Linear in-plane and out-of-plane lateral vibrations of a horizontally rotating fluid-tube cantilever

The case of simultaneous in-plane and out-of-plane lateral vibrations of sm all amplitude of a horizontally rotating fluid-tube cantilever conveying fl uid is investigated. The rotation is with respect to a vertical axis at the fixed end at a constant angular velocity. The diameter of the tube is cons tant along its length and much smaller than the length. There is no nozzle attached at the free end. The fluid-tube cantilever is inextensible. Two in ter-dependent equations of motion in the two directions of lateral displace ment of the system are derived by means of Newton's second law on a fluid-t ube element. The same system of equations is derived by means of Hamilton's principle. An approximate solution is sought in the case of linear vibrati ons in the form of a series of normalized eigenfunctions from the linear ca ntilever beam theory using Galerkin's method. The critical nondimensional c ircular frequency of lateral vibration and critical nondimensional speed of flow of the fluid-tube cantilever system are investigated for the in-plane and the out-of-plane case. Comparisons between the in-plane and out-of-pla ne case, between the rotating and the nonrotating case, as well as between the rotating with internal flow and the rotating case without flow are disc ussed. (C) 2000 Academic Press.