Cantilevered Pipe Research Articles

Nonlinear vibration control of a cantilevered fluid-conveying pipe using the idea of nonlinear energy sink

Different from the both-end supported pipe conveying fluid as a conservative system, the cantilevered fluid-transporting pipe is a non-conservative system and its dynamic behavior is more complex with flutter instabilities when the flow velocity is beyond the critical value. Indeed, controlling such a flutter system is always challenging in engineering applications. This study presents nonlinear vibrations of cantilevered pipe conveying fluid passively controlled via a nonlinear energy sink (NES). Based on the Hamilton principle, the nonlinear dynamic equations coupling with the NES are derived and discretized using high-order Galerkin method. It is indicated that increasing the mass and damping of NES results in an increase in critical flow velocity. Importantly, the optimal placed position of NES where the critical flow velocity is highest has a strong relationship with the pipe’s flutter mode. In the following, the nonlinear analysis shows the dynamic controlling effect on vibration amplitude of the pipe can be classified to three suppression regions with increasing the flow velocity. Varying the mass, damping and stiffness of NES is followed by variations of the suppression regions which are associated with controlling effects and dynamic behaviors of the pipe system.

Cantilevered pipe ejecting fluid under VIV: an investigation based on a planar nonlinear reduced-order model

This paper aims to investigate the use of the modular modeling methodology for deriving mathematical models for a class of problems hardly treated in the fluid–structure interaction literature: the combined effects of internal and external flow in a slender flexible pipe. As first attempt to address this issue, this paper proposes a planar nonlinear reduced-order model for a submerged cantilevered pipe ejecting fluid under VIV. The adopted strategy has already proved to be very effective for the corresponding problems in which only internal flow effects are considered. Within this methodology, not only subsystems might be treated individually but also, in the first approach, compatibility conditions can be relaxed and nonlinear terms can be replaced by redundant variables, once an algorithm allows to enforce constraints a posteriori. Such a modular approach proves to be particularly adequate for coupling, under the adopted hypotheses, phenomenological nonlinear wake oscillator models to the previously introduced formulation for the pipe ejecting fluid. Numerical simulations provide further discussions on the response of the nonlinear reduced-order model proposed.

Bending-torsional instability of a viscoelastic cantilevered pipe conveying pulsating fluid with an inclined terminal nozzle

In the present study, dynamic stability of a viscoelastic cantilevered pipe conveying fluid which fluctuates harmonically about a mean flow velocity is considered; while the fluid flow is exhausted through an inclined end nozzle. The Euler-Bernoulli beam theory is used to model the pipe and fluid flow effects are modelled as a distributed load along the pipe which contains the inertia, Coriolis, centrifugal and induced pulsating fluid flow forces. Moreover, the end nozzle is modelled as a follower force which couples bending vibrations with torsional ones. The extended Hamilton's principle and the Galerkin method are used to derive the bending-torsional equations of motion. The coupled equations of motion are solved using Runge-Kutta algorithm with adaptive time step and the instability boundary is determined using the Floquet theory. Numerical results present effects of some parameters such as fluid flow fluctuation, bending-to-torsional rigidity ratio, nozzle inclination angle, nozzle mass and viscoelastic material on the stability margin of the system and some conclusions are drawn.

Dynamics and optimal control of an electromagnetically actuated cantilever pipe conveying fluid

This paper deals with the problem of applying electromagnetic devices of the motional type to improve the dynamic stability of a pipe conveying air. When the flow velocity reaches a critical value, the steady equilibrium position becomes unstable, and self-excited lateral vibrations arise. In contrast, electromagnetic devices of the transformer type have been demonstrated to be highly effective in the passive stabilization of such a system, as well as the active stabilization of similar non-conservative systems with a follower force. In the present paper, we apply a pair of motional devices made of a conducting plate which is attached to the pipe and moves together with it within the perpendicular magnetic field generated by the controlled electromagnets. This motion generates eddy currents in the plates and a drag force of a viscous character. In this setting, we first investigate the possibility of designing a stabilizing control within the region of the magnetic field where every passive solution results in an unstable or conservative state. For that purpose, we determine a practical condition justifying the existence of a stabilizing control for a given set of system parameters. Later we pose and solve an optimal control problem aiming at stabilizing the system with the optimal rate of decrease of the system's energy. The solution is examined by means of numerical simulations performed within the three regions of the flow velocity: low subcritical, where the Coriolis acceleration of the conveyed fluid generates the predominate damping force; high subcritical, where the inertia of the fluid begins to dominate the dynamics of the system; and low supercritical, where unstable flutter vibrations start to arise. The effectiveness of the designed optimal controller is validated by comparisons with the corresponding passive solutions.

