Cantilevered Pipe Research Articles

Dynamic Stability of Periodic Pipes Conveying Fluid

In this paper, the stability of a periodic cantilevered pipe conveying fluid is studied theoretically by means of a novel transfer matrix method. This method is first validated by comparing the results to those available in the literature for a uniform pipe, showing that it is capable of high accuracy and displaying good convergence characteristics. Then, the stability of periodic pipes is investigated, with geometric, material-properties periodicity, and a combination of the two, showing that a considerable stabilizing effect may be achieved over different ranges of the mass parameter β (β=mf/(mf+mp), where mf and mp are the fluid and pipe masses per unit length). The effect of other different system parameters is also probed.

Nonlinear dynamics of cantilevered extensible pipes conveying fluid

In this paper the nonlinear planar dynamics of a fluid-conveying cantilevered pipe is investigated. The centreline of the pipe is considered to be extensible; i.e., coupled longitudinal and transverse displacements are considered. The extended version of the Lagrange equations for systems containing non-material volumes is employed to derive the equations of motion, resulting directly in a set of coupled nonlinear ordinary differential equations. The pseudo-arclength continuation technique along with direct time integration are used to solve these equations. Bifurcation diagrams of the system are constructed as the flow velocity is increased; these diagrams are supplemented by time traces, phase-plane portraits, and fast Fourier transforms for some sets of system parameters. As opposed to the case of an inextensible pipe, an extensible pipe elongates in the axial direction as the flow velocity is increased from zero; depending on the system parameters, this static elongation can be considerable. At the critical flow velocity, the system loses stability via a supercritical Hopf bifurcation, emerging from the trivial solution for the transverse displacement and leading to a flutter.

유체유동에 의한 유연한 파이프의 진동 저감을 위한 파이프 지지용 MR댐퍼의 설계 및 성능 평가

This paper presents design and performance evaluation of MR damper for the reducing vibration of a flexible pipe conveying fluid. A novel type of MR damper which is suitable for pipe vibration characteristics is proposed and the MR damper is mathematically modeled and its damping force characteristics are evaluated. The vibration control performance of the MR damper associated with The cantilever pipe system is evaluated.

Bending–torsional flutter of a cantilevered pipe conveying fluid with an inclined terminal nozzle

Stability analysis of a horizontal cantilevered pipe conveying fluid with an inclined terminal nozzle is considered in this paper. The pipe is modelled as a cantilevered Euler–Bernoulli beam, and the flow-induced inertia, Coriolis and centrifugal forces along the pipe as well as the follower force induced by the jet-flow are taken into account. The governing equations of the coupled bending–torsional vibrations of the pipe are obtained using extended Hamilton's principle and are then discretized via the Galerkin method. The resulting eigenvalue problem is then solved, and several cases are examined to determine the effect of nozzle inclination angle, nozzle aspect ratio, mass ratio and bending-to-torsional rigidity ratio on flutter speed of the system.

VIV Analysis of a Cantilevered Pipe Discharging Fluid with a Nozzle in the Sea

Vortex induced vibration (VIV) should be taken into account for the vibration analysis of a cantilevered marine riser. This paper is dedicated to the VIV for the nonlinear vibration analysis of a cantilevered pipe discharging fluid with a nozzle at sea. In this study, three dimensional (3-D) motion equations are derived based on modified Hamilton Principle in a rigorous way. The Morison formula is used to calculate the external fluid-induced static drag force and viscous damping forces in the in-line (IL) and cross-flow (CF) directions. A double-van der Pol oscillator model is adopted employing a set of calibrated wake coefficients in Facchinetti et al. (2004) for the VIV effect. A new hydrodynamic force model is proposed considering the coupling effect of IL and CF motions. The mathematical model together with proposed solution methods are validated after comparisons with experimental results.

The dynamics of a cantilevered pipe aspirating fluid studied by experimental, numerical and analytical methods

This paper investigates the dynamics of a slender, flexible, aspirating cantilevered pipe, ingesting fluid at its free end and conveying it towards its clamped end. The problem is interesting not only from a fundamental perspective, but also because applications exist, notably in ocean mining. First, the need for the present work is demonstrated through a review of previous research into the topic – spanning many years and yielding often contradictory results – most recently suggesting that the system loses stability by flutter at relatively low flow velocities. In the present paper, that conclusion is refined and expanded upon by exploring the problem in three ways: experimentally, numerically and analytically. First, air-flow experiments were conducted using different elastomer pipes and intake shapes, in which the flow velocity of the fluid was varied and the frequency and amplitude of oscillation of the pipe were measured. Second, a fully coupled Computational Fluid Dynamics (CFD) and Computational Structural Mechanics (CSM) model was developed in ANSYS™ in order to simulate experiments and corroborate experimental results. Finally, using a Newtonian analytical approach, a new linear equation of motion describing the system was derived, and then solved via the Galerkin method in order to determine its stability characteristics. Heavily influenced by the CFD analysis, the proposed analytical model is different from previous ones, most notably because of the inclusion of a two-part fluid depressurisation at the intake. In general, both the actual and numerical experiments suggest a first-mode loss of stability by flutter at flow velocities comparable to those for the discharging case, which agrees with the results from the new analytical model.

