Three routes to chaos for a three-degree-of-freedom articulated cylinder system subjected to annular flow and impacting on the outer pipe

Abstract

In this paper, the dynamics of a cantilevered articulated system of rigid cylinders interconnected by rotational springs, within a pipe containing fluid flow is studied. Although the formulation is generalized to any number of degrees-of-freedom (articulations), the present work is restricted to three-degree-of-freedom systems. The motions are considered to be planar, and the equations of motion, apart from impacting terms, are linearized. Impacting of the articulated cylinder system on the outer pipe is modelled by either a cubic spring (for analytical convenience) or, more realistically, by a trilinear spring model. The critical flow velocities, for which the system loses stability, by flutter (Hopf bifurcation) or divergence (pitchfork bifurcation) are determined by an eigenvalue analysis. Beyond these first bifurcations, it is shown that, for different values of the system parameters, chaos is obtained through three different routes as the flow is incremented: a period-doubling cascade, the quasiperiodic route, and type III intermittency. The dynamical behaviour of the system and differing routes to chaos are illustrated by phase-plane portraits, bifurcation diagrams, power spectra, Poincar6 sections, and Lyapunov exponent calculations.