Numerical investigation of the influence of gravity on flutter of cantilevered pipes conveying fluid

We have carried out a numerical investigation of the three dimensional nonlinear dynamics of a cantilevered pipe conveying fluid in the presence of gravity. The pipe may be misaligned at the clamped end with respect to gravity, and the effects of this misalignment are the main objects of the present investigation. The problem has been formulated using the Cosserat rod model. First, we have computed the equilibrium solutions and used them to experimentally validate both the Cosserat model and the constitutive law. Then, we have analyzed the occurrence of flutter, via Hopf bifurcation, for critical values of the relevant parameters of the problem, such as fluid to total mass ratio, dimensionless flow rate, dimensionless gravity and misalignment angle. The influence of the equilibrium solution on flutter has been explored, and the results of the linear stability analysis show that the stabilizing or destabilizing effect of fluid flow, either in or out of the plane of the pipe, depend crucially on the misalignment. We have also computed the non-linear periodic behavior after flutter instability by two different methods: the first one is by solving the full nonlinear equations by direct integration in time and space, while the second one is by assuming the time dependence given by an appropriate ansatz. Circular periodic orbits have then been studied and found that its loss of stability via Hopf bifurcation gives rise to stable planar periodic orbits. Finally, we have also computed the multiply periodic and chaotic behaviors which take place for sufficiently large values of the flow rate.     

THE DYNAMIC BEHAVIORS OF VISCOELASTIC PIPE CONVEYING FLUID WITH THE KELVIN MODEL

Based on the differential constitutive relationship of linear viscoelastic, material, a solid-liquid coupling vibration equation for viscoelastic pipe conveying fluid is derived by the D'Alembert's principle. The critical flow velocities and natural frequencies of the cantilever pipe conveying fluid with the Kelvin model (flutter instability) are calculated with the modified finite difference method in the form of the recurrence formula. The curves between the complex frequencies of the first, second and third mode and flow velocity of the pipe are plotted. On the basis of the numerical, calculation results, the dynamic behaviors and stability of the pipe are discussed. It should be pointed out that the delay time of viscoelastic material with the Kelvin model has a remarkable effect on the dynamic characteristics and stability behaviors of the cantilevered pipe conveying fluid, which is a gyroscopic non-conservative system.     

Analysis of coupled-mode flutter of pipes conveying fluid on the elastic foundation

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Natural Frequency and Stability Tuning of Cantilevered CNTs Conveying Fluid in Magnetic Field

This paper investigates the dynamics of cantilevered CNTs conveying fluid in longitudinal magnetic field and presents the possibility of controlling/tuning the stability of the CNT system with the aid of magnetic field. The slender CNT is treated as an Euler-Bernoulli beam. Based on nonlocal elasticity theory, the equation of motion with consideration of magnetic field effect is developed. This partial differential equation is then discretized using the differential quadrature method(DQM). Numerical results show that the nonlocal small-scale parameter makes the fluid-conveying CNT more flexible and can shift the unstable mode in which flutter instability occurs first at sufficiently high flow velocity from one to another. More importantly,the addition of a longitudinal magnetic field leads to much richer dynamical behaviors of the CNT system. Indeed, the presence of longitudinal magnetic field can significantly affect the evolution of natural frequency of the dynamical system when the flow velocity is successively increased.With increasing magnetic field parameter, it is shown that the CNT system behaves stiffer and hence the critical flow velocity becomes higher. It is of particular interest that when the magnetic field parameter is equal to or larger than the flow velocity, the cantilevered CNT conveying fluid becomes unconditionally stable, indicating that the dynamic stability of the system can be controlled due to the presence of a longitudinal magnetic field.     

