Nonlinear dynamics of cantilevered extensible pipes conveying fluid

In this paper the nonlinear planar dynamics of a fluid-conveying cantilevered pipe is investigated. The centreline of the pipe is considered to be extensible; i.e., coupled longitudinal and transverse displacements are considered. The extended version of the Lagrange equations for systems containing non-material volumes is employed to derive the equations of motion, resulting directly in a set of coupled nonlinear ordinary differential equations. The pseudo-arclength continuation technique along with direct time integration are used to solve these equations. Bifurcation diagrams of the system are constructed as the flow velocity is increased; these diagrams are supplemented by time traces, phase-plane portraits, and fast Fourier transforms for some sets of system parameters. As opposed to the case of an inextensible pipe, an extensible pipe elongates in the axial direction as the flow velocity is increased from zero; depending on the system parameters, this static elongation can be considerable. At the critical flow velocity, the system loses stability via a supercritical Hopf bifurcation, emerging from the trivial solution for the transverse displacement and leading to a flutter.