Three-dimensional oscillations of a cantilever pipe conveying fluid

The aim of the study described in this paper is to investigate the two-dimensional (2-D) and three-dimensional (3-D) flutter of cantilevered pipes conveying fluid. Specifically, by means of a complete set of non-linear equations of motion, two questions are addressed: (i) whether for a system losing stability by either 2-D or 3-D flutter the motion remains of the same type as the flow velocity is increased substantially beyond the Hopf bifurcation precipitating the flutter; (ii) whether the bifurcational behaviour of a horizontal system and a vertical one (sufficiently long for gravity to have an important effect on the dynamics) are substantially similar. Stability maps and tables are used to delineate areas in a flow velocity versus mass parameter plane where 2-D or 3-D motions occur, and limit-cycle motions are illustrated by phase-plane plots, PSDs and cross-sectional diagrams showing whether the motion is circular (3-D) or planar (2-D).     

Dynamics of a pipe conveying fluid flexibly restrained at the ends

The subject of this paper is the study of dynamics and stability of a pipe flexibly supported at its ends and conveying fluid. First, the equation of motion of the system is derived via the extended form of Hamilton׳s principle for open systems. In the derivation, the effect of flexible supports, modelled as linear translational and rotational springs, is appropriately considered in the equation of motion rather than in the boundary conditions. The resulting equation of motion is then discretized via the Galerkin method in which the eigenfunctions of a free-free Euler–Bernoulli beam are utilized. Thus, a general set of second-order ordinary differential equations emerges, in which, by setting the stiffness of the end-springs suitably, one can readily investigate the dynamics of various systems with either classical or non-classical boundary conditions. Several numerical analyses are initially performed, in which the eigenvalues of a simplified system (a beam) with flexible end-supports are obtained and then compared with numerical results, so as to verify the equation of motion, in its simplified form. Then, the dynamics of a pinned-free pipe conveying fluid is systematically investigated, in which it is found that a pinned-free pipe conveying fluid is generally neutrally stable until it becomes unstable via a Hopf bifurcation leading to flutter. The next part of the paper is devoted to studying the dynamics of a pinned-free pipe additionally constrained at the pinned end by a rotational spring. A wide range of dynamical behaviour is seen as the mass ratio of the system (mass of the fluid to the fluid+pipe mass) varies. It is surprising to see that for a range of rotational spring stiffness, an increase in the stiffness makes the pipe less stable. Finally, a pipe conveying fluid supported only by a translational and a rotational spring at the upstream end is considered. For this system, the critical flow velocity is determined for various values of spring constants, and several Argand diagrams along with modal shapes of the unstable modes are presented. The dynamics of this system is found to be very complex and often unpredictable (unexpected).     

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。     

In-plane and out-of-plane dynamics of a curved pipe conveying pulsating fluid

This paper investigates the in-plane and out-of-plane dynamics of a curved pipe conveying fluid. Considering the extensibility, von Karman nonlinearity, and pulsating flow, the governing equations are derived by the Newtonian method. First, according to the modified inextensible theory, only the out-of-plane vibration is investigated based on a Galerkin method for discretizing the partial differential equations. The instability regions of combination parametric resonance and principal parametric resonance are determined by using the method of multiple scales (MMS). Parametric studies are also performed. Then the differential quadrature method (DQM) is adopted to discretize the complete pipe model and the nonlinear dynamic equations are carried out numerically with a fourth-order Runge–Kutta technique. The nonlinear dynamic responses are presented to validate the out-of-plane instability analysis and to demonstrate the influence of von Karman geometric nonlinearity. Further, some numerical results obtained in this work are compared with previous experimental results, showing the validity of the theoretical model developed in this paper.     

Bifurcations of a cantilevered pipe conveying steady fluid with a terminal nozzle

This paper studies interactions of pipe and fluid and deals with bifurcations of a cantilevered pipe conveying a steady fluid, clamped at one end and having a nozzle subjected to nonlinear constraints at the free end. Either the nozzle parameter or the flow velocity is taken as a variable parameter. The discrete equations of the system are obtained by the Ritz-Galerkin method. The static stability is studied by the Routh criteria. The method of averaging is employed to examine the analytical results and the chaotic motions. Three critical values are given. The first one makes the system lose the static stability by pitchfork bifurcation. The second one makes the system lose the dynamical stability by Hopf bifurcation. The third one makes the periodic motions of the system lose the stability by doubling-period bifurcation. The project supported by the Science Foundation of Tongji University and Tongji University and National Key Projects of China under Grant No. PD9521907.     