STABILITY AND MODE EVOLUTION CHARACTERISTICS OF A CANTILEVERED FLUID-CONVEYING PIPE ATTACHED WITH THE LUMPED MASS 1)

本文主要研究通过调控集中质量对悬臂输流管稳定性和振动模态特性的影响规律,为输流管动力学性能的可控性提供理论指导和实验依据. 首先基于扩展的哈密顿原理,建立了含集中质量悬臂输流管的非线性动力学理论模型. 基于线性动力学特性分析,研究发现集中质量沿管道轴向位置变化对输流管发生颤振失稳的临界流速有重要影响.并通过伽辽金前四阶模态截断处理线性矩阵方程式,定性地分析了集中质量位置与质量比的变化对于输流管稳定性影响的变化.实验结果表明, 输流管的颤振失稳模态随集中质量位置的变化发生了转迁. 此外,基于动力学理论分析, 发现集中质量比值对失稳临界流速也有重要的影响,且主要取决于集中质量的安装位置. 基于非线性特性,进一步分析了集中质量对输流管振动幅值的影响. 实验和理论研究发现,集中质量位置从固定端向自由端变化时, 输流管振幅表现出先增大后减小趋势,且振动模态也从二阶转迁到三阶.本研究有望为输流管振动驱动应用提供理论支撑与指导意义.     

含集中质量悬臂输流管的稳定性与模态演化特性研究

本文主要研究通过调控集中质量对悬臂输流管稳定性和振动模态特性的影响规律,为输流管动力学性能的可控性提供理论指导和实验依据. 首先基于扩展的哈密顿原理,建立了含集中质量悬臂输流管的非线性动力学理论模型. 基于线性动力学特性分析,研究发现集中质量沿管道轴向位置变化对输流管发生颤振失稳的临界流速有重要影响.并通过伽辽金前四阶模态截断处理线性矩阵方程式,定性地分析了集中质量位置与质量比的变化对于输流管稳定性影响的变化.实验结果表明, 输流管的颤振失稳模态随集中质量位置的变化发生了转迁. 此外,基于动力学理论分析, 发现集中质量比值对失稳临界流速也有重要的影响,且主要取决于集中质量的安装位置. 基于非线性特性,进一步分析了集中质量对输流管振动幅值的影响. 实验和理论研究发现,集中质量位置从固定端向自由端变化时, 输流管振幅表现出先增大后减小趋势,且振动模态也从二阶转迁到三阶.本研究有望为输流管振动驱动应用提供理论支撑与指导意义.      

Stability Analysis of Maxwell Viscoelastic Pipes Conveying Fluid with Both Ends Simply Supported

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分析弹性支承输流管道的失稳临界流速

研究了两端弹性支承输流管道静态失稳和动态失稳临界流速. 根据梁模型横向弯曲振动模态 函数，由两端弹性支承的边界条件得到了其模态函数的一般表达式. 根据特征方程具体分析 了弹性支承刚度、质量比、流体压力和管截面轴向力等主要参数对失稳临界流速的影响. 数 值计算结果表明，管道在弹性支承下的动力稳定性比较复杂，在较小的弹性支承刚度和较小 的参数范围内，管道主要表现为动态颤振失稳；在较大的弹性支承刚度和较大的参数作用下， 管道的失稳形式主要表现为静态失稳；并且失稳临界流速随流体压力和管截面轴向压力的增 加而下降，随管截面轴向拉力的增加而上升.     

FLOW-INDUCED INTERNAL RESONANCES AND MODE EXCHANGE IN HORIZONTAL CANTILEVERED PIPE CONVEYING FLUID （Ⅱ）

Based on the nonlinear mathematical model of motion of a horizontally can-tilevered rigid pipe conveying fluid, the 3:1 internal resonance induced by the minimum critical velocity is studied in details. With the detuning parameters of internal and primary resonances and the amplitude of the external disturbing excitation varying, the flow in the neighborhood of the critical flow velocity yields that some nonlinearly dynamical behaviors occur in the system such as mode exchange, saddle-node, Hopf and co-dimension 2 bifurcations. Correspondingly, the periodic motion losses its stability by jumping or flutter, and more complicated motions occur in the pipe under consideration. The good agreement between the analytical analysis and the numerical simulation for several parameters ensures the validity and accuracy of the present analysis.     

