Spatial nonlinear dynamics near principal parametric resonance for a fluid-conveying cantilever pipe

Flow-induced vibrations of a fluid-conveying cantilever pipe are examined theoretically under the condition that the fluid velocity has a small harmonic pulsatile component. More specifically, the case of principal parametric resonance is considered for the pipe free to undergo three-dimensional motions. The mean flow is considered to be near the critical flow rate at which the tube undergoes a Hopf bifurcation into self-excited oscillations. When the governing equations of motion for the tube with steady flow are reduced to those on the center manifold in the neighborhood of Hopf bifurcation, the normal form equations are O(2)-equivariant. The weak harmonic fluctuations due to pulsatile flow result in symmetry-breaking terms in the normal form. The eigenvalues of an O(2)-equivariant system undergoing a symmetry-breaking Hopf bifurcation have multiplicity two. When an additive linear term, arising from time-periodic modulations of the original dynamic system, is introduced into the normal form, the symmetry-breaking bifurcation structure for the trivial solution splits into three categories: a steady-state bifurcation giving rise to standing wave fixed-point solutions, a Hopf bifurcation giving rise to two-frequency solutions, and an O(2)-Takens-Bogdanov bifurcation. The resulting dynamics in each case are studied along with secondary and tertiary bifurcations. The dynamics of the tube system are studied as a function of the mean flow rate and the frequency of flow fluctuations. Amplitude response diagrams constructed for a specific example tube system using the continuation and bifurcation analysis software package AUTO illustrate the variety of possible behaviors.