A cantilever conveying fluid: coherent modes versus beam modes

A semi-analytical approach to obtain the proper orthogonal modes is described for the non-linear oscillation of a cantilevered pipe conveying fluid. Theoretically, while the spatial coherent structures are the eigenfunctions of the time-averaged spatial autocorrelation functions, it emerges that once the Galerkin projection of the proper orthogonal modes is carried out using the uniform cantilever-beam modes, the spatial dependency of the integral eigenvalue problem can be eliminated by analytical manipulation which avoids any spatial discretization error. As the solution of the integral equation is obtained semi-analytically by linearly projecting the proper orthogonal modes on the cantilever-beam modes, any linear or non-linear operation can be carried out analytically on the proper orthogonal modes. Furthermore, the reduced-order eigenvalue problem minimizes the numerical pollution which leads to spurious eigenvectors, as may arise in the case of a large-scale eigenvalue problem (without the Galerkin projection of the eigenvectors on the cantilever-beam modes). This methodology can conveniently be used to study the convergence of the numerically calculated proper orthogonal modes obtained from the full-scale eigenvalue problem.     

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.     

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.     

Research on solid-liquid coupling dynamics of pipe conveying fluid

I.IntroductionSolid-fluidcouplingvibrationproblemofpipesconveyingfluidarepresencegenerallyinthedomainofastronomic,energysources,chemicalindustryetc..Notonlytheoreticallytheproblemhaswideresearchvalue,butpracticallytheproblemhaswideengineeringbackground.Therefore,itisimportantreseachproblemihsciencedomainspang.Thefirstrightequationofsolid-liquidcouplingvibrationofpipeconveyingfluidwaspiovidedbyG.W.Housner,andV.Y.Feodosievil'2].Thebasicfrequencycharacteristicofpipesconveyingfluidwasstudiedre…     

Random uncertainty modeling and vibration analysis of a straight pipe conveying fluid

A new procedure on random uncertainty modeling is presented for vibration analysis of a straight pipe conveying fluid when the pipe is fixed at both ends. Taking real conveying condition into account, several randomly uncertain loads and a motion constraint are imposed on the pipe and its corresponding equations of motion, which are established from the Euler–Bernoulli beam theory and the nonlinear Lagrange strain theory previously. Based on the stochastically nonlinear dynamic theory and the Galerkin method, the equations of motion are reduced to the finite discretized ones with randomly uncertain excitations, from which the vibration characteristics of the pipe are investigated in more detail by some previously developed numerical methods and a specific Poincaré map. It is shown that, the vibration modes change not only with the frequency of the harmonic excitation but also with the strength and spectrum width of the randomly uncertain excitations, quasi-periodic-dominant responses can be observed clearly from the point sets in the Poincaré’s cross-section. Moreover, the nonlinear elastic coefficient and location of the motion constraint can be adjusted properly to reduce the transverse vibration amplitude of the pipe.     

Dynamic stability of a pipe conveying fluid with an uncertain computational model

This paper deals with the problem of a pipe conveying fluid of interest in several engineering applications, such as micro-systems or drill-string dynamics. The deterministic stability analysis developed by Paidoussis and Issid (1974) is extended to the case for which there are model uncertainties induced by modeling errors in the computational model. The aim of this work is twofold: (1) to propose a probabilistic model for the fluid–structure interaction considering modeling errors and (2) to analyze the stability and reliability of the stochastic system. The Euler–Bernoulli beam model is used to model the pipe and the plug flow model is used to take into account the internal flow in the pipe. The resulting differential equation is discretized by means of the finite element method and a reduced-order model is constructed from some eigenmodes of the beam. A probabilistic approach is used to model uncertainties in the fluid–structure interaction. The proposed strategy takes into account global uncertainties related to the noninertial coupled fluid forces (related to damping and stiffness). The resulting random eigenvalue problem is used to analyze flutter and divergence unstable modes of the system for different values of the dimensionless flow speed. The numerical results show the random response of the system for different levels of uncertainty, and the reliability of the system for different dimensionless speeds and levels of uncertainty.     

Dynamics of a long tubular cantilever conveying fluid downwards,which then flows upwards around the cantilever as a confined annular flow

A theoretical model is developed for the dynamics of a hanging tubular cantilever conveying fluid downwards; the fluid, after exiting from the free end, is pushed upwards in the outer annular region contained by the cantilever and a rigid cylindrical channel. This configuration thus resembles that of a drill-string with a floating fluid-powered drill-bit. The linear equation of motion is solved by means of a hybrid Galerkin–Fourier method, as well as by a conventional Galerkin method. Calculations are conducted for a very slender system with parameters appropriate for a drill-string, for different degrees of confinement of the outer annular channel; and also for another, bench-top-size experiment. For wide annuli, the dynamics is dominated by the internal flow and, for low flow velocities, the flow increases the damping associated with the presence of the annular fluid. For narrow annuli, however, the annular flow is dominant, tending to destabilize the system, giving rise to flutter at remarkably low flow velocities. The mechanisms underlying the dynamics are also considered, in terms of energy transfer from the fluid to the cantilever and vice versa, as are possible applications of this work.     

