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

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Influence of a Wieghardt foundation on the dynamic stability of a fluid conveying pipe

This article considers the behaviour of a fluid conveying pipe on a partial elastic foundation. The model of the pipe is that of a Timoshenko beam; the foundation response is of Wieghardt type. Both material and environmental damping are taken into account. The critical value of the velocity of the fluid inducing dynamical instability of the system is evaluated as a function of the attachment ratio of the foundation for various values of the physical quantities involved. It is shown that this dependance is not always monotonic.     

ABSOLUTE AND CONVECTIVE BENDING INSTABILITIES IN FLUID-CONVEYING PIPES

The effect of internal plug flow on the lateral stability of fluid conveying pipes is investigated by determining the absolute or convective nature of the instability from the analytically derived linear dispersion relation. The fluid–structure interaction is modelled by following the work of Gregory & Paı̈doussis. The formulation of the fluid-conveying pipe problem is shown to be related to previous studies of a flat plate in the presence of uniform flow by Brazier-Smith & Scott and Crigthon & Oswell. The different domains of stability, convective instability, and absolute instability are explicitly derived in control parameter space. The effects of flow velocity, fluid–structure mass ratio, stiffness of the elastic foundation, bending rigidity and axial tension are considered. Absolute instability in flexural pipes prevails over a wide range of parameters. Convective instability is mostly found in tensioned pipes, which are modelled by a generalized linear Klein–Gordon equation. The impulse response is given in closed form or as an integral approximation and its behaviour confirms the results found directly from the dispersion equation.     

DIVERGENCE INSTABILITY OF VARIABLE-ARC-LENGTH ELASTICA PIPES TRANSPORTING FLUID

A flexible elastic pipe transporting fluid is held by an elastic rotational spring at one end, while at the other end, a portion of the pipe may slide on a frictional support. Regardless of the gravity loads, when the internal flow velocity is higher than the critical velocity, large displacements of static equilibrium and divergence instability can be induced. This problem is highly nonlinear. Based on the inextensible elastica theory, it is solved herein via the use of elliptic integrals and the shooting method. Unlike buckling with stable branching of a simply supported elastica pipe with constant length, the variable arc-length elastica pipe buckles with unstable branching. The friction at the support has an influence in shifting the critical locus over the branching point. Alteration of the flow history causes jumping between equilibrium paths due to abrupt changes of direction of the support friction. The elastic rotational restraint brings about unsymmetrical bending configurations; consequently, snap-throughs and snap-backs can occur on odd and even buckling modes, respectively. From the theoretical point of view, the equilibrium configurations could be formed like soliton loops due to snapping instability.     

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.     

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

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三参量固体模型粘弹性输流管道的动力特性分析

推导了三参量固体模型粘弹性输流管道的振动微分方程,计算了在不同无量纲松弛系数和弹性常数比下管道的无量纲临界流速和无量纲自振复频率,并给出了前三阶复频率与流速的关系.计算结果表明,质量比、无量纲松弛系数及无量纲弹性常数比对输流管道的动力特性均有影响.     

Stability of fluid-conveying periodic shells on an elastic foundation with external loads

This paper presents the beam-mode stability of a fluid-conveying periodic shell on an elastic foundation subjected to external loading. A transfer matrix (TM) method was developed to investigate the characteristics of steady-state waves in the system and the dynamic response of the periodic shell system. When subjected to external perturbations, including either a moving load or a stationary one, the shell may be subjected to instability for flow velocities exceeding a certain critical velocity. The system can also become unstable when a travelling load exceeds a certain critical value. The coupled effects of the speed of a moving load and the flow velocity of a fluid on the stability of the shell system were also investigated. A periodic structure was designed for such a shell system to enhance its dynamic stability. The periodic shell system produces innumerable velocity band gaps (VBGs), which could raise the critical velocity and extend the stable range for both the moving load and the flowing fluid. Finally, the formation mechanism of the VBGs was studied, as well as the effects of the thickness, length of the shell cells, Young׳s modulus and stiffness of the elastic foundation on modulating the VBGs.     

