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 Houbolt's 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.