Semi-active stabilisation of a pipe conveying fluid using eddy-current dampers: state-feedback control design,experimental validation

An application of electromagnetic devices of the motional type (i.e. eddy-current dampers) to improve the dynamic stability of a cantilever pipe discharging fluid is proposed. When the flow velocity reaches a critical value, this system loses stability through the flutter. A contactless damping device is used. This actuator is made of a conducting plate attached to the pipe that moves together with it within the perpendicular magnetic field that is generated by the controlled electromagnets. During the motion the eddy currents in the plate and a resultant drag force of a viscous character are generated. First, an optimal control problem that aims to stabilise the system with the optimal rate of decrease of the system’s energy is posed and solved. Then a state-feedback parametrization of the obtained optimal control, which can be used in a closed-loop scheme is proposed. The effectiveness of the designed optimal controller is validated by making a comparison with the corresponding passive solutions on the specially designed and constructed experimental test stand of a pipe conveying air.

     

Nonlinear vibration of slightly curved pipe with conveying pulsating fluid

The nonlinear governing motion equation of slightly curved pipe with conveying pulsating fluid is set up by Hamilton’s principle. The motion equation is discretized into a set of low dimensional system of nonlinear ordinary differential equations by the Galerkin method. Linear analysis of system is performed upon this set of equations. The effect of amplitude of initial deflection and flow velocity on linear dynamic of system is analyzed. Curves of the resonance responses about \(\varOmega \approx {\omega _\mathrm{{1}}}\) and \(\varOmega \approx \mathrm{{2}}{\omega _\mathrm{{1}}}\) are performed by means of the pseudo-arclength continuation technique. The global nonlinear dynamic of system is analyzed by establishing the bifurcation diagrams. The dynamical behaviors are identified by the phase diagram and Poincare maps. The periodic motion, chaotic motion and quasi-periodic motion are found in this system.     

Internal-external resonance of a curved pipe conveying fluid resting on a nonlinear elastic foundation

In this study, the forced vibration of a curved pipe conveying fluid resting on a nonlinear elastic foundation is considered. The governing equations for the pipe system are formed with the consideration of viscoelastic material, nonlinearity of foundation, external excitation, and extensibility of centre line. Equations governing the in-plane vibration are solved first by the Galerkin method to obtain the static in-plane equilibrium configuration. The out-of-plane vibration is simplified into a constant coefficient gyroscopic system. Subsequently, the method of multiple scales (MMS) is developed to investigate external first and second primary resonances of the out-of-plane vibration in the presence of three-to-one internal resonance between the first two modes. Modulation equations are formed to obtain the steady state solutions. A parametric study is carried out for the first primary resonance. The effects of damping, nonlinear stiffness of the foundation, internal resonance detuning parameter, and the magnitude of the external excitation are investigated through frequency response curves and force response curves. The characteristics of the single mode response and the relationship between single and two mode steady state solutions are revealed for the second primary resonance. The stability analysis is carried out for these plots. Finally, the approximately analytical results are confirmed by the numerical integrations.     

Nonlinear dynamics of a pipe conveying pulsating fluid with parametric and internal resonances

In this paper, the nonlinear planar vibration of a pipe conveying pulsatile fluid subjected to principal parametric resonance in the presence of internal resonance is investigated. The pipe is hinged to two immovable supports at both ends and conveys fluid at a velocity with a harmonically varying component over a constant mean velocity. The geometric cubic nonlinearity in the equation of motion is due to stretching effect of the pipe. The natural frequency of the second mode is approximately three times the natural frequency of the first mode for a range of mean flow velocity, resulting in a three-to-one internal resonance. The analysis is done using the method of multiple scales (MMS) by directly attacking the governing nonlinear integral-partial-differential equations and the associated boundary conditions. The resulting set of first-order ordinary differential equations governing the modulation of amplitude and phase is analyzed numerically for principal parametric resonance of first mode. Stability, bifurcation, and response behavior of the pipe are investigated. The results show new zones of instability due to the presence of internal resonance. A wide array of dynamical behavior is observed, illustrating the influence of internal resonance.