Dynamic analysis of a cantilevered pipe conveying fluid with density variation

The fluid flow conveyed by a cantilevered pipe might be a multi-phase flow, the density of which can fluctuate with time and space. In this paper, variable density fluid is simulated by a new mathematical model that satisfies the continuity of the fluid flow. The fluid forces acting on the pipe are derived from Newton’s second law. Combined with a Bernoulli–Euler beam model, a new dynamic model for a cantilevered pipe conveying a variable density fluid is established. The numerical results of the present model can be obtained by utilizing finite difference methods and agree with the classical theoretical results. The influences of the fluctuating amplitude, wave number and initial phase angle of the fluid density on the stability and dynamics of the cantilevered pipe system are analysed in detail. It can be found that when the fluid density varies with a large amplitude, the cantilevered pipe system is prone to losing its stability, which consequently leads to flutter. A small wave number of the fluid density has a notable influence on the system’s stability. Moreover, the stability of this system is seldom affected by the initial phase angle of the fluid density. Based on the current research work, an improved stability criterion is proposed with which the stability of the present system can be more precisely determined.

Stability analysis of thin-walled spinning reinforced pipes conveying fluid in thermal environment

Abstract The divergence and flutter instabilities of the thin-walled spinning pipes reinforced by single-walled carbon nanotubes in thermal environment are investigated. The material properties of carbon nanotube-reinforced composites are assumed to be uniform distribution as well as two types of functionally graded distribution patterns. The thermal effects are also considered and the material properties of carbon nanotube-reinforced composites are assumed to be temperature-dependent. The cantilever pipe conveying fluid is spinning along its longitudinal axis and subjected to an axial force at the free end. Based on the thin-walled Timoshenko beam theory, the governing equations of motion are derived using the extended Hamilton's principle and discretized via the Galerkin method. The resulting thermal-structural-fluid eigenvalue problem is solved and the frequency and the critical fluid velocities are calculated. The effects of carbon nanotubes distributions, volume fraction of carbon nanotubes, compressive axial force, spinning speed, gravity and fluid mass ratio on the critical divergence and flutter velocities of the thin-walled spinning pipe conveying fluid are studied.

Stability analysis of composite thin-walled pipes conveying fluid

Stability analysis of a composite thin-walled cantilever pipe conveying fluid supported at free end by linear translational and rotational springs is considered in this paper. The governing equations of the system are developed by extended Hamilton's principle for open systems. Applying extended Galerkin technique, eigenvalue analysis is implemented and the critical fluid velocity and consequently stability of the system are obtained. The critical velocity and eigenvalue are validated by comparison with the results in available literature. The numerical results are presented to investigate the effects of fiber orientation angle, volume fraction of fiber, composite lay-up, size of fiber, structural damping coefficient, mass fluid ratio and elastic boundary conditions on eigenvalue and critical fluid velocity of the system. It is revealed that by increasing the value of fiber orientation angle, the critical fluid velocity of the system is decreased. Furthermore, it is found that the volume fraction of fiber and size of fiber have stabilizing effects on the dynamic behavior of the system. Moreover, it is demonstrated that by increasing the value of linear spring constant at the free end, the pipe loses stability by either divergence or flutter.

