?s method using a sinusoidal-based deflection solution. This insensitivity of the deflection solution of a simply supported nonlocal elastic plate with respect to the small length terms of Eringen?s model is not correct, as already detailed in the literature (for example, see [2] for beam problems). In fact, the deflection of the nonlocal plate (in the Eringen sense) is larger than the one of the local case, as shown in many papers available in the literature. We prove in this reply to the authors that the Navier?s method has to be correctly applied for highlighting the specific sensitivity phenomenon of the deflection solution, as compared to exact analytical solution.     

Dynamic stability of elastically supported pipes conveying pulsating fluid

The effect of support flexibility on the dynamic behaviour of pipes conveying fluid is investigated for both steady and pulsatile flows. The pipes are built-in at the upstream end and supported at the other by both a translational and a rotational spring. For the steady flow condition, the critical flow velocities, the frequencies and flow induced damping patterns that are associated with the different vibration modes of selected pipe systems are determined as functions of the flow velocity. The results from steady flow cases show that the pipes may first lose stability by either buckling or flutter, depending on the values of the rotational and translational spring constants and their relative magnitudes. In the case of pulsatile flow, the Floquet theory is utilized for the stability analysis of the selected pipe-fluid systems. Numerical results are presented to illustrate the effects of the amount of translational and rotational resiliences at the elastic support on the regions of parametric and combination resonances of the pipes. The results more of the interesting aspects of the behaviour of non-conservative systems.