Dynamics of cantilevered pipes conveying fluid. Part 3: Three-dimensional dynamics in the presence of an end-mass

In this part, the last in a three-part study, the three-dimensional (3-D) dynamics of a cantilevered pipe conveying fluid is explored when an additional “point” mass is attached at the free end. For a typical case, the dynamical behaviour of this system is presented in the form of a bifurcation diagram, along with the time traces, phase-plane plots, PSD plots and Poincaré maps, showing planar periodic, quasiperiodic and chaotic oscillations, followed by 3-D quasiperiodic and chaotic motions. The effect on the results of the number of beam modes used in the Galerkin solution scheme is studied in some detail. The theoretical results are then compared with the results of a set of experiments done previously and good qualitative and quantitative agreement is observed.