Reduced order modeling of a cantilevered pipe conveying fluid applying a modular methodology

In this paper, a modular methodology, in which constraints associated to compatibility conditions can be included a posteriori, is applied for the derivation of a non-linear reduced order model of a cantilevered pipe conveying fluid, a classical problem in the area of Fluid–Structure Interaction (FSI). The dynamic response of this model is detailed in the neighborhood of the equilibrium state, by describing the dependency of the eigenvalues of its linearized form with respect to its parameters. For two scenarios, these responses are compared to benchmark results available in the literature. Also, two numerical simulations are proposed to compare how accurately the reduced order model can reproduce the results computed by a non-linear FEM model, previously proposed by the authors for this problem. The assessment of the numerical analyses indicates good agreement in the performed comparisons, which illustrates the good quality of the non-linear reduced order model proposed and suggests that the adopted modeling approach might be successfully applied for other problems in FSI.

Insights on the point of contact analysis and characterization of constrained pipelines conveying fluid

Modern methods of nonlinear dynamics including time histories, phase portraits, power spectra, and Poincare sections are used to characterize the stability and bifurcation regions of a cantilevered pipe conveying fluid with symmetric constraints at the point of contact. In this study, efforts are made to demonstrate the importance of characterizing the system at the arbitrarily positioned symmetric constraints rather than at the tip of the cantilevered pipe. Using the full nonlinear equations of motion and the Galerkin discretization, a nonlinear analysis is performed. After validating the model with previous results using the bifurcation diagrams and achieving full agreement, the bifurcation diagram at the point of contact is further investigated to select key flow velocities of interest. In addition to demonstrating the progression of the selected regions using primarily phase portraits, a detailed comparison is made between the tip and the point of contact at the key flow velocities. In doing so, period doubling, pitchfork bifurcations, grazing bifurcations, sticking, and chaos that occur at the point of contact are found to not always occur at the tip for the same flow speed. Thus, it is shown that in the case of cantilevered pipes with constraints, more accurate characterization of the system is obtained in a specified range of flow velocities by characterizing the system at the point of contact rather than at the tip.

Non-planar responses of cantilevered pipes conveying fluid with intermediate motion constraints

In this paper, the nonlinear responses of a loosely constrained cantilevered pipe conveying fluid in the context of three-dimensional (3-D) dynamics are investigated. The pipe is allowed to oscillate in two perpendicular principal planes, and hence its 3-D motions are possible. Two types of motion constraints are considered. One type of constraints is the tube support plate (TSP) which comprises a plate with drilled holes for the pipe to pass through. A second type of constraints consists of two parallel bars (TPBs). The restraining force between the pipe and motion constraints is modeled by a smoothened-trilinear spring. In the theoretical analysis, the 3-D version of nonlinear equations is discretized via Galerkin’s method, and the resulting set of equations is solved using a fourth-order Runge–Kutta integration algorithm. The dynamical behaviors of the pipe system for varying flow velocities are presented in the form of bifurcation diagrams, time traces, power spectra diagrams and phase plots. Results show that both types of motion constraints have a significant influence on the dynamic responses of the cantilevered pipe. Compared to previous work dealing with the loosely constrained pipe with motions restricted to a plane, both planar and non-planar oscillations are explored in this 3-D version of pipe system. With increasing flow velocity, it is shown that both periodic and quasi-periodic motions can occur in the system of a cantilever with TPBs constraints. For a cantilevered pipe with TSP constraints, periodic, chaotic, quasi-periodic and sticking behaviors are detected. Of particular interest of this work is that quasi-periodic motions may be induced in the pipe system with either TPBs or TSP constraints, which have not been observed in the 2-D version of the same system. The results obtained in this work highlight the importance of consideration of the non-planar oscillations in cantilevered pipes subjected to loose constraints.

