C. Semler

Non-linear dynamics of a fluid-conveying cantilevered pipe with a small mass attached at the free end

Abstract In this paper, the planar dynamics of a fluid-conveying cantilevered pipe with a small mass attached at the free end (‘end-mass’, for short) are examined theoretically and experimentally. An experimental study is undertaken with elastomer pipes conveying water and with end-masses made of brass, aluminum or plastic. The main purpose is to extend the work of Copeland and Moon on a modified configuration: the motion is constrained to be planar instead of three-dimensional and the pipe is modelled as a beam having a non-negligible flexural rigidity instead of a string hanging under gravity. As in previous studies, it is demonstrated that for the system with no end-mass, only one stable periodic solution exists, at least for the parameters considered. On the other hand, in the presence of a small end-mass, the dynamics are much richer and different types of periodic solutions are found to exist. Jump phenomena as well as chaotic oscillations are observed in the experiments, revealing therefore the importance of even a small mass on the dynamics. In parallel, a theoretical/numerical investigation is undertaken. The non-linear equations for planar motions of a vertical pipe are modified to take into account the small lumped mass at the free end. The resultant discretized equations contain non-linear inertial terms and are integrated using two methods developed specifically to treat such a case: a Finite Difference Method based on Houbolts scheme (FDM), which leads to a set of non-linear algebraic equations that is solved with a Newton-Raphson approach; and an Incremental Harmonic Balance method (IHB), which enables the construction of bifurcation diagrams of periodic solutions and the determination of their stability. For a constant (non-zero) end-mass and an increasing flow velocity, it is shown that after the first Hopf bifurcation, the system undergoes a series of bifurcations leading again to a wide diversity of dynamical behaviour. As in the experiments, two different periodic solutions are detected; also jump phenomena, quasiperiodic and chaotic oscillations are found for different end-masses and are investigated in detail. Particular attention is paid to the emergence of new solutions, showing why a linear analysis for this system is not very useful. Even though both theory and experiment have certain limitations, the agreement between the two is rather good, from both qualitative and quantitative points of view. This confirms (i) the validity of the present model, (ii) the necessity of taking account in the analysis of even small modifications to the system, and (iii) the richness of the system of a cantilevered fluid-conveying pipe from a dynamical point of view.

Experiments on vertical slender flexible cylinders clamped at both ends and subjected to axial flow

Three series of experiments were conducted on vertical clamped–clamped cylinders in order to observe experimentally the dynamical behaviour of the system, and the results are compared with theoretical predictions. In the first series of experiments, the downstream end of the clamped–clamped cylinder was free to slide axially, while in the second, the downstream end was fixed; the influence of externally applied axial compression was also studied in this series of experiments. The third series of experiments was similar to the second, except that a considerably more slender, hollow cylinder was used. In these experiments, the cylinder lost stability by divergence at a sufficiently high flow velocity and the amplitude of buckling increased thereafter. At higher flow velocities, the cylinder lost stability by flutter (attainable only in the third series of experiments), confirming experimentally the existence of a post-divergence oscillatory instability, which was previously predicted by both linear and nonlinear theory. Good quantitative agreement is obtained between theory and experiment for the amplitude of buckling, and for the critical flow velocities.

Nonlinear Dynamics of Slender Cylinders Supported at Both Ends and Subjected to Axial Flow

In this paper the weakly nonlinear equations of motion, correct to third order of magnitude, are presented for a slender cylinder subjected to axial flow. The cylinder is considered to be extensible and two coupled nonlinear equations describe its motions, involving both longitudinal and transverse displacements. The inviscid component of the fluid force is modeled by an extension of Lighthill’s slender-body work, and viscous, hydrostatic, gravity and pressure-loss forces are added in a similar manner as for cantilevered inextensible cylinders. However, both the derivation and the final equations have many different and distinctive features. The equations are discretized via Galerkin’s method and solved by Houbolt’s finite difference method. Bifurcation diagrams with flow velocity as the independent variable, supported by phase- plane plots, show that the system loses stability via a supercritical pitchfork bifurcation leading to divergence. At higher flow velocities, a secondary Hopf bifurcation leads to flutter and, at higher flow velocities, the limit cycle evolves into chaotic oscillation. In some cases, an oscillatory large-amplitude limit cycle is found with no clear physical origination.

An Analytical Study of the Dynamics of Pipes Conveying Fluid of Axially Varying Density

This paper investigates the dynamics of a slender, flexible pipe, conveying a fluid whose density varies axially along the length of the pipe. Specific applications for this system have appeared in the mining of submerged methane crystals [1], but a general interest also exists due to more common situations in which fluid density changes along the length of the pipe, such as when a gas is conveyed at high velocity. Therefore, following a brief review of related work and of the well-established theory concerning pipes conveying fluid of constant density, the current problem is approached from an analytical perspective. In particular, a linear model describing the system is derived using a Hamiltonian approach, for the cases of (i) a pipe clamped at both ends and (ii) a cantilevered pipe, and results obtained using a Galerkin approach. Ultimately, it is shown that, in both the cantilevered and clamped-clamped cases, the behaviour of the system is similar to that of a pipe conveying fluid of constant density — that is, loss of stability by flutter and buckling respectively — save for two crucial differences. The first and most important is that it is the density at the discharging end which has the most significant effect on the critical flow velocities, rather than any other. Second, in the case of a cantilevered pipe, the magnitude of the density change can strongly influence in which mode the system loses stability, thereby also impacting the critical flow velocities. The specifics of both these effects are addressed in the paper.Copyright