Bifurcations in Three-Dimensional Motions of Articulated Tubes, Part 1: Linear Systems and Symmetry

Three-dimensional motions of a two-segment articulated tube system carrying a fluid and having rotational symmetry about the vertical axis are examined for bifurcating periodic solutions. As the flow rate through the tubes is increased past a critical value, the downward vertical position of equilibrium gets unstable and bifurcates into two qualitatively different kinds of periodic motions. The mathematical problem is more general than that occurring in the Hopf bifurcations and the method of analysis used is the method of Alternate Problems. Since physical systems invariably have some asymmetry, the analysis takes into account these symmetry-breaking perturbations. In Part 1 of this two-part paper, symmetry properties of the system and the linear stability are discussed.