Dynamics of axially functionally graded cantilevered pipes conveying fluid

The linear dynamics of axially functionally graded (AFG) cantilevered pipes conveying fluid is studied, aiming at enhancing the dynamic stability of such fluid-interaction systems. Either the elastic modulus or the density of the AFG cantilevered pipe is assumed to be varied from the clamped to the free ends. The governing equation of the AFG pipe is derived first and then discretized by using the differential quadrature method (DQM). The effects of elastic modulus gradient and density gradient on the critical flow velocity for flutter instability are analyzed. It is found that, compared with uniform pipes, the decrease of density along the pipe length leads to a more stable system, while the opposite result may be obtained for the decrease of elastic modulus for small values of mass ratio. From the boundary curves of critical flow velocity versus density gradient and of critical flow velocity versus elastic modulus gradient, it is shown that the occurrence of Z-shape segments is possible when the mass ratio becomes large. Furthermore, the phenomenon of mode exchange has also been detected with increasing density gradient or elastic modulus gradient within a certain range of mass ratio.

Nonlinear Forced Vibration of Cantilevered Pipes Conveying Fluid

The nonlinear forced vibrations of a cantilevered pipe conveying fluid under base excitations are explored by means of the full nonlinear equation of motion, and the fourth-order Runge–Kutta integration algorithm is used as a numerical tool to solve the discretized equations. The self-excited vibration is briefly discussed first, focusing on the effect of flow velocity on the stability and post-flutter dynamical behavior of the pipe system with parameters close to those in previous experiments. Then, the nonlinear forced vibrations are examined using several concrete examples by means of frequency response diagrams and phase-plane plots. It shows that, at low flow velocity, the resonant amplitude near the first-mode natural frequency is larger than its counterpart near the second-mode natural frequency. The second-mode frequency response curve clearly displays a softening-type behavior with hysteresis phenomenon, while the first-mode frequency response curve almost maintains its neutrality. At moderate flow velocity, interestingly, the first-mode resonance response diminishes and the hysteresis phenomenon of the second-mode response disappears. At high flow velocity beyond the flutter threshold, the frequency response curve would exhibit a quenching-like behavior. When the excitation frequency is increased through the quenching point, the response of the pipe may shift from quasiperiodic to periodic. The results obtained in the present work highlight the dramatic influence of internal fluid flow on the nonlinear forced vibrations of slender pipes.

Nonlinear dynamics analysis of pipe conveying fluid by Riccati absolute nodal coordinate transfer matrix method

Based on the absolute nodal coordinate (ANC) formulation and transfer matrix method (TMM), a novel efficient Riccati ANC–TMM is applied to nonlinear dynamics analysis of pipe conveying fluid with large deformations. By deducing the new transfer equations of pipe elements in the ANC frame and introducing Riccati transform technology into the dynamics solution algorithm, the nonlinear dynamics of pipe conveying fluid can be calculated highly efficient and stable. An illustrative comparison simulation of a cantilever pipe conveying fluid by using this method and ordinary ANC method is presented, and its critical fluid velocity and other nonlinear dynamics behavior are analyzed. Compared with ordinary ANC method, this method avoids the higher-order global differential or differential-algebra equations of the system, therefore, its computational efficiency could be improved essentially. It may be further extended to study nonlinear dynamic characteristics of a pipe conveying fluid under the condition of arbitrarily large overall motions and commonly elastic supports easily.

Modular methodology applied to the nonlinear modeling of a pipe conveying fluid

This paper proposes an extension of a modular modeling approach, originally developed for lumped parameter systems, to the derivation of FEM-discretized equations of motion of one-dimensional distributed parameter systems. This methodology is characterized by the use of a recursive algorithm based on projection operators that allows any constraint condition to be enforced a posteriori. This leads to a modular approach in which a system can be conceived as the top member of a hierarchy in which the increase of complexity from one level to the parent one is associated to the enforcement of constraints. For lumped parameter systems this allows the implementation of modeling procedures starting from already known mathematical models of subsystems. In the case of distributed parameter systems, such a novel methodology not only allows to explore subsystem-based modeling strategies, but also makes it possible to propose formulations in which compatibility and boundary conditions can be enforced a posteriori. The benchmark chosen to explore these further possibilities is the classical problem of a cantilevered pipe conveying fluid. Taking a pipe made of a linear-elastic material, allowing geometric nonlinearities and assuming an internal plug-flow, a Hamiltonian derivation of FEM-discretized equations of motion is performed according to this novel approach. Numerical simulations are carried out to address the nonlinear model obtained.