两端弹性支承输流管道固有特性研究

输流管道广泛应用于航天航空、石油化工、海洋等重要的工程领域, 其振动特性尤其是系统固有特性一直是国内外学者研究的热点问题. 本文研究了两端弹性支承输流管道横向振动的固有特性, 尤其是在非对称弹性支承下的系统固有特性. 使用哈密顿原理得到了输流管道的控制方程及边界条件, 通过复模态法得到了静态管道的模态函数, 以其作为伽辽金法的势函数和权函数对线性派生系统控制方程进行截断处理. 分析了两端对称支承刚度、两端非对称支承刚度、管道长度以及流体质量比对系统固有频率的影响规律, 重点讨论了管道两端可能形成的非对称支承条件下固有频率的变化规律. 结果表明, 较大的对称支承刚度下管道的第一阶固有频率下降较快; 当管道两端支承刚度变化时, 管道的各阶固有频率在两端支承刚度相等时取得最值; 对于两端非对称支承的管道而言, 两端支承刚度越接近, 第一阶固有频率下降的越快, 而且相应的临界流速越小; 流体的流速越大, 其对两端非对称弹簧支承的管道固有频率的影响更为明显.      

输液曲管平面内振动的波动方法研究

采用Flügge曲粱模拟弯曲管道,推导了管内流体的加速度,在总体轴线不可伸长假定的基础上建立了曲管平面内振动的动力学方程;采用波动方法,获得了曲管内振动波的传播和反射矩阵,提出了计算曲管平面内振动固有频率的数值方法.算例分析中,通过计算两端固定半圆形曲管的临界流速并与已有文献结果对比,验证了论文方法的正确性.最后,计算了两端固定半圆形曲管在四种不同流速下的前四阶固有频率,结果表明,管内流速的增大会降低管道的固有频率,当流速增大到某一特定值时,管道的一阶固有频率消失.     

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 flutter instability of cantilever carbon nanotubes

Carbon nanotubes are finding significant application to nanofluidic devices. This work studies the influence of internal moving fluid on free vibration and flow-induced flutter instability of cantilever carbon nanotubes based on a continuum elastic model. Since the flow-induced vibration of cantilever pipes is non-conservative in nature, cantilever carbon nanotubes conveying fluid are damped with decaying amplitude for flow velocity below a certain critical value. Beyond this critical flow velocity, flutter instability occurs and vibration becomes amplified with growing amplitude. Our results indicate that internal moving fluid substantially affects vibrational frequencies and the decaying rate of amplitude especially for longer cantilever carbon nanotubes of larger innermost radius at higher flow velocity, and the critical flow velocity for flutter instability in some cases may fall within the practical range. On the other hand, a moderately stiff surrounding elastic medium (such as polymers) can significantly suppress the effect of internal moving fluid on vibrational frequencies and suppress or eliminate flutter instability within the practical range of flow velocity.     

Extremely large-amplitude oscillation of soft pipes conveying fluid under gravity

In this work, the nonlinear behaviors of soft cantilevered pipes containing internal fluid flow are studied based on a geometrically exact model, with particular focus on the mechanism of large-amplitude oscillations of the pipe under gravity. Four key parameters, including the flow velocity, the mass ratio, the gravity parameter, and the inclination angle between the pipe length and the gravity direction, are considered to affect the static and dynamic behaviors of the soft pipe. The stability ...     

Over-reflection and instabilities in shear flows of shallow water with bottom friction

In a two-dimensional shear flow of shallow water, the bottom friction relates uniquely the spanwise profile of the depth-averaged velocity to the bottom topography. If the basic flow varies weakly in the spanwise direction, the local analysis of stability at every spanwise position gives the region of the flow parameters for which the classic hydraulic instability due to the bottom friction cannot occur. In this region, the linear analyses of the waves scattering and instability due to the lateral shear can be performed effectively by means of the frictionless linearized equations if both the bottom slope and friction are equally small.The energy of the total perturbed flow can be split into three main parts that correspond to the basic flow, small amplitude wave motion and induced mean flow. The waves can be either amplified or damped near the critical layers, where their streamwise phase velocity equals the velocity of the basic flow. Two physical mechanisms of this amplification exist. The first one is similar to that suggested by Takehiro and Hayashi for a linear frictionless shallow water flow. The incident and transmitted waves carry energy of opposite signs, which results in an increase in the amplitude of the reflected wave compared to that of the incident one. This mechanism of over-reflection operates for any combination of the flow parameters. The other mechanism is similar to Landau damping in plasma flows; it is related to the energy exchange between the waves and fluid particles at the critical layers due to the velocity synchronism. It may lead to either additional amplification or damping of the waves for different flow conditions. In particular, its significance can be reduced by stronger bottom friction. If the basic flow has uniform potential vorticity, Landau damping is negligible, and over-reflection always occurs. If the feed-back is provided by another critical layer, the net over-reflection results in the formation of trapped modes.     