Articulated Pipes Conveying Fluid Pulsating with High Frequency

Stability and nonlinear dynamics of two articulated pipes conveying fluid with a high-frequency pulsating component is investigated. The non-autonomous model equations are converted into autonomous equations by approximating the fast excitation terms with slowly varying terms. The downward hanging pipe position will lose stability if the mean flow speed exceeds a certain critical value. Adding a pulsating component to the fluid flow is shown to stabilize the hanging position for high values of the ratio between fluid and pipe-mass, and to marginally destabilize this position for low ratios. An approximate nonlinear solution for small-amplitude flutter oscillations is obtained using a fifth-order multiple scales perturbation method, and large-amplitude oscillations are examined by numerical integration of the autonomous model equations, using a path-following algorithm. The pulsating fluid component is shown to affect the nonlinear behavior of the system, e.g. bifurcation types can change from supercritical to subcritical, creating several coexisting stable solutions and also anti-symmetrical flutter may appear.     

In-plane dynamics of a fluid-conveying corrugated pipe supported at both ends

The dynamics and stability of fluid-conveying corrugated pipes are investigated. The flow velocity is assumed to harmonically vary along the pipe rather than with time. The dimensionless equation is discretized with the differential quadrature method(DQM). Subsequently, the effects of the mean flow velocity and two key parameters of the corrugated pipe, i.e., the amplitude of the corrugations and the total number of the corrugations, are studied. The results show that the corrugated pipe will lose stability by flutter even if it has been supported at both ends. When the total number of the corrugations is sufficient, this flutter instability occurs at a micro flow velocity. These phenomena are verified via the Runge-Kutta method. The critical flow velocity of divergence is analyzed in detail. Compared with uniform pipes, the critical velocity will be reduced due to the corrugations, thus accelerating the divergence instability. Specifically,the critical flow velocity decreases if the amplitude of the corrugations increases. However, the critical flow velocity cannot be monotonously reduced with the increase in the total number of the corrugations. An extreme point appears, which can be used to realize the parameter optimization of corrugated pipes in practical applications.     

FLOW-INDUCED INTERNAL RESONANCES AND MODE EXCHANGE IN HORIZONTAL CANTILEVERED PIPE CONVEYING FLUID (Ⅱ)

Based on the nonlinear mathematical model of motion of a horizontally cantilevered rigid pipe conveying fluid, the 3:1 internal resonance induced by the minimum critical velocity is studied in details. With the detuning parameters of internal and primary resonances and the amplitude of the external disturbing excitation varying, the flow in the neighborhood of the critical flow velocity yields that some nonlinearly dynamical behaviors occur in the system such as mode exchange, saddle-node, Hopf and co-dimension 2 bifurcations. Correspondingly, the periodic motion losses its stability by jumping or flutter, and more complicated motions occur in the pipe under consideration.The good agreement between the analytical analysis and the numerical simulation for several parameters ensures the validity and accuracy of the present analysis.     

Nonlinear dynamics of functionally graded pipes conveying hot fluid

For improved stability of fluid-conveying pipes operating under the thermal environment, functionally graded materials (FGMs) are recommended in a few recent studies. Besides this advantage, the nonlinear dynamics of fluid-conveying FG pipes is an important concern for their engineering applications. The present study is carried out in this direction, where the nonlinear dynamics of a vertical FG pipe conveying hot fluid is studied thoroughly. The FG pipe is considered with pinned ends while the internal hot fluid flows with the steady or pulsatile flow velocity. Based on the Euler–Bernoulli beam theory and the plug-flow model, the nonlinear governing equation of motion of the fluid-conveying FG pipe is derived in the form of the nonlinear integro-partial-differential equation that is subsequently reduced as the nonlinear temporal differential equation using Galerkin method. The solutions in the time or frequency domain are obtained by implementing the adaptive Runge–Kutta method or harmonic balance method. First, the divergence characteristics of the FG pipe are investigated and it is found that buckling of the FG pipe arises mainly because of temperature of the internal fluid. Next, the dynamic characteristics of the FG pipe corresponding to its pre- and post-buckled equilibrium states are studied. In the pre-buckled equilibrium state, higher-order parametric resonances are observed in addition to the principal primary and secondary parametric resonances, and thus the usual shape of the parametric instability region deviates. However, in the post-buckled equilibrium state of the FG pipe, its chaotic oscillations may arise through the intermittent transition route, cyclic-fold bifurcation, period-doubling bifurcation and subcritical bifurcation. The overall study reveals complex dynamics of the FG pipe with respect to some system parameters like temperature of fluid, material properties of FGM and fluid flow velocity.     

