Planar and non-planar vibrations of a fluid-conveying cantilevered pipe subjected to axial base excitation

The main aim of the present study is to explore the nonlinear vibrations of a fluid-conveying cantilevered pipe under an axial base excitation, by considering both two-dimensional (2-D) and three-dimensional (3-D) responses of the pipe. For this purpose, the extended Hamilton’s principle is applied to derive the 3-D nonlinear governing equations of the pipe with an axial base excitation. The resulting partial differential equations are then discretized by employing a Galerkin method. The Floquet theory is adopted to determine the stability regions of the pipe system. A linear analysis shows that the internal flow velocity, excitation amplitude and excitation frequency have great influences on the stability regions of the pipe system, indicating that both subharmonic and combination resonances can occur. The planar and non-planar nonlinear responses of the pipe are calculated by using a fourth-order Runge–Kutta integration algorithm. These nonlinear responses of the pipe are displayed in the form of bifurcation diagrams, phase-plane plots, power spectrum diagrams, Poincare maps, oscillating shapes and oscillation trajectories of the pipe tip. Our numerical results demonstrate that, for a pipe conveying fluid with a subcritical flow velocity, the presence of an axial excitation is able to generate resonant responses, while for a pipe conveying fluid with a supercritical flow velocity, the axial excitation can even make the pipe stable in some specific cases. Furthermore, it is interesting that a self-excited non-planar periodic oscillation can evolve to a planar quasi-periodic or periodic oscillation by adding the axial excitation. In such a fluid–structure interaction system, however, it is shown that a self-excited planar motion cannot be shifted to a non-planar one by adding the axial excitation, at least for the system parameters considered in this work.