Natural frequencies, modes and critical velocities of top tensioned cantilever pipes conveying pressurized steady two-phase flow under thermal loading

Natural frequencies, modes and critical velocities of top tensioned cantilever pipes conveying pressurized steady two-phase flow under thermal loading

The Analysis of Nonlinear Vibrations of Top-Tensioned Cantilever Pipes Conveying Pressurized Steady Two-Phase Flow under Thermal Loading

This paper studied the nonlinear vibrations of top-tensioned cantilevered pipes conveying pressurized steady two-phase flow under thermal loading. The coupled axial and transverse governing partial differential equations of motion of the system were derived based on Hamilton’s mechanics, with the centerline assumed to be extensible. Using the multiple-scale perturbation technique, natural frequencies, mode shapes, and first order approximate solutions of the steady-state response of the pipes were obtained. The multiple-scale assessment reveals that at some frequencies the system is uncoupled, while at some frequencies a 1:2 coupling exists between the axial and the transverse frequencies of the pipe. Nonlinear frequencies versus the amplitude displacement of the cantilever pipe, conveying two-phase flow at super-critical mixture velocity for the uncoupled scenario, exhibit a nonlinear hardening behavior; an increment in the void fractions of the two-phase flow results in a reduction in the pipe’s transverse vibration frequencies and the coupled amplitude of the system. However, increases in the temperature difference, pressure, and the presence of top tension were observed to increase the pipe’s transverse vibration frequencies without a significant change in the coupled amplitude of the system.

Three-dimensional nonlinear dynamics of a cantilevered pipe conveying fluid subjected to loose constraints

The dynamics of pipes conveying fluid has become a hot topic in the research field of fluid-structure interactions. Perhaps one of the main reasons why the dynamics of pipes conveying fluid has remained of intense interest to dynamicists is the fact that it displays interesting and sometimes unexpected nonlinear dynamical behavior and it has become a handy tool in developing or testing modern dynamics theory. In physical terms, the dynamical system of a loosely supported pipe conveying fluid is an example of large class of problems involving self-excited oscillations and interactions with loose constraints. Hence, understanding and modeling the dynamics of such systems is of both fundamental and practical interests. The two-dimensional (2-D) vibration of a cantilevered pipe conveying fluid has been studied for a long time, but for the moment, only few studies discussed the three-dimensional (3-D) planar and non-planar dynamics of cantilevered pipes subjected to nonlinear loose constraints. This paper establishes the 3-D governing equations and explores the 3-D dynamics of a cantilevered pipe with loose constraints somewhere along its length, with consideration of the friction effect during impacting between the pipe and the loose constraints. The loose constraints consist of two parallel bars (TPBs) on both sides of the pipe in one fixed lateral direction, with free gaps between the pipe and the restraining bars. The impacting force between the pipe and loose constraints is depicted by a smoothened-trilinear spring. In the theoretical analysis, the two governing equations were discretized via Galerkin′s approach and solved using a fourth-order Runge-Kutta method. As the flow velocity becomes sufficiently high, flutter instability of the pipe occurs and limit cycle motions would be generated. When the lateral displacement of the pipe exceeds the free gap, effective impacting occurs. During impacting between the pipe and loose constraints, either static or dynamic friction forces would occur along the other lateral direction. The nonlinear behavior in two perpendicular planes and the influence of friction coefficient were analyzed with special attention. The dynamic responses of the pipe system for various internal flow velocities are exhibited in the form of bifurcation diagrams, time traces and phase plots. Comparisons of the planar and non-planar motions of the pipe with or without loose constraints are conducted for various internal flow velocity ranges. Results show that the constrained pipe is capable of displaying interesting dynamics in the presence of nonlinear impacting force induced by the loose constraints. Both 3-D periodic and quasi-periodic oscillations are observed in a wide range of internal flow velocities. It is found that the introducing of nonlinear impacting constraints slightly enlarges the internal flow velocity range for non-planar motions. It is also shown that the orientation of 2-D planar vibrations of the pipe may be changed with the increase of the friction coefficient. The results obtained in this work may be useful for further understanding other problems in fluid-structure interactions involving slender structures and axial flows, such as the dynamics of slender cylinders in axial flow and deep water risers concurrently subjected to axial and cross flows.