分布式运动约束下悬臂输液管的参数共振研究

输液管道结构在航空、航天、机械、海洋、水利和核电等工程领域都有广泛应用,其稳定性、振动与安全评估备受关注.针对具有分布式运动约束悬臂输液管的非线性动力学模型,分别采用立方非线性弹簧和修正三线性弹簧来模拟运动约束的作用力,研究了管道在脉动内流激励下的参数共振行为.首先,从输液管系统的非线性控制方程出发,利用Galerkin方法进行离散化;然后,由Floquet理论得出线性系统在失稳前两个不同平均流速下脉动幅值和脉动频率变化时的共振参数区域;最后,考虑系统的几何非线性项和分布式非线性运动约束力的影响,求解了管道的非线性动力学响应,讨论了非线性项及运动约束力对管道参数共振行为的影响.研究结果表明,系统非线性共振响应的参数区域与线性系统的共振参数区域是一致的,分布式运动约束力对发生参数共振时管道的位移响应有显著影响;立方非线性弹簧和修正三线性弹簧模型所预测的分岔路径存有较大差异,但都可诱发管道在一定的参数激励下出现混沌运动.      

Nonplanar post-buckling analysis of simply supported pipes conveying fluid with an axially sliding downstream end

In this study,the nonplanar post-buckling behavior of a simply supported fluid-conveying pipe with an axially sliding downstream end is investigated within the framework of a three-dimensional(3 D)theoretical model.The complete nonlinear governing equations are discretized via Galerkin’s method and then numerically solved by the use of a fourth-order Runge-Kutta integration algorithm.Different initial conditions are chosen for calculations to show the nonplanar buckling characteristics of the pipe in two perpendicular lateral directions.A detailed parametric analysis is performed in order to study the influence of several key system parameters such as the mass ratio,the flow velocity,and the gravity parameter on the post-buckling behavior of the pipe.Typical results are presented in the form of bifurcation diagrams when the flow velocity is selected as the variable parameter.It is found that the pipe will stay at its original straight equilibrium position until the critical flow velocity is reached.Just beyond the critical flow velocity,the pipe would lose stability by static divergence via a pitchfork bifurcation,and two possible nonzero equilibrium positions are generated.It is shown that the buckling and post-buckling behaviors of the pipe cannot be influenced by the mass ratio parameter.Unlike a pipe with two immovable ends,however,the pinned-pinned pipe with an axially sliding downstream end shows some different features regarding post-buckling behaviors.The most important feature is that the buckling amplitude of the pipe with an axially sliding downstream end would increase first and then decrease with the increase in the flow velocity.In addition,the buckled shapes of the pipe varying with the flow velocity are displayed in order to further show the new post-buckling features of the pipe with an axially sliding downstream end.     

Stability and local bifurcation of parameter-excited vibration of pipes conveying pulsating fluid under thermal loading

The parametric excited vibration of a pipe under thermal loading may occur because the fluid is often transported heatedly. The effects of thermal loading on the pipe stability and local bifurcations have rarely been studied. The stability and the local bifurcations of the lateral parametric resonance of the pipe induced by the pulsating fluid velocity and the thermal loading are studied. A mathematical model for a simply supported pipe is developed according to the Hamilton principle. Two partial differential equations describing the lateral and longitudinal vibration are obtained. The singularity theory is utilized to analyze the stability and the bifurcation of the system solutions. The transition sets and the bifurcation diagrams are obtained both in the unfolding parameter space and the physical parameter space, which can reveal the relationship between the thermal field parameter and the dynamic behaviors of the pipe. The frequency response and the relationship between the critical thermal rate and the pulsating fluid velocity are obtained. The numerical results demonstrate the accuracy of the single-mode expansion of the solution and the stability and local bifurcation analyses. It also confirms the existence of the chaos. The presented work can provide valuable information for the design of the pipeline and the controllers to prevent the structural instability.     