Study on Adaptive Control of Restrained Cantilever Pipe Conveying Fluid

The cantilever fluid conveying pipe which restrained by linear spring and cubic nonlinear spring at its free end was studied. Firstly, the model reference adaptive control system with parallel structural style was designed for the restrained cantilever pipe based on Lyapunov theory of stability. Then the vibration of the pipe system was under positive controlled making use of this adaptive control scheme. Lastly, the numerical simulating method was used to verify the effectivity of this control method and the influence of system parameters on control performance and control input energy were also investigated. The simulation results showed that the control effect of the control system was extremely distinct. In addition, the larger the Lyapunov function state feedback declinational gains were, the better the control effect had, and the shorter the required time that the controlled system entirely settled down needed. With the position of piezoelectric actuator moved to the pipe free end the required time that the controller entirely settled down would become longer and longer. Moreover, the effect of the length of piezoelectric actuator on the controlled system was relatively complicated, the required time that fluid conveying pipe system settled down and the control energy took on complicated variation regularities.

Out-of-Plane Pipe Whip for a Bent Cantilever Pipe: Comparison Between Experiment and FEM Models

The three-dimensional, dynamic, elastic-plastic response of a right-angle bent cantilever pipe, with an initially uniform, circular cross section, subjected to out-of-plane loading is examined using finite element beam and shell models in ABAQUS. The large-deflection behavior involves both bending and torsional elastoplastic deformations of the pipe, phenomena which have not been previously studied in the context of the dynamic problem of pipe whip. Initially, neglecting ovalization and local collapse (kinking), the bent pipe is modeled as a beam, using spatial beam elements in ABAQUS. This enables the basic three-dimensional kinematic behavior of the pipe to be analyzed. A similar, but potentially more accurate, analysis was then performed using shell elements. It is shown that there is no significant difference in the global dynamic plastic response. However the ovalization of the pipe cross section and formation and movement of the plastic zones (hinges) can be captured by using shell elements. This provides data which could form the basis for examining local failures in the pipe run. Previously unpublished experimental results, obtained in an earlier study by some of the present authors, are compared with the simulated results. Good agreement is observed and it is concluded that a nonlinear dynamic model using finite elements provides a rigorous approach for estimating the hazard zone (HZ) and, also, for treating the kinematics of a whipping pipe for this complex three-dimensional situation.

Fatigue Strength Simulation of Boom System Structure of Pump Truck

The motion equation of boom system of pump truck carrying concrete flow is set up based on the coupled vibration theory of cantilevered pipe and the finite element theory. Dynamic stress history of boom system with parameters of dip angle α=0°is simulated combining MATLAB simulation analyses with ANSYS finite element model. The feasibility of this method is verified through field experiment. Stress histories are simulated by adjusting parameter of dip angle as follows: α=20°,52°and 80°.Great stress points are painted on the Goodman Fatigue Limit Diagram measured by experiment. And then fatigue strength assessment of boom system is done. The research method can be used for the design calculation of similar products.

Dynamic modeling of mass-flowing linear medium with large amplitude displacement and rotation

In this paper, a dynamic model of a linear medium with mass flow, such as traveling strings, cables, belts, beams or pipes conveying fluids, is proposed, in the framework of Arbitrary-Lagrange–Euler (ALE) description. The material coordinate is introduced to characterize the mass-flow of the medium, and the Absolute Nodal Coordinate Formulation (ANCF) is employed to capture geometric nonlinearity of the linear media under large displacement and rotation. The governing equations are derived in terms of d'Alembert's principle. When using an ALE description, complex mass-flowing boundary conditions can be easily enforced. Numerical examples are presented to validate the proposed method by comparison with analytical results of simplified models. The computed critical fluid velocity for the stability of a cantilevered pipe conveying fluid is correlated with the available theory in literature. The large amplitude limit-cycle oscillations of flexible pipes conveying fluid are presented, and the effect of the velocity of the fluid on the static equilibrium of the pipe under gravity is investigated.