Nonlinear dynamics of cantilevered pipes conveying fluid: Towards a further understanding of the effect of loose constraints

The nonlinear dynamics of a fluid-conveying cantilevered pipe with loose constraints placed somewhere along its length is investigated. The main objective of this study is to determine the effects of several geometrical and physical parameters of the loose constraints on the characteristics and behavior of pipes conveying fluid. Based on the full nonlinear equation of motion, the dynamical behavior of the pipe system is investigated. Phase portraits and bifurcation diagrams are constructed for a selected set of system parameters. Typical results are firstly compared to numerical ones reported previously and excellent agreement is obtained. Then, the threshold flow velocities for several key bifurcations including pitchfork, period doubling, chaos, and sticking behaviors are predicted, showing that in many cases, the gap size, stiffness, and asymmetry of the loose constraints have remarkable effects on the nonlinear responses of the cantilevered pipe conveying fluid. For a pipe system with small/large constraint gap sizes, small constraint stiffness, or large constraint offset, some of the complex dynamical behaviors including chaos and period-doubling bifurcations would disappear, at least in the flow velocity range of interest.

Dynamical modeling and multi-pulse chaotic dynamics of cantilevered pipe conveying pulsating fluid in parametric resonance

The dynamical modeling, multi-pulse orbits and chaotic dynamics of cantilevered pipe conveying pulsating fluid with harmonic external force are studied using the energy-phase method for the first time. The nonlinear geometric deformation of the pipe and the Kelvin constitutive relation of the pipe material are considered. The nonlinear governing equation of motion for cantilevered pipe conveying pulsating fluid is determined using Hamilton principle. The four-dimensional averaged equation in the case of primary parametric resonance-1/2 subharmonic resonance and 1:2 internal resonance is obtained by directly using the method of multiple scales and the Galerkin approach to the partial differential governing equation of motion for cantilevered pipe conveying pulsating fluid. From the averaged equation, the normal form theory is used to find the explicit formulas of normal form. Based on normal form obtained here, the energy-phase method is utilized to analyze the multi-pulse global bifurcations and chaotic dynamics of cantilevered pipe conveying pulsating fluid. The analysis of global dynamics indicates that the multi-pulse jumping orbits exist in the perturbed phase space of the averaged equation. Based on the averaged equation, the chaotic motions and the multi-pulse orbits for the cantilevered pipe are found by using numerical simulations. The results described above indicate the existence of chaos in the Smale horseshoe sense for cantilevered pipe conveying pulsating fluid. It has been concluded that it is dangerous to induce the chaotic vibrations of the pipes conveying pulsating fluid because the amplitudes of the chaotic vibrations are larger than those of the periodic motions.

Internal flow effect on the cross-flow vortex-induced vibration of a cantilevered pipe discharging fluid

Motivated by the application of ocean CCS (Carbon dioxide Capture and Storage) proposal and the fact that a cantilevered pipe can gain or lose energy due to the discharging flow, this study devotes efforts to investigate the IFE (Internal Flow Effect) on the CF (Cross-Flow) VIV (Vortex-Induced Vibration) of a cantilevered pipe discharging fluid. In this study, a simple fluid-structure interaction model is proposed which employs a linear structural equation to describe the transverse vibrations and utilizes a distributed wake oscillator to create the VIV effect. The simulations are justified by comparing with experiments for VIV effect and the non-conservative IFE of flexible risers. Then simulations of a flexible riser are performed by varying incoming currents and internal flow rates. The IFE on VIVs are examined by the space-time modifications of riser responses and dominant vibration frequency for which the mode switching and sharing can be identified. When the internal flow velocity is small, the pipe is losing energy to the inner flow and the VIVs can be depressed significantly. However, the pipe would lose the stability when the internal flow exceeds a critical value. The critical internal flow velocity depends on the current velocity and VIV dominant mode.