Stability, double degeneracy and chaos in cantilevered pipes conveying fluid

Abstract

The non-linear dynamics of planar motions of cantilevered pipes conveying fluid is studied via a two-mode discretization of the governing partial differential equation, after first replacing non-linear inertial terms by equivalent ones in the equations of motion through a perturbation procedure. For hanging cantilevers, as the flow velocity U is incremented to a critical value, the undeformed vertical configuration of the pipe becomes unstable and bifurcates into stable periodic orbits through a Hopf bifurcation. For a “standing” cantilever, in which the flow discharges from the free, upper end, the pipe is statically unstable for small U if the pipe is sufficiently long; it regains stability through a subcritical pitchfork bifurcation at higher U, and this is followed by a Hopf bifurcation and periodic motions at still higher U. For some parameter values these two bifurcations occur simultaneously (double degeneracy). By using centre manifold theory and normal forms, it is shown that heteroclinic cycles exist in the reduced subsystem, suggesting the possible existence of chaotic behaviour. Melnikov computations give guidance as to the likely location of chaotic regimes in the parameter space. The numerical results, in which the existence of chaos is demonstrated, are presented in the form of phase portraits, bifurcation diagrams and Lyapunov exponents.