固—液耦合Timoshenko管道的稳定性分析

根据Ｈａｍｉｌｔｏｎ原理的固－液耦合振同分方程用幂级数法计算了Ｔｉｍｏｓｈｅｎｋｏ管道的固有频率和临界流速。给出了管道前三阶固有频率－流速的关系曲线,分析了转动惯量对该输流管道的稳定特性的影响。计算结果表明,转动一对两端简支的固－液合Ｔｉｍｏｓｈｅｎｋｏ管道的静力失稳没有影响,但对其频率特性和动力失稳有影响。     

Experimental investigation of dynamic stability of a cantilever pipe aspirating fluid

The dynamic stability of a submerged cantilever pipe conveying fluid from the free end to the fixed one is considered as one of the unresolved issues in the area of fluid–structure interaction. There is a contradiction between theoretical predictions and experiments. Reported experiments did not show any instability, while theory predicts instability beyond a critical fluid velocity. Recently, several papers appeared, improving the theoretical modelling of pipe dynamics. All theories predict instability, either oscillatory or static, referred to here as flutter and divergence, respectively. A new test set-up was designed to investigate the hypothesis that previous experimental set-ups could not allow observations of pipe instability or the pipe aspirating water is unconditionally stable. In this new test set-up, the fluid velocity could exceed the theoretically predicted critical velocities. A cantilever pipe of about 5 m length was partly submerged in water. The free open end of the pipe was in the water, whereas the fixed end was above the waterline. The experiments clearly showed that the cantilever pipe aspirating water is unstable beyond a critical velocity of water convection through the pipe. Below this velocity the pipe is stable, whereas above it the pipe shows a complex motion that consists of two alternating phases. The first phase is a nearly periodic orbital motion with maximum amplitude of a few pipe diameters, whereas the second one is a noise-like vibration with very small amplitudes. Increasing the internal fluid velocity results in a larger amplitude of the orbital motion, but does not change the pipe motion qualitatively.     

磁场对不同温度场中输流悬臂碳纳米管动态特性的影响

本文在采用经典欧拉-伯努利梁模型的基础上,引入考虑小尺度效应的非局部弹性理论,着重研究不同温度场中输流悬臂单层碳纳米管系统（SWCNT）在外加纵向磁场作用下的颤振失稳问题。基于哈密顿原理获得了该流固耦合系统的振动控制方程及相应的边界条件,应用微分变换法（DTM法）求解此高阶偏微分方程,通过数值计算研究了不同温度场中施加纵向磁场对系统动力学特性的影响。结果表明：施加纵向磁场在不同温度场中都将增强输流悬臂碳纳米管的动态稳定性。然而,这种增强程度却与温度场的变化量有关,在不同温度变化量下,磁场对系统稳定性的增强程度有一个峰值,这意味着,实际应用中,为了提高这类流固耦合系统的动态稳定性,一味提高纵向磁场强度并不可取。     

Three-dimensional dynamics of supported pipes conveying fluid

This paper deals with the three-dimensional dynamics and postbuckling behavior of flexible supported pipes conveying fluid, considering flow velocities lower and higher than the critical value at which the buckling instability occurs. In the case of low flow velocity, the pipe is stable with a straight equilibrium position and the dynamics of the system can be examined using linear theory. When the flow velocity is beyond the critical value, any motions of the pipe could be around the postbuckling configuration (non-zero equilibrium position) rather than the straight equilibrium position, so nonlinear theory is required. The nonlinear equations of perturbed motions around the postbuckling configuration are derived and solved with the aid of Galerkin discretization. It is found, for a given flow velocity, that the first-mode frequency for in-plane motions is quite different from that for out-of-plane motions. However, the second- or third-mode frequencies for in-plane motions are approximately equal to their counterparts for out-of-plane motions, keeping almost constant values with increasing flow velocity. Moreover, the orientation angle of the postbuckling configuration plane for a buckled pipe can be significantly affected by initial conditions, displaying new features which have not been observed in the same pipe system factitiously supposed to deform in a single plane.