THERMOMECHANICAL STABILITY ANALYSIS OF FUNCTIONALLY GRADED THIN-WALLED CANTILEVER PIPE WITH FLOWING FLUID SUBJECTED TO AXIAL LOAD

In this study, the thermomechanical stability of functionally graded thin-walled cantilever pipes conveying flow and loading by compressive axial force is investigated. The governing equations of motion and boundary conditions are derived via the Hamilton's variational principle. The thin-walled structure is formulated based on Rayleigh's theory. Moreover, quasi-steady flow pressure loadings and steady surface temperature are considered and the temperature gradient through the wall thickness of the pipe is included. The partial differential equations of the pipe are transformed into a set of ordinary differential equations using the extended Galerkin method. Finally, having solved the resulting thermal-structural-fluid eigenvalue system of equations, the effects of the compressive axial force, fluid speed, fluid mass ratio, volume fraction index of functionally graded materials (FGMs), and temperature change through the thickness of the pipe on the stability boundary are investigated. Numerical comparisons are also performed with the available data in the literature and good agreement is observed.

Three-dimensional dynamics of a fluid-conveying cantilevered pipe fitted with an additional spring-support and an end-mass

The three-dimensional (3-D) dynamics of a fluid-conveying cantilevered pipe fitted with an end-mass and additional intra-span spring-support is investigated in this paper, both theoretically and experimentally. The main objective is to examine how the dynamics of a cantilevered pipe with additional spring-support is modified by the presence of a small mass attached at the free end. In the theoretical study, the nonlinear three-dimensional equations of motion are discretized via Galerkin's method, and the resulting equations are solved by a finite difference method. For the cases studied, the system was found to lose stability by planar flutter; as the flow velocity is increased beyond that point, a sequence of higher order bifurcations ensue, involving 2-D and 3-D periodic and quasiperiodic motions, as well as chaotic ones. In the experiments, performed with elastomer pipes and water flow, similarly 2-D and 3-D periodic, quasiperiodic and chaotic oscillations were observed. Theory and experiments have been shown to be in good qualitative and quantitative agreement. The experimental behaviour is illustrated by video clips (electronic annexes). Moreover, the effects of (i) small stiffness imperfections and (ii) excitation by a point-force are explored theoretically in a preliminary way.

Analytical modelling of a pipe with flowing medium subjected to an impulse load

Pipes transporting gas or liquid can be subjected to impulse loading as a result of an accident. In many previous studies the effects of the flowing media on the transient response of pipes were largely ignored or considered to be negligible. This paper aims to shed light on the significance of these effects. First, we present a simplified mathematical model of a pipe filled with a flowing liquid or gas and subjected to dynamic loading. This model is based on the well-known Bernoulli–Euler beam theory, but also incorporates forces generated by the internal flow which follows the pipe lateral deflections and generates the inertia, centrifugal and Coriolis flow accelerations. Further, we develop asymptotic analytical and numerical methods for the analysis of the governing equation. Finally, as example, we investigate in detail the effect of the flow parameters on the dynamic response of a cantilever pipe subjected to impulse loading. Special attention is given to the pipe whip phenomenon, which causes growing unlimited displacements of the pipe regardless of how small the value of the applied loading is. The developed theoretical methods provide a framework for analysis of many other dynamic problems of pipes with flowing media subjected to arbitrary boundary and loading conditions.

EFFECTS OF TIP MASS ON STABILITY OF ROTATING CANTILEVER PIPE CONVEYING FLUID WITH CRACK

In this paper, the dynamic stability of a rotating cantilever pipe conveying fluid with a crack and tip mass is investigated by numerical method. That is, the effects of the rotating the rotating angular velocity, the mass ratio, the crack and tip mass on the critical flow velocity for flutter instability of system are studied. The equations of motion of rotating pipe are derived by using the extended Hamilton's principle. The crack section of pipe is represented by a local flexibility matrix connecting two undamaged pipe segments. The crack is assumed to be in the first mode of fracture and always opened during the vibrations. Finally, the stability maps of the cracked rotating pipe system as a rotating angular velocity and mass ratio β are presented.

Dynamics of pipes conveying fluid with non-uniform turbulent and laminar velocity profiles

Previous analytical work on stability of fluid-conveying pipes assumed a uniform velocity profile for the conveyed fluid. In real fluid flows, the presence of viscosity leads to a sheared region near the wall. Earlier studies correctly note that viscous forces do not affect the dynamics of the system since these forces are balanced by pressure drop in the conveyed fluid. Although viscous shear has not been ignored in these studies, a uniform velocity profile assumes that the sheared region is infinitely thin. Prior analysis was extended to account for a fully developed non-uniform profile such as would be encountered in real fluid flows. A modified, highly tractable equation of motion was derived, which includes a single additional parameter to account for the true momentum of the fluid. This empirical parameter was determined by numerical analysis over the Reynolds number range of interest. The stability of cantilever pipes conveying fluid with two types of non-uniform velocity profile was assessed. In the first case, the profile was a function of Reynolds number and transition to turbulence occurred before the onset of flutter instability. This case had stability properties similar to the uniform velocity case except in specific narrow regions of the parameter space. The second case required that the Reynolds number be such that the flow was always laminar. For this case, lower fluid velocity was required to achieve instability, and the oscillation frequency at instability was considerably lower over much of the parameter space, compared to the uniform case.

