Dynamic analysis of a vertically deploying/retracting cantilevered pipe conveying fluid

Based on Euler-Bernoulli beam theory and Hamilton's principle, the differential equation of a vertical cantilevered pipe conveying fluid is derived when the pipe has deploying or retracting motion. The resulting equation is discretized via the Galerkin method in which the eigenfunctions of a clamped-free Euler-Bernoulli beam are utilized. Then, the dynamic responses and stability are discussed with regard to the deploying or retracting speed, mass ratio, and fluid velocity. Numerical results reveal that the dynamical behavior of the system is mainly affected by the flow velocity, instantaneous length of pipe, gravity, and mass ratio. For the small flow velocity, the fluid and higher mass ratio helps to stabilize the transverse vibration of the cantilevered pipe conveying fluid in both deployment and retraction modes, and the system will lose stability with the further increase of flow velocity. The critical flow velocity is mainly influenced by the instantaneous length of pipe. The additional restoring force due to gravity causes critical flow velocity to be higher for the vertically cantilevered pipe conveying fluid. Therefore, gravity is conducive to the stability the transverse vibration of the system in both deployment and retraction modes.