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.     

A further study on the non-linear dynamics of simply supported pipes conveying pulsating fluid

In this paper, the non-linear dynamics of simply supported pipes conveying pulsating fluid is further investigated, by considering the effect of motion constraints modeled as cubic springs. The partial differential equation, after transformed into a set of ordinary differential equations (ODEs) using the Galerkin method with N=2, is solved by a fourth order Runge-Kutta scheme. Attention is concentrated on the possible motions of the system with a higher mean flow velocity. Phase portraits, bifurcation diagrams and power spectrum diagrams are presented, showing some interesting and sometimes unexpected results. The analytical model is found to exhibit rich and variegated dynamical behaviors that include quasi-periodic and chaotic motions. The route to chaos is shown to be via period-doubling bifurcations. Finally, the cumulative effect of two non-linearities on the dynamics of the system is discussed.     

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.     

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 ...     

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.     

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…     

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).     

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

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Evolution of the double-jumping in pipes conveying fluid flowing at the supercritical speed

Non-linear vibration of viscoelastic pipes conveying fluid around curved equilibrium due to the supercritical flow is investigated with the emphasis on steady-state response in external and internal resonances. The governing equation, a non-linear integro-partial-differential equation, is truncated into a perturbed gyroscopic system via the Galerkin method. The method of multiple scales is applied to establish the solvability condition in the first primary resonance and the 2:1 internal resonance. The approximate analytical expressions are derived for the frequency–amplitude curves of the steady-state responses. The stabilities of the steady-state responses are determined. The generation and the vanishing of a double-jumping phenomenon on the frequency–amplitude curves are examined. The analytical results are supported by the numerical integration results.     

Nonlinear dynamics of extensible viscoelastic cantilevered pipes conveying pulsatile flow with an end nozzle

Nonlinear dynamics of an extensible cantilevered pipe conveying pulsating flow is considered in this paper. The fluid flow fluctuates harmonically and exhausts via a nozzle attached to the end of the pipe. Taking into account the extensibility assumption, the coupled nonlinear lateral–longitudinal equations of motion are derived using Hamilton's principle and discretized via Galerkin's method. The adaptive time step Adams algorithm is applied to extract the time response, and then the bifurcation, power spectral density and phase plane maps are plotted for some case studies. Effects of some geometrical parameters such as flow mass, pulsating flow frequency, gravity, nozzle mass and nozzle aspect ratio parameters are studied on the dynamics of such system and the validity of extensibility assumption is investigated and some conclusions are drawn.     

输液管道临界流速的傅里叶级数法

傅里叶级数法被用于计算输液管道的临界流速，与有限元等数值法相比，更为简单可靠。     

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.     

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.     

粘弹性地基上粘弹性输流管道的稳定性分析

从Winkler假设和单轴线性粘弹性本构方程出发,推导了Kelvin-Voigt粘弹性地基上三参量固体模型输流管道的运动微分方程,采用改进的有限差分法,分析了管道和地基的粘弹性参数对输流管道无量纲复频率和无量纲流速之间的变化关系的影响。     

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.     

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).     

基于WDQ法的粘弹性输流管道稳定性分析

在微分求积法（DQ法）基础上,根据多分辨分析理论,以尺度函数为基础构造插值基函数,形成小波微分求积法（WDQ法）,用该方法研究了简支Kelvin型粘弹性输流管道的稳定性问题,给出了不同参数下管道复频率随内部流速的变化关系,分析了外部流速对Kelvin型粘弹性输流管道在不同延滞时间下的振动特性及稳定性的影响。     

Nonlinear steady states of hyperelastic membrane tubes conveying a viscous non-Newtonian fluid

We study possible steady states of an infinitely long tube made of a hyperelastic membrane and conveying either an inviscid, or a viscous fluid with power-law rheology. The tube model is geometrically and physically nonlinear; the fluid model is limited to smooth changes in the tube’s radius. For the inviscid case, we analyse the tube’s stretch and flow velocity range at which standing solitary waves of both the swelling and the necking type exist. For the viscous case, we first analyse the tube’s upstream and downstream limit states that are balanced by infinitely growing upstream (and decreasing downstream) fluid pressure and axial stress caused by fluid viscosity. Then we investigate conditions that can connect these limit states by a single solution. We show that such a solution exists only for sufficiently small flow speeds and that it has a form of a kink wave; solitary waves do not exist. For the case of a semi-infinite tube (infinite either upstream or downstream), there exist both kink and solitary wave solutions. For finite-length tubes, there exist solutions of any kind, i.e. in the form of pieces of kink waves, solitary waves, and periodic waves.     

Vortex-induced vibration of pipes conveying fluid using the method of multiple scales

The nonlinear dynamics of supported pipes conveying fluid subjected to vortex-induced vibration is evaluated using the method of multiple scales. Frequency response portraits for different internal fluid velocities under lock-in conditions are obtained and the stability of steady-state responses is discussed. Results show that the internal fluid velocity has a prominent effect on the oscillation amplitude and that the steady-state responses incorporating unstable solutions in the lock-in region are also obtained. In addition, the effects of two kinds of fluctuating lift coefficients on the steady-state responses are compared with each other.     

Differential quadrature method to stability analysis of pipes conveying fluid with spring support

It is a new attempt to extend the differential quadrature method (DQM) to stability analysis of the straight and curved centerline pipes conveying fluid. Emphasis is placed on the study of the influences of several parameters on the critical flow velocity. Compared to other methods, this method can more easily deal with the pipe with spring support at its boundaries and asks for much less computing effort while giving aceptable precision in the numerical results. Supported by National Key Project of China (No. PD9521907) and the National Science Foundation of China (No. 19872025).