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

The Newtonian method is employed to obtain nonlinear mathematical model of motion of a horizontally cantilevered and inflexible pipe conveying fluid. The order magnitudes of relevant physical parameters are analyzed qualitatively to establish a foundation on the further study of the model. The method of multiple scales is used to obtain eigenfunctions of the linear free-vibration modes of the pipe. The boundary conditions yield the characteristic equations from which eigenvalues can be derived. It is found that flow velocity in the pipe may induced the 3:1, 2:1 and 1:1 internal resonances between the first and second modes such that the mechanism of flow-induced internal resonances in the pipe under consideration is explained theoretically. The 3:1 internal resonance first occurs in the system and is, thus, the most important since it corresponds to the minimum critical velocity.     

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

The Newtonian method is employed to obtain nonlinear mathematical model of motion of a horizontally cantilevered and inflexible pipe conveying fluid. The order magnitudes of relevant physical parameters are analyzed qualitatively to establish a foundation on the further study of the model. The method of multiple scales is used to obtain eigenfunctions of the linear free-vibration modes of the pipe. The boundary conditions yield the characteristic equations from which eigenvalues can be derived. It is found that flow velocity in the pipe may induced the 3:1, 2:1 and 1:1 internal resonances between the first and second modes such that the mechanism of flow-induced internal resonances in the pipe under consideration is explained theoretically. The 3:1 internal resonance first occurs in the system and is, thus, the most important since it corresponds to the minimum critical velocity.     

温度场作用下输流悬臂碳纳米管的颤振失稳分析

以非局部弹性理论为基础,采用欧拉-伯努利梁模型,考虑碳纳米管的小尺度效应,应用哈密顿原理获得了温度场作用下的输流悬臂单层碳纳米管（SWCNT）的振动控制方程以及边界条件,依靠微分变换法（DTM法）对此高阶偏微分方程进行求解,通过数值计算研究了温度场中悬臂单层输流碳纳米管的振动与颤振失稳问题。结果表明：管内流体流速、温度场中温度变化情况与小尺度参数都会对系统振动频率以及颤振失稳临界流速产生影响。其中,小尺度效应将会降低悬臂输流系统的稳定性,使系统更为柔软;而高温场与低温场对系统动态失稳的影响不同,低温场中随温度变化值的增加,系统的稳定性提高;高温场这一作用效果恰好与之相反。     

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.     

The role of boundary conditions in the instability of one-dimensional systems

We investigate the instability properties of one-dimensional systems of finite length that can be described by a local wave equation and a set of boundary conditions. A method to quantify the respective contributions of the local instability and of wave reflections in the global instability is proposed. This allows to differentiate instabilities that emanate from wave propagation from instabilities due to wave reflections. This is illustrated on three different systems, that exhibit three different behaviors. The first one is a model system in fluid mechanics (Ginzburg–Landau equation), the second one is the fluid-conveying pipe (Bourrières equation), the third one is the fluid-conveying pipe resting on an elastic foundation (Roth equation).     

Stability analysis of viscoelastic curved pipes conveying fluid

Based on the Hamilton' s principle for elastic systems of changing mass, a differential equation of motion for viscoelastic curved pipes conveying fluid was derived using variational method, and the complex characteristic equation for the viscoelastic circular pipe conveying fluid was obtained by normalized power series method. The effects of dimensionless delay time on the variation relationship between dimensionless complex frequency of the clamped-clamped viscoelastic circular pipe conveying fluid with the Kelvin-Voigt model and dimensionless flow velocity were analyzed. For greater dimensionless delay time, the behavior of the viscoelastic pipe is that the first, second and third mode does not couple, while the pipe behaves divergent instability in the first and second order mode, then single-mode flutter takes place in the first order mode.