Dynamics of a cantilevered pipe discharging fluid, fitted with a stabilizing end-piece

In this communication, the dynamics of a flexible cantilevered pipe fitted with a special end-piece is considered, both theoretically and experimentally. This end-piece can be configured in two ways: (i) with the flow going straight through, unimpeded, and emerging at the free end as a jet, and (ii) with the straight-through path blocked, so that the flow is discharged radially from a number of holes perpendicular to the pipe. The dynamics in the first case is similar to that of a pipe with no end-piece: the system loses stability by flutter via a Hopf bifurcation, though the dynamics becomes more complex at higher flow velocities. The dynamics in the second case is entirely different: the system remains stable over the full range of flow velocities considered. This study provides insight into the mechanism of flutter of cantilevered pipes conveying fluid and the key role played by the mathematically obvious but physically counter-intuitive compressive follower force generated by the straight-through discharging jet.

Three-dimensional dynamics of a cantilevered pipe conveying fluid, additionally supported by an intermediate spring array

In this paper, the three-dimensional (3-D) non-linear dynamics of a cantilevered pipe conveying fluid, constrained by arrays of four springs attached at a point along its length is investigated. In the theoretical analysis, the 3-D equations are discretized via Galerkin's technique. The resulting coupled non-linear differential equations are solved numerically using a finite difference method. The dynamic behaviour of the system is presented in the form of bifurcation diagrams, along with phase-plane plots, time-histories, PSD plots, and Poincaré maps for five different spring configurations. Interesting dynamical phenomena, such as 2-D or 3-D flutter, divergence, quasiperiodic and chaotic motions, have been observed with increasing flow velocity. Experiments were performed for the cases studied theoretically, and good qualitative and quantitative agreement was observed. The experimental behaviour is illustrated by video clips (electronic annexes). The effect of the number of beam modes in the Galerkin discretization on accuracy of the results and on convergence of the numerical solutions is discussed.

The Flutter Response of a Piezoelectrically Damped Cantilever Pipe

The investigation of passively damped piezoelectric structures within fluid flows is important for two reasons: (a) to increase the critical flutter speed and (b) conversely to generate electrical energy to power small scale electronic systems. In this article, a cantilever pipe with resistive piezoelectric damping is chosen as the model structure to demonstrate behavior over a range of fluid/structure mass ratios. The effects of piezoelectric coupling on the critical flutter velocity, capacitance, load resistance, and piezoelectric location are investigated. The modeling shows that depending on the piezoelectric parameters chosen and the attached electrical load resistance, the addition of a passive piezoelectric element can either increase or decrease stability of the system (i.e., the critical flutter speed of the cantilever pipe can be altered and hence controlled).

유동유체에 의한 파이프의 동적안정성에 미치는 외부감쇠와 말단질량의 영향

The paper presents the influences of the external damping and the tip mass on dynamic stability of a vertical cantilevered pipe conveying fluid. In general, real pipe systems may have some valves and attached mechanical parts, which can be regarded as attached lumped masses and support-dampers. The support-dampers can be assumed as viscous dampers. The equations of motion are derived by energy expressions using extended Hamilton's principle, and some numerical results using Galerkin's method are presented. Critical flow velocities and stability maps of the pipe with external dampers and tip mass are obtained for various tip mass ratios, external damping coefficients and positions of the viscous dampers.

A Large Deformation Planar Finite Element for Pipes Conveying Fluid Based on the Absolute Nodal Coordinate Formulation

A novel planar pipe finite element conveying fluid with steady flow, suitable for modeling large deformations in the framework of the Bernoulli–Euler beam theory, is presented. The element is based on a third order planar beam finite element, introduced by Berzeri and Shabana (2000, “Development of Simple Models for the Elastic Forces in the Absolute Nodal Co-Ordinate Formulation,” J. Sound Vib., 235(4), pp. 539–565), applying the absolute nodal coordinate formulation. The equations of motion of the pipe finite element are derived using an extended version of Lagrange’s equations of the second kind taking into account the flow of fluid; in contrast, most derivations in the literature are based on Hamilton’s principle or the Newtonian approaches. The advantage of this element in comparison to classical large deformation beam elements, which are based on rotations, is the direct interpolation of position and directional derivatives, which simplifies the equations of motion considerably. As an advantage, Lagrange’s equations of the second kind offer a convenient connection for introducing fluids into multibody dynamic systems. Standard numerical examples show the convergence of the deformation for increasing number of elements. For a cantilever pipe, the critical flow velocities for increasing number of pipe elements are compared with existing works, based on Euler elastica beams and moving discrete masses. The results show good agreement with the reference solutions applying only a small